{"id":556,"date":"2021-04-30T21:10:20","date_gmt":"2021-04-30T21:10:20","guid":{"rendered":"https:\/\/opentextbc.ca\/businesstechnicalmath\/chapter\/solving-problems\/"},"modified":"2021-08-31T21:18:09","modified_gmt":"2021-08-31T21:18:09","slug":"solving-problems","status":"publish","type":"chapter","link":"https:\/\/opentextbc.ca\/businesstechnicalmath\/chapter\/solving-problems\/","title":{"raw":"2.3 Use a Problem Solving Strategy","rendered":"2.3 Use a Problem Solving Strategy"},"content":{"raw":"[latexpage]\n
\n

Learning Objectives<\/p>\n\n<\/header>\n

\n\nBy the end of this section it is expected that you will be able to:\n
    \n \t
  • Translate to an equation and solve<\/li>\n \t
  • Translate and solve applications<\/li>\n<\/ul>\n<\/div>\n<\/div>\n

    Translate to an Equation and Solve<\/h1>\n

    To solve applications algebraically, we will begin by translating from English sentences into equations. Our first step is to look for the word (or words) that would translate to the equals sign<\/span>.\u00a0In the next few examples, we will translate sentences into equations and then solve the equations.<\/p>\n\n

    \n

    EXAMPLE 1<\/p>\n\n<\/header>\n

    \n
    \n

    Translate and solve: Eleven more than x<\/em> is equal to 54.<\/p>\n\n<\/div>\n

    \n
    <\/div>\n
    Solution<\/strong><\/div>\n\n\n\n\n\n\n
    Translate.<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
    Subtract 11 from both sides.<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
    Simplify.<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
    Check: Is 54 eleven more than 43?\n<\/span> \\(\\begin{array}{ccc}\\hfill 43+11& \\stackrel{?}{=}\\hfill & 54\\hfill \\\\ \\hfill 54& =\\hfill & 54\u2713\\hfill \\end{array}\\)<\/td>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n
    \n

    TRY IT 1<\/p>\n\n<\/header>\n

    \n
    \n
    \n
    \n

    Translate and solve: Ten more than x<\/em> is equal to 41.<\/p>\n\n<\/div>\n

    Show answer<\/summary>\n

    \\(x+10=41; x=31\\)<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

    \n

    EXAMPLE 2<\/p>\n\n<\/header>\n

    \n
    \n

    Translate and solve: The number 143 is the product of \\(-11\\) and y<\/em>.<\/p>\n\n<\/div>\n

    \n
    <\/div>\n
    Solution<\/strong><\/div>\n
    <\/div>\n

    Begin by translating the sentence into an equation.<\/p>\n\n\n\n\n\n\n\n
    Translate.<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
    Divide by \\(-11\\).<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
    Simplify.<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
    Check:<\/td>\n\\(\\begin{array}{cccc}& \\hfill 143& =& -11y\\hfill \\\\ & \\hfill 143& \\stackrel{?}{=}& -11\\left(-13\\right)\\hfill \\\\ & \\hfill 143& =& 143\u2713\\hfill \\end{array}\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n
    \n

    TRY IT 2<\/p>\n\n<\/header>\n

    \n
    \n
    \n
    \n

    Translate and solve: The number 132 is the product of \u221212<\/em> and y<\/em>.<\/p>\n\n<\/div>\n

    Show answer<\/summary>\n

    \\(132=-12y;y=-11\\)<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

    \n
    \n
    \n
    \n
    \n

    EXAMPLE 3<\/p>\n\n<\/header>\n

    \n
    \n

    Translate and solve: The quotient of \\(y\\) and \\(-4\\) is \\(68\\).<\/p>\n\n<\/div>\n

    \n
    <\/div>\n
    Solution<\/strong><\/div>\n
    <\/div>\n

    Begin by translating the sentence into an equation.<\/p>\n\n\n\n\n\n\n\n\n\n\n
    Translate.<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
    Multiply both sides by \\(-4\\).<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
    Simplify.<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
    Check:<\/td>\nIs the quotient of \\(y\\) and \\(-4\\) equal to \\(68\\)?<\/td>\n<\/tr>\n
    Let \\(y=-272\\).<\/td>\nIs the quotient of \\(-272\\) and \\(-4\\) equal to \\(68\\)?<\/td>\n<\/tr>\n
    Translate.<\/td>\n\\(\\frac{-272}{-4}\\stackrel{?}{=}68\\)<\/td>\n<\/tr>\n
    Simplify.<\/td>\n\\(\\phantom{\\rule{1.3em}{0ex}}68=68\u2713\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n
    \n

    TRY IT 3<\/p>\n\n<\/header>\n

    \n
    \n
    \n
    \n

    Translate and solve: The quotient of \\(q\\) and \\(-8\\) is 72.<\/p>\n\n<\/div>\n

    Show answer<\/summary>\n

    \\(\\frac{q}{-8}=72;q=-576\\)<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

    \n
    \n
    \n
    \n

    EXAMPLE 4<\/p>\n\n<\/header>\n

    \n
    \n

    Translate and solve: Three-fourths of \\(p\\) is 18.<\/p>\n\n<\/div>\n

    \n
    <\/div>\n
    Solution<\/strong><\/div>\n
    <\/div>\n

    Begin by translating the sentence into an equation. Remember, \u201cof\u201d translates into multiplication.<\/p>\n\n\n\n\n\n\n\n\n\n\n
    Translate.<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
    Multiply both sides by \\(\\frac{4}{3}.\\)<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
    Simplify.<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
    Check:<\/td>\nIs three-fourths of p<\/em> equal to 18?<\/td>\n<\/tr>\n
    Let \\(p=24.\\)<\/td>\nIs three-fourths of 24 equal to 18?<\/td>\n<\/tr>\n
    Translate.<\/td>\n\\(\\frac{3}{4}\\cdot\\phantom{\\rule{0.2em}{0ex}}24\\phantom{\\rule{0.2em}{0ex}}\\stackrel{?}{=}18\\)<\/td>\n<\/tr>\n
    Simplify.<\/td>\n\\(\\phantom{\\rule{1.6em}{0ex}}18=18\u2713\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n
    \n

    TRY IT 4<\/p>\n\n<\/header>\n

    \n
    \n
    \n
    \n

    Translate and solve: Two-fifths of \\(f\\) is 16.<\/p>\n\n<\/div>\n

    Show answer<\/summary>\n

    \\(\\frac{2}{5}f=16;f=40\\)<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

    Translate and Solve Applications<\/h1>\n

    Most of the time a question that requires an algebraic solution comes out of a real life situation. To begin, that question is asked in English (or the language of the person asking) and not in math symbols. Because of this, it is an important skill to be able to translate an everyday situation into algebraic language.<\/p>\n

    We will start by restating the problem in just one sentence, assign a variable, and then translate the sentence into an equation to solve. When assigning a variable, choose a letter that reminds you of what you are looking for. For example, you might use q<\/em> for the number of quarters if you were solving a problem about coins.<\/p>\n\n

    \n

    EXAMPLE 5<\/p>\n\n<\/header>\n

    \n
    How to Translate and Solve Applications<\/div>\n
    \n
    \n

    The Alec family recycled newspapers for two months. The two months of newspapers weighed a total of 57 pounds. The second month, the newspapers weighed 28 pounds. How much did the newspapers weigh the first month?<\/p>\n\n<\/div>\n

    \n
    <\/div>\n
    Solution<\/strong><\/div>\n
    <\/div>\n\"This<\/span>\"In<\/span>\"In<\/span>\"In<\/span>\"In<\/span>\"In<\/span>\"In<\/span>\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n

    TRY IT 5<\/p>\n\n<\/header>\n

    \n
    \n
    \n
    \n

    Translate into an algebraic equation and solve:<\/p>\n

    The Snider family has two cats, Zeus and Athena. Together, they weigh 23 pounds. Zeus weighs 16 pounds. How much does Athena weigh?<\/p>\n\n<\/div>\n

    Show answer<\/summary>\n

    7 pounds<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

    \n

    HOW TO: Solve an application<\/p>\n\n<\/header>\n

    \n
      \n \t
    1. Read<\/strong> the problem. Make sure all the words and ideas are understood.<\/li>\n \t
    2. Identify<\/strong> what we are looking for.<\/li>\n \t
    3. Name<\/strong> what we are looking for. Choose a variable to represent that quantity.<\/li>\n \t
    4. Translate<\/strong> into an equation. It may be helpful to restate the problem in one sentence with the important information.<\/li>\n \t
    5. Solve<\/strong> the equation using good algebra techniques.<\/li>\n \t
    6. Check<\/strong> the answer in the problem and make sure it makes sense.<\/li>\n \t
    7. Answer<\/strong> the question with a complete sentence.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n
      \n
      \n
      \n
      \n

      EXAMPLE 6<\/p>\n\n<\/header>\n

      \n
      \n

      Abdullah paid \\$28,675 for his new car. This was \\$875 less than the sticker price. What was the sticker price of the car?<\/p>\n\n<\/div>\n

      \n
      <\/div>\n
      Solution<\/strong><\/div>\n\n\n\n\n\n\n\n\n\n
      Step 1. Read<\/strong> the problem.<\/td>\n<\/td>\n<\/tr>\n
      Step 2. Identify<\/strong> what we are looking for.<\/td>\n\"What was the sticker price of the car?\"<\/td>\n<\/tr>\n
      Step 3. Name<\/strong> what we are looking for. \n<\/span>Choose a variable to represent that quantity.<\/td>\nLet \\(s=\\) the sticker price of the car.<\/td>\n<\/tr>\n
      Step 4. Translate<\/strong> into an equation. Restate the problem in one sentence.<\/td>\n\\$28,675 is \\$875 less than the sticker price<\/td>\n<\/tr>\n
      Step 5. Solve<\/strong> the equation.<\/td>\n\\$28,675 is \\$875 less than s\n\n\\(\\begin{array}{c}\\hfill \\begin{array}{ccc}\\hfill 28,675& \\stackrel{}{=}\\hfill & s-875\\hfill \\\\ \\hfill 28,675 + 875& =\\hfill & s - 875 + 875 \\hfill \\\\ \\hfill 29,550 & = \\hfill & s \\end{array}\\hfill \\end{array}\\)<\/td>\n<\/tr>\n
      Step 6. Check<\/strong> the answer.<\/td>\nIs \\$875 less than \\$29,550 equal to \\$28,675? \n<\/span>\\(\\begin{array}{c}\\hfill \\begin{array}{ccc}\\hfill 29,550-875& \\stackrel{?}{=}\\hfill & 28,675\\hfill \\\\ \\hfill 28,675& =\\hfill & 28,675\u2713\\hfill \\end{array}\\hfill \\end{array}\\)<\/td>\n<\/tr>\n
      Step 7. Answer<\/strong> the question with a complete sentence.<\/td>\nThe sticker price of the car was \\$29,550.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n
      \n

      TRY IT 6<\/p>\n\n<\/header>\n

      \n
      \n
      \n
      \n

      Translate into an algebraic equation and solve:<\/p>\n

      Jaffrey paid \\$19,875 for her new car. This was \\$1,025 less than the sticker price. What was the sticker price of the car?<\/p>\n\n<\/div>\n

      Show answer<\/summary>\n

      \\$20,900<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

      \n
      \n
      \n
      \n

      Now that we have a problem solving strategy, we will use it on several different types of word problems. The first type we will work on is \u201cnumber problems\".<\/p>\n\n

      \n
      \n
      \n
      \n

      EXAMPLE 7<\/p>\n\n<\/header>\n

      \n
      \n

      The sum of twice a number and seven is 15. Find the number.<\/p>\n\n<\/div>\n

      \n
      <\/div>\n
      Solution<\/strong><\/div>\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
      Step 1. Read<\/strong> the problem.<\/td>\n<\/td>\n<\/tr>\n
      Step 2. Identify<\/strong> what we are looking for.<\/td>\nthe number<\/td>\n<\/tr>\n
      Step 3. Name.<\/strong> Choose a variable to represent the number.<\/td>\nLet \\(n=\\) the number.<\/td>\n<\/tr>\n
      Step 4. Translate.<\/strong><\/td>\n<\/td>\n<\/tr>\n
      Restate the problem as one sentence.<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
      Translate into an equation.<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
      Step 5. Solve<\/strong> the equation.<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
      Subtract 7 from each side and simplify.<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
      Divide each side by 2 and simplify.<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
      Step 6. Check.<\/strong><\/td>\n<\/td>\n<\/tr>\n
      Is the sum of twice 4 and 7 equal to 15?<\/td>\n\\(\\begin{array}{ccc}\\hfill 2\\cdot 4+7& \\stackrel{?}{=}\\hfill & 15\\hfill \\\\ \\hfill 15& =\\hfill & 15\u2713\\hfill \\end{array}\\)<\/td>\n<\/tr>\n
      Step 7. Answer<\/strong> the question.<\/td>\nThe number is 4.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

      Did you notice that we left out some of the steps as we solved this equation? If you\u2019re not yet ready to leave out these steps, write down as many as you need.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

      \n
      \n
      \n
      \n

      TRY IT 7<\/p>\n\n<\/header>\n

      \n
      \n

      The sum of four times a number and two is 14. Find the number.<\/p>\n\n<\/div>\n

      Show answer<\/summary>\n

      3<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

      Some number word problems ask us to find two or more numbers. It may be tempting to name them all with different variables, but so far we have only solved equations with one variable. In order to avoid using more than one variable, we will define the numbers in terms of the same variable. Be sure to read the problem carefully to discover how all the numbers relate to each other.<\/p>\n\n

      \n

      EXAMPLE 8<\/p>\n\n<\/header>\n

      \n
      \n

      One number is five more than another. The sum of the numbers is 21. Find the numbers.<\/p>\n\n<\/div>\n

      \n
      <\/div>\n
      Solution<\/strong><\/div>\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
      Step 1. Read<\/strong> the problem.<\/td>\n<\/td>\n<\/tr>\n
      Step 2. Identify<\/strong> what we are looking for.<\/td>\nWe are looking for two numbers.<\/td>\n<\/tr>\n
      Step 3. Name.<\/strong> We have two numbers to name and need a name for each.<\/td>\n<\/td>\n<\/tr>\n
      Choose a variable to represent the first number.<\/td>\nLet \\(n={1}^{\\mathrm{st}}\\) number.<\/td>\n<\/tr>\n
      What do we know about the second number?<\/td>\nOne number is five more than another.<\/td>\n<\/tr>\n
      <\/td>\n\\(n+5={2}^{\\mathrm{nd}}\\) number<\/td>\n<\/tr>\n
      Step 4. Translate. <\/strong>Restate the problem as one sentence with all the important information.<\/td>\nThe sum of the 1st<\/sup> number and the 2nd<\/sup> number is 21.<\/td>\n<\/tr>\n
      Translate into an equation.<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
      Substitute the variable expressions.<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
      Step 5. Solve<\/strong> the equation.<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
      Combine like terms.<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
      Subtract 5 from both sides and simplify.<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
      Divide by 2 and simplify.<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
      Find the second number, too.<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
      <\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
      <\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
      Step 6. Check.<\/strong><\/td>\n<\/td>\n<\/tr>\n
      Do these numbers check in the problem?<\/td>\n<\/td>\n<\/tr>\n
      Is one number 5 more than the other?<\/td>\n\\(\\phantom{\\rule{1.6em}{0ex}}13\\stackrel{?}{=}8+5\\)<\/td>\n<\/tr>\n
      Is thirteen 5 more than 8? Yes.<\/td>\n\\(\\phantom{\\rule{1.6em}{0ex}}13=13\u2713\\)<\/td>\n<\/tr>\n
      Is the sum of the two numbers 21?<\/td>\n\\(8+13\\stackrel{?}{=}21\\)<\/td>\n<\/tr>\n
      <\/td>\n\\(\\phantom{\\rule{1.6em}{0ex}}21=21\u2713\\)<\/td>\n<\/tr>\n
      Step 7. Answer<\/strong> the question.<\/td>\nThe numbers are 8 and 13.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      \n
      \n
      \n

      TRY IT 8<\/p>\n\n<\/header>\n

      \n
      \n

      One number is six more than another. The sum of the numbers is twenty-four. Find the numbers.<\/p>\n\n<\/div>\n

      Show answer<\/summary>\n

      9, 15<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

      \n
      \n
      \n\nNow, we will use the problem solving strategy to solve some geometry problems.\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      \n
      \n
      \n

      EXAMPLE 9<\/p>\n\n<\/header>\n

      \n
      \n

      The length of a rectangle is \\(32\\) metres and the width is \\(20\\) metres. Find a) the perimeter, and b) the area.<\/p>\n\n<\/div>\n

      \n
      <\/div>\n
      Solution<\/strong><\/div>\n\n\n\n\n\n\n\n\n\n\n
      a)<\/td>\n<\/td>\n<\/tr>\n
      Step 1. Read<\/strong> the problem. Draw the figure and label it with the given information.<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
      Step 2. Identify<\/strong> what you are looking for.<\/td>\nthe perimeter of a rectangle<\/td>\n<\/tr>\n
      Step 3. Name.<\/strong> Choose a variable to represent it.<\/td>\nLet P<\/em> = the perimeter<\/td>\n<\/tr>\n
      Step 4. Translate.<\/strong>\nWrite the appropriate formula.\nSubstitute.<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
      Step 5. Solve<\/strong> the equation.<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
      Step 6. Check:<\/strong><\/td>\n\".\"<\/td>\n<\/tr>\n
      Step 7. Answer<\/strong> the question.<\/td>\nThe perimeter of the rectangle is 104 metres.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\n\n\n\n\n\n\n\n\n\n
      b)<\/td>\n<\/td>\n<\/tr>\n
      Step 1. Read<\/strong> the problem. Draw the figure and label it with the given information.<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
      Step 2. Identify<\/strong> what you are looking for.<\/td>\nthe area of a rectangle<\/td>\n<\/tr>\n
      Step 3. Name.<\/strong> Choose a variable to represent it.<\/td>\nLet A<\/em> = the area<\/td>\n<\/tr>\n
      Step 4. Translate.<\/strong>\nWrite the appropriate formula.\nSubstitute.<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
      Step 5. Solve<\/strong> the equation.<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
      Step 6. Check:<\/strong><\/td>\n\".\"<\/td>\n<\/tr>\n
      Step 7. Answer<\/strong> the question.<\/td>\nThe area of the rectangle is 60 square metres.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n
      \n

      TRY IT 9<\/p>\n\n<\/header>\n

      \n
      \n

      The length of a rectangle is \\(120\\) yards and the width is \\(50\\) yards. Find a) the perimeter and b) the area.<\/p>\n\n<\/div>\n

      Show answer<\/summary>\n

      a) 340 yd<\/p>\n

      b) 6000 sq. yd<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

      \n
      \n
      \n
      \n

      EXAMPLE 10<\/p>\n\n<\/header>\n

      \n
      \n

      Find the length of a rectangle with perimeter \\(50\\) inches and width \\(10\\) inches.<\/p>\n\n<\/div>\n

      \n
      <\/div>\n
      Solution<\/strong><\/div>\n\n\n\n\n\n\n\n\n\n
      Step 1. Read<\/strong> the problem. Draw the figure and label it with the given information.<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
      Step 2. Identify<\/strong> what you are looking for.<\/td>\nthe length of the rectangle<\/td>\n<\/tr>\n
      Step 3. Name.<\/strong> Choose a variable to represent it.<\/td>\nLet L<\/em> = the length<\/td>\n<\/tr>\n
      Step 4. Translate.<\/strong>\nWrite the appropriate formula.\nSubstitute.<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
      Step 5. Solve<\/strong> the equation.<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
      Step 6. Check:<\/strong><\/td>\n\".\"<\/td>\n<\/tr>\n
      Step 7. Answer<\/strong> the question.<\/td>\nThe length is 15 inches.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n
      \n

      TRY IT 10<\/p>\n\n<\/header>\n

      \n
      \n

      Find the length of a rectangle with a perimeter of \\(80\\) inches and width of \\(25\\) inches.<\/p>\n\n<\/div>\n

      Show answer<\/summary>\n

      15 in.<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

      \n
      \n
      \n
      \n

      EXAMPLE 11<\/p>\n\n<\/header>\n

      \n
      \n

      The area of a rectangular room is \\(168\\) square feet. The length is \\(14\\) feet. What is the width?<\/p>\n\n<\/div>\n

      \n
      <\/div>\n
      Solution<\/strong><\/div>\n\n\n\n\n\n\n\n\n\n
      Step 1. Read<\/strong> the problem.<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
      Step 2. Identify<\/strong> what you are looking for.<\/td>\nthe width of a rectangular room<\/td>\n<\/tr>\n
      Step 3. Name.<\/strong> Choose a variable to represent it.<\/td>\nLet W<\/em> = width<\/td>\n<\/tr>\n
      Step 4.Translate.<\/strong>\nWrite the appropriate formula and substitute in the given information.<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
      Step 5. Solve<\/strong> the equation.<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
      Step 6. Check:<\/strong><\/td>\n\".\"<\/td>\n<\/tr>\n
      Step 7. Answer<\/strong> the question.<\/td>\nThe width of the room is 12 feet.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n
      \n

      TRY IT 11<\/p>\n\n<\/header>\n

      \n
      \n\nThe area of a rectangle is \\(598\\) square feet. The length is \\(23\\) feet. What is the width?\n\n<\/div>\n
      Show answer<\/summary>\n

      26 ft<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

      \n
      \n
      \n
      \n

      EXAMPLE 12<\/p>\n\n<\/header>\n

      \n
      \n

      The perimeter of a rectangular swimming pool is \\(150\\) feet. The length is \\(15\\) feet more than the width. Find the length and width.<\/p>\n\n<\/div>\n

      \n
      <\/div>\n
      Solution<\/strong><\/div>\n\n\n\n\n\n\n\n\n\n
      Step 1. Read<\/strong> the problem. Draw the figure and label it with the given information.<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
      Step 2. Identify<\/strong> what you are looking for.<\/td>\nthe length and width of the pool<\/td>\n<\/tr>\n
      Step 3. Name.<\/strong> Choose a variable to represent it.\nThe length is 15 feet more than the width.<\/td>\nLet \\(W=\\text{width}\\)\n\\(W+15=\\text{length}\\)<\/td>\n<\/tr>\n
      Step 4.Translate.<\/strong>\nWrite the appropriate formula and substitute.<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
      Step 5. Solve<\/strong> the equation.<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
      Step 6. Check:<\/strong><\/td>\n\".\"<\/td>\n<\/tr>\n
      Step 7. Answer<\/strong> the question.<\/td>\nThe length of the pool is 45 feet and the width is 30 feet.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      \n
      \n
      \n

      TRY IT 12<\/p>\n\n<\/header>\n

      \n
      \n

      The perimeter of a rectangular swimming pool is \\(200\\) feet. The length is \\(40\\) feet more than the width. Find the length and width.<\/p>\n\n<\/div>\n

      Show answer<\/summary>\n

      30 ft, 70 ft<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

      \n
      \n

      The formula for the area of a triangle is \\(A=\\frac{1}{2}bh\\), where \\(b\\) is the base and \\(h\\) is the height.<\/p>\n

      To find the area of the triangle, you need to know its base and height.<\/p>\n\n

      \n
      \n
      \n

      EXAMPLE 13<\/p>\n\n<\/header>\n

      \n
      \n

      Find the area of a triangle whose base is \\(11\\) inches and whose height is \\(8\\) inches.<\/p>\n\n<\/div>\n

      \n
      <\/div>\n
      Solution<\/strong><\/div>\n\n\n\n\n\n\n\n\n\n
      Step 1. Read<\/strong> the problem. Draw the figure and label it with the given information.<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
      Step 2. Identify<\/strong> what you are looking for.<\/td>\nthe area of the triangle<\/td>\n<\/tr>\n
      Step 3. Name.<\/strong> Choose a variable to represent it.<\/td>\nlet A<\/em> = area of the triangle<\/td>\n<\/tr>\n
      Step 4.Translate.<\/strong>\nWrite the appropriate formula.\nSubstitute.<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
      Step 5. Solve<\/strong> the equation.<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
      Step 6. Check:<\/strong><\/td>\n\".\"<\/td>\n<\/tr>\n
      Step 7. Answer<\/strong> the question.<\/td>\nThe area is 44 square inches.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n
      \n

      TRY IT 13<\/p>\n\n<\/header>\n

      \n
      \n

      Find the area of a triangle with base \\(13\\) inches and height \\(2\\) inches.<\/p>\n\n<\/div>\n

      Show answer<\/summary>\n

      13 sq. in.<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<\/div>\n

      \n
      \n
      \n
      \n

      EXAMPLE 14<\/p>\n\n<\/header>\n

      \n
      \n

      The perimeter of a triangular garden is \\(24\\) feet. The lengths of two sides are \\(4\\) feet and \\(9\\) feet. How long is the third side?<\/p>\n\n<\/div>\n

      \n
      <\/div>\n
      Solution<\/strong><\/div>\n\n\n\n\n\n\n\n\n\n
      Step 1. Read<\/strong> the problem. Draw the figure and label it with the given information.<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
      Step 2. Identify<\/strong> what you are looking for.<\/td>\nlength of the third side of a triangle<\/td>\n<\/tr>\n
      Step 3. Name.<\/strong> Choose a variable to represent it.<\/td>\nLet c<\/em> = the third side<\/td>\n<\/tr>\n
      Step 4.Translate.<\/strong>\nWrite the appropriate formula.\nSubstitute in the given information.<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
      Step 5. Solve<\/strong> the equation.<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
      Step 6. Check:<\/strong><\/td>\n\".\"<\/td>\n<\/tr>\n
      Step 7. Answer<\/strong> the question.<\/td>\nThe third side is 11 feet long.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      \n

      TRY IT 14<\/p>\n\n<\/header>\n

      \n
      \n

      The perimeter of a triangular garden is \\(24\\) feet. The lengths of two sides are \\(18\\) feet and \\(22\\) feet. How long is the third side?<\/p>\n\n<\/div>\n

      Show answer<\/summary>\n

      8 ft<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

      \n
      \n
      \n
      \n

      EXAMPLE 15<\/p>\n\n<\/header>\n

      \n
      \n

      The area of a triangular church window is \\(90\\) square metres. The base of the window is \\(15\\) metres. What is the window\u2019s height?<\/p>\n\n<\/div>\n

      \n
      <\/div>\n
      Solution<\/strong><\/div>\n\n\n\n\n\n\n\n\n\n
      Step 1. Read<\/strong> the problem. Draw the figure and label it with the given information.<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
      Step 2. Identify<\/strong> what you are looking for.<\/td>\nheight of a triangle<\/td>\n<\/tr>\n
      Step 3. Name.<\/strong> Choose a variable to represent it.<\/td>\nLet h<\/em> = the height<\/td>\n<\/tr>\n
      Step 4.Translate.<\/strong>\nWrite the appropriate formula.\nSubstitute in the given information.<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
      Step 5. Solve<\/strong> the equation.<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
      Step 6. Check:<\/strong><\/td>\n\".\"<\/td>\n<\/tr>\n
      Step 7. Answer<\/strong> the question.<\/td>\nThe height of the triangle is 12 metres.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      \n
      \n
      \n

      TRY IT 15<\/p>\n\n<\/header>\n

      \n
      \n

      The area of a triangular painting is \\(126\\) square inches. The base is \\(18\\) inches. What is the height?<\/p>\n\n<\/div>\n

      Show answer<\/summary>\n

      14 in.<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

      Key Concepts<\/h1>\n<\/div>\n<\/div>\n<\/div>\n
        \n \t
      • To translate a sentence to an equation<\/strong>\n
          \n \t
        1. Locate the \u201cequals\u201d word(s). Translate to an equal sign (=).<\/li>\n \t
        2. Translate the words to the left of the \u201cequals\u201d word(s) into an algebraic expression.<\/li>\n \t
        3. Translate the words to the right of the \u201cequals\u201d word(s) into an algebraic expression.<\/li>\n<\/ol>\n<\/li>\n \t
        4. To solve an application<\/strong>\n
            \n \t
          1. Read the problem. Make sure all the words and ideas are understood.<\/li>\n \t
          2. Identify what we are looking for.<\/li>\n \t
          3. Name what we are looking for. Choose a variable to represent that quantity.<\/li>\n \t
          4. Translate into an equation. It may be helpful to restate the problem in one sentence with the important information.<\/li>\n \t
          5. Solve the equation using good algebra techniques.<\/li>\n \t
          6. Check the answer in the problem and make sure it makes sense.<\/li>\n \t
          7. Answer the question with a complete sentence.<\/li>\n<\/ol>\n<\/li>\n<\/ul>\n

            2.3 Exercise Set<\/h1>\n

            In the following exercises, translate to an equation and then solve it.<\/p>\n\n

              \n \t
            1. Nine more than \\(x\\) is equal to 52.<\/li>\n \t
            2. Ten less than m is \\(-14\\).<\/li>\n \t
            3. The sum of y and \\(-30\\) is 40.<\/li>\n \t
            4. The difference of \\(n\\) and \\(\\frac{1}{6}\\) is \\(\\frac{1}{2}\\).<\/li>\n \t
            5. The sum of \\(-4n\\) and \\(5n\\) is \\(-82\\).<\/li>\n \t
            6. 133 is the product of \\(-19\\) and n<\/em>.<\/li>\n \t
            7. The quotient of \\(b\\) and \\(-6\\) is 18.<\/li>\n \t
            8. Three-tenths of x<\/em> is 15.<\/li>\n \t
            9. The sum of two-fifths and f<\/em> is one-half.<\/li>\n \t
            10. The difference of q<\/em> and one-eighth is three-fourths<\/li>\n<\/ol>\n

              In the following exercises, translate into an equation and solve.<\/p>\n\n

                \n \t
              1. Avril rode her bike a total of 18 miles, from home to the library and then to the beach. The distance from Avril\u2019s house to the library is 7 miles. What is the distance from the library to the beach?<\/li>\n \t
              2. Eva\u2019s daughter is 15 years younger than her son. Eva\u2019s son is 22 years old. How old is her daughter?<\/li>\n \t
              3. For a family birthday dinner, Celeste bought a turkey that weighed 5 pounds less than the one she bought for Thanksgiving. The birthday turkey weighed 16 pounds. How\u00a0much did the Thanksgiving turkey weigh?<\/li>\n \t
              4. Arjun\u2019s temperature was 0.7 degrees higher this morning than it had been last night. His temperature this morning was 101.2 degrees. What was his temperature last night?<\/li>\n \t
              5. \u00a0Ron\u2019s paycheck this week was \\$17.43 less than his paycheck last week. His paycheck this week was \\$103.76. How much was Ron\u2019s paycheck last week?<\/li>\n<\/ol>\n

                In the following exercises, solve each number word problem<\/p>\n\n

                  \n \t
                1. The sum of a number and eight is 12. Find the number.<\/li>\n \t
                2. The difference of twice a number and seven is 17. Find the number.<\/li>\n \t
                3. Three times the sum of a number and nine is 12. Find the number.<\/li>\n \t
                4. One number is six more than the other. Their sum is 42. Find the numbers.<\/li>\n \t
                5. The sum of two numbers is \\(-45.\\) One number is nine more than the other. Find the numbers.<\/li>\n \t
                6. One number is 14 less than another. If their sum is increased by seven, the result is 85. Find the numbers.<\/li>\n \t
                7. One number is one more than twice another. Their sum is \\(-5.\\) Find the numbers.<\/li>\n<\/ol>\n

                  In the following exercises, find the a) perimeter and b) area of each rectangle.<\/p>\n\n

                    \n \t
                  1. The length of a rectangle is \\(85\\) feet and the width is \\(45\\) feet.<\/li>\n \t
                  2. A rectangular room is \\(15\\) feet wide by \\(14\\) feet long.<\/li>\n<\/ol>\n

                    In the following exercises, solve.<\/p>\n\n

                      \n \t
                    1. Find the length of a rectangle with perimeter \\(124\\) inches and width \\(38\\) inches.<\/li>\n \t
                    2. Find the width of a rectangle with perimeter \\(92\\) metres and length \\(19\\) metres.<\/li>\n \t
                    3. The area of a rectangle is \\(414\\) square metres. The length is \\(18\\) metres. What is the width?<\/li>\n \t
                    4. The length of a rectangle is \\(9\\) inches more than the width. The perimeter is \\(46\\) inches. Find the length and the width.<\/li>\n \t
                    5. The perimeter of a rectangle is \\(58\\) metres. The width of the rectangle is \\(5\\) metres less than the length. Find the length and the width of the rectangle.<\/li>\n \t
                    6. The width of the rectangle is \\(0.7\\) metres less than the length. The perimeter of a rectangle is \\(52.6\\) metres. Find the dimensions of the rectangle.<\/li>\n \t
                    7. The perimeter of a rectangle is \\(150\\) feet. The length of the rectangle is twice the width. Find the length and width of the rectangle.<\/li>\n \t
                    8. The length of a rectangle is \\(3\\) metres less than twice the width. The perimeter is \\(36\\) metres. Find the length and width.<\/li>\n \t
                    9. The width of a rectangular window is \\(24\\) inches. The area is \\(624\\) square inches. What is the length?<\/li>\n \t
                    10. The area of a rectangular roof is \\(2310\\) square metres. The length is \\(42\\) metres. What is the width?<\/li>\n \t
                    11. The perimeter of a rectangular courtyard is \\(160\\) feet. The length is \\(10\\) feet more than the width. Find the length and the width.<\/li>\n \t
                    12. The width of a rectangular window is \\(40\\) inches less than the height. The perimeter of the doorway is \\(224\\) inches. Find the length and the width.<\/li>\n<\/ol>\n

                      In the following exercises, solve using the properties of triangles.<\/p>\n\n

                        \n \t
                      1. A triangular flag has base of \\(8\\) foot and height of \\(1.5\\) feet. What is its area?<\/li>\n \t
                      2. What is the base of a triangle with an area of \\(207\\) square inches and height of \\(18\\) inches?<\/li>\n \t
                      3. The perimeter of a triangular reflecting pool is \\(36\\) yards. The lengths of two sides are \\(10\\) yards and \\(15\\) yards. How long is the third side?<\/li>\n \t
                      4. The perimeter of a triangle is \\(39\\) feet. One side of the triangle is \\(1\\) foot longer than the second side. The third side is \\(2\\) feet longer than the second side. Find the length of each side.<\/li>\n<\/ol>\n

                        Answers<\/h1>\n
                          \n \t
                        1. \\(x+9=52;x=43\\)<\/li>\n \t
                        2. \\(m-10=-14;m=-4\\)<\/li>\n \t
                        3. \\(y+\\left(-30\\right)=40;y=70\\)<\/li>\n \t
                        4. \\(n-\\frac{1}{6}=\\frac{1}{2};\\frac{2}{3}\\)<\/li>\n \t
                        5. \\(-4n+5n=-82;-82\\)<\/li>\n \t
                        6. 133=-19n; n=-7<\/li>\n \t
                        7. \\(\\frac{b}{-6}=18;b=-108\\)<\/li>\n \t
                        8. 3\/10x=15; x=50<\/li>\n \t
                        9. \\(\\frac{2}{5}+f=\\frac{1}{2};f=\\frac{1}{10}\\)<\/li>\n \t
                        10. \\(q-\\frac{1}{8}=\\frac{3}{4};q=\\frac{7}{8}\\)<\/li>\n \t
                        11. \\(11\\) miles<\/li>\n \t
                        12. \\(7\\) years old<\/li>\n \t
                        13. \\(21\\) pounds<\/li>\n \t
                        14. \\(100.5\\) degrees<\/li>\n \t
                        15. \\(\\$121.19\\)<\/li>\n \t
                        16. 4<\/li>\n \t
                        17. 12<\/li>\n \t
                        18. -5<\/li>\n \t
                        19. 18,24<\/li>\n \t
                        20. \\(-18,-27\\)<\/li>\n \t
                        21. 32,46<\/li>\n \t
                        22. \\(-2,-3\\)<\/li>\n \t
                        23. \n
                            \n \t
                          1. 260 ft<\/li>\n \t
                          2. 3825 sq. ft<\/li>\n<\/ol>\n<\/li>\n \t
                          3. \n
                              \n \t
                            1. 58 ft<\/li>\n \t
                            2. 210 sq. ft<\/li>\n<\/ol>\n<\/li>\n \t
                            3. 24 inches<\/li>\n \t
                            4. 27 metres<\/li>\n \t
                            5. 23 m<\/li>\n \t
                            6. 7 in., 16 in.<\/li>\n \t
                            7. 17 m, 12 m<\/li>\n \t
                            8. 13.5 m, 12.8 m<\/li>\n \t
                            9. 25 ft, 50 ft<\/li>\n \t
                            10. 7 m, 11 m<\/li>\n \t
                            11. 26 in.<\/li>\n \t
                            12. 55 m<\/li>\n \t
                            13. 35 ft, 45 ft<\/li>\n \t
                            14. 76 in., 36 in.<\/li>\n \t
                            15. 6 sq. ft<\/li>\n \t
                            16. 23 in.<\/li>\n \t
                            17. 11 ft<\/li>\n \t
                            18. 12 ft, 13 ft, 14 ft<\/li>\n<\/ol>","rendered":"
                              \n
                              \n

                              Learning Objectives<\/p>\n<\/header>\n

                              \n

                              By the end of this section it is expected that you will be able to:<\/p>\n

                                \n
                              • Translate to an equation and solve<\/li>\n
                              • Translate and solve applications<\/li>\n<\/ul>\n<\/div>\n<\/div>\n

                                Translate to an Equation and Solve<\/h1>\n

                                To solve applications algebraically, we will begin by translating from English sentences into equations. Our first step is to look for the word (or words) that would translate to the equals sign<\/span>.\u00a0In the next few examples, we will translate sentences into equations and then solve the equations.<\/p>\n

                                \n
                                \n

                                EXAMPLE 1<\/p>\n<\/header>\n

                                \n
                                \n

                                Translate and solve: Eleven more than x<\/em> is equal to 54.<\/p>\n<\/div>\n

                                \n
                                <\/div>\n
                                Solution<\/strong><\/div>\n\n\n\n\n\n\n
                                Translate.<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
                                Subtract 11 from both sides.<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
                                Simplify.<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
                                Check: Is 54 eleven more than 43?
                                \n<\/span> \"\begin{array}{ccc}\hfill 43+11& \stackrel{?}{=}\hfill & 54\hfill \\ \hfill 54& =\hfill & 54✓\hfill \end{array}\"<\/td>\n                                		<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n
                                \n
                                \n

                                TRY IT 1<\/p>\n<\/header>\n

                                \n
                                \n
                                \n
                                \n

                                Translate and solve: Ten more than x<\/em> is equal to 41.<\/p>\n<\/div>\n

                                \n
                                \nShow answer<\/summary>\n

                                \"x+10=41; x=31\"<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                                \n
                                \n

                                EXAMPLE 2<\/p>\n<\/header>\n

                                \n
                                \n

                                Translate and solve: The number 143 is the product of \"-11\" and y<\/em>.<\/p>\n<\/div>\n

                                \n
                                <\/div>\n
                                Solution<\/strong><\/div>\n
                                <\/div>\n

                                Begin by translating the sentence into an equation.<\/p>\n\n\n\n\n\n\n
                                Translate.<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
                                Divide by \"-11\".<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
                                Simplify.<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
                                Check:<\/td>\n\"\begin{array}{cccc}& \hfill 143& =& -11y\hfill \\ & \hfill 143& \stackrel{?}{=}& -11\left(-13\right)\hfill \\ & \hfill 143& =& 143✓\hfill \end{array}\"<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n
                                \n
                                \n

                                TRY IT 2<\/p>\n<\/header>\n

                                \n
                                \n
                                \n
                                \n

                                Translate and solve: The number 132 is the product of \u221212<\/em> and y<\/em>.<\/p>\n<\/div>\n

                                \n
                                \nShow answer<\/summary>\n

                                \"132=-12y;y=-11\"<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                                \n
                                \n
                                \n
                                \n
                                \n
                                \n

                                EXAMPLE 3<\/p>\n<\/header>\n

                                \n
                                \n

                                Translate and solve: The quotient of \"y\" and \"-4\" is \"68\".<\/p>\n<\/div>\n

                                \n
                                <\/div>\n
                                Solution<\/strong><\/div>\n
                                <\/div>\n

                                Begin by translating the sentence into an equation.<\/p>\n\n\n\n\n\n\n\n\n\n
                                Translate.<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
                                Multiply both sides by \"-4\".<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
                                Simplify.<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
                                Check:<\/td>\nIs the quotient of \"y\" and \"-4\" equal to \"68\"?<\/td>\n<\/tr>\n
                                Let \"y=-272\".<\/td>\nIs the quotient of \"-272\" and \"-4\" equal to \"68\"?<\/td>\n<\/tr>\n
                                Translate.<\/td>\n\"\frac{-272}{-4}\stackrel{?}{=}68\"<\/td>\n<\/tr>\n
                                Simplify.<\/td>\n\"\phantom{\rule{1.3em}{0ex}}68=68✓\"<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n
                                \n
                                \n

                                TRY IT 3<\/p>\n<\/header>\n

                                \n
                                \n
                                \n
                                \n

                                Translate and solve: The quotient of \"q\" and \"-8\" is 72.<\/p>\n<\/div>\n

                                \n
                                \nShow answer<\/summary>\n

                                \"\frac{q}{-8}=72;q=-576\"<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                                \n
                                \n
                                \n
                                \n
                                \n

                                EXAMPLE 4<\/p>\n<\/header>\n

                                \n
                                \n

                                Translate and solve: Three-fourths of \"p\" is 18.<\/p>\n<\/div>\n

                                \n
                                <\/div>\n
                                Solution<\/strong><\/div>\n
                                <\/div>\n

                                Begin by translating the sentence into an equation. Remember, \u201cof\u201d translates into multiplication.<\/p>\n\n\n\n\n\n\n\n\n\n
                                Translate.<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
                                Multiply both sides by \"\frac{4}{3}.\"<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
                                Simplify.<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
                                Check:<\/td>\nIs three-fourths of p<\/em> equal to 18?<\/td>\n<\/tr>\n
                                Let \"p=24.\"<\/td>\nIs three-fourths of 24 equal to 18?<\/td>\n<\/tr>\n
                                Translate.<\/td>\n\"\frac{3}{4}\cdot\phantom{\rule{0.2em}{0ex}}24\phantom{\rule{0.2em}{0ex}}\stackrel{?}{=}18\"<\/td>\n<\/tr>\n
                                Simplify.<\/td>\n\"\phantom{\rule{1.6em}{0ex}}18=18✓\"<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n
                                \n
                                \n

                                TRY IT 4<\/p>\n<\/header>\n

                                \n
                                \n
                                \n
                                \n

                                Translate and solve: Two-fifths of \"f\" is 16.<\/p>\n<\/div>\n

                                \n
                                \nShow answer<\/summary>\n

                                \"\frac{2}{5}f=16;f=40\"<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                                Translate and Solve Applications<\/h1>\n

                                Most of the time a question that requires an algebraic solution comes out of a real life situation. To begin, that question is asked in English (or the language of the person asking) and not in math symbols. Because of this, it is an important skill to be able to translate an everyday situation into algebraic language.<\/p>\n

                                We will start by restating the problem in just one sentence, assign a variable, and then translate the sentence into an equation to solve. When assigning a variable, choose a letter that reminds you of what you are looking for. For example, you might use q<\/em> for the number of quarters if you were solving a problem about coins.<\/p>\n

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                                EXAMPLE 5<\/p>\n<\/header>\n

                                \n
                                How to Translate and Solve Applications<\/div>\n
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                                \n

                                The Alec family recycled newspapers for two months. The two months of newspapers weighed a total of 57 pounds. The second month, the newspapers weighed 28 pounds. How much did the newspapers weigh the first month?<\/p>\n<\/div>\n

                                \n
                                <\/div>\n
                                Solution<\/strong><\/div>\n
                                <\/div>\n

                                \"This<\/span>\"In<\/span>\"In<\/span>\"In<\/span>\"In<\/span>\"In<\/span>\"In<\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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                                TRY IT 5<\/p>\n<\/header>\n

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                                Translate into an algebraic equation and solve:<\/p>\n

                                The Snider family has two cats, Zeus and Athena. Together, they weigh 23 pounds. Zeus weighs 16 pounds. How much does Athena weigh?<\/p>\n<\/div>\n

                                \n
                                \nShow answer<\/summary>\n

                                7 pounds<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                                \n
                                \n

                                HOW TO: Solve an application<\/p>\n<\/header>\n

                                \n
                                  \n
                                1. Read<\/strong> the problem. Make sure all the words and ideas are understood.<\/li>\n
                                2. Identify<\/strong> what we are looking for.<\/li>\n
                                3. Name<\/strong> what we are looking for. Choose a variable to represent that quantity.<\/li>\n
                                4. Translate<\/strong> into an equation. It may be helpful to restate the problem in one sentence with the important information.<\/li>\n
                                5. Solve<\/strong> the equation using good algebra techniques.<\/li>\n
                                6. Check<\/strong> the answer in the problem and make sure it makes sense.<\/li>\n
                                7. Answer<\/strong> the question with a complete sentence.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n
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                                  EXAMPLE 6<\/p>\n<\/header>\n

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                                  \n

                                  Abdullah paid $28,675 for his new car. This was $875 less than the sticker price. What was the sticker price of the car?<\/p>\n<\/div>\n

                                  \n
                                  <\/div>\n
                                  Solution<\/strong><\/div>\n\n\n\n\n\n\n\n\n\n
                                  Step 1. Read<\/strong> the problem.<\/td>\n<\/td>\n<\/tr>\n
                                  Step 2. Identify<\/strong> what we are looking for.<\/td>\n“What was the sticker price of the car?”<\/td>\n<\/tr>\n
                                  Step 3. Name<\/strong> what we are looking for.
                                  \n<\/span>Choose a variable to represent that quantity.<\/td>\n                                  		Let \"s=\" the sticker price of the car.<\/td>\n<\/tr>\n
                                  Step 4. Translate<\/strong> into an equation. Restate the problem in one sentence.<\/td>\n$28,675 is $875 less than the sticker price<\/td>\n<\/tr>\n
                                  Step 5. Solve<\/strong> the equation.<\/td>\n$28,675 is $875 less than s<\/p>\n

                                  \"\begin{array}{c}\hfill \begin{array}{ccc}\hfill 28,675& \stackrel{}{=}\hfill & s-875\hfill \\ \hfill 28,675 + 875& =\hfill & s - 875 + 875 \hfill \\ \hfill 29,550 & = \hfill & s \end{array}\hfill \end{array}\"<\/td>\n<\/tr>\n

                                  Step 6. Check<\/strong> the answer.<\/td>\nIs $875 less than $29,550 equal to $28,675?
                                  \n<\/span>\"\begin{array}{c}\hfill \begin{array}{ccc}\hfill 29,550-875& \stackrel{?}{=}\hfill & 28,675\hfill \\ \hfill 28,675& =\hfill & 28,675✓\hfill \end{array}\hfill \end{array}\"<\/td>\n<\/tr>\n
                                  Step 7. Answer<\/strong> the question with a complete sentence.<\/td>\nThe sticker price of the car was $29,550.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n
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                                  TRY IT 6<\/p>\n<\/header>\n

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                                  \n

                                  Translate into an algebraic equation and solve:<\/p>\n

                                  Jaffrey paid $19,875 for her new car. This was $1,025 less than the sticker price. What was the sticker price of the car?<\/p>\n<\/div>\n

                                  \n
                                  \nShow answer<\/summary>\n

                                  $20,900<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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                                  Now that we have a problem solving strategy, we will use it on several different types of word problems. The first type we will work on is \u201cnumber problems”.<\/p>\n

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                                  EXAMPLE 7<\/p>\n<\/header>\n

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                                  The sum of twice a number and seven is 15. Find the number.<\/p>\n<\/div>\n

                                  \n
                                  <\/div>\n
                                  Solution<\/strong><\/div>\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
                                  Step 1. Read<\/strong> the problem.<\/td>\n<\/td>\n<\/tr>\n
                                  Step 2. Identify<\/strong> what we are looking for.<\/td>\nthe number<\/td>\n<\/tr>\n
                                  Step 3. Name.<\/strong> Choose a variable to represent the number.<\/td>\nLet \"n=\" the number.<\/td>\n<\/tr>\n
                                  Step 4. Translate.<\/strong><\/td>\n<\/td>\n<\/tr>\n
                                  Restate the problem as one sentence.<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
                                  Translate into an equation.<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
                                  Step 5. Solve<\/strong> the equation.<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
                                  Subtract 7 from each side and simplify.<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
                                  Divide each side by 2 and simplify.<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
                                  Step 6. Check.<\/strong><\/td>\n<\/td>\n<\/tr>\n
                                  Is the sum of twice 4 and 7 equal to 15?<\/td>\n\"\begin{array}{ccc}\hfill 2\cdot 4+7& \stackrel{?}{=}\hfill & 15\hfill \\ \hfill 15& =\hfill & 15✓\hfill \end{array}\"<\/td>\n<\/tr>\n
                                  Step 7. Answer<\/strong> the question.<\/td>\nThe number is 4.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                                  Did you notice that we left out some of the steps as we solved this equation? If you\u2019re not yet ready to leave out these steps, write down as many as you need.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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                                  TRY IT 7<\/p>\n<\/header>\n

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                                  The sum of four times a number and two is 14. Find the number.<\/p>\n<\/div>\n

                                  \n
                                  \nShow answer<\/summary>\n

                                  3<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                                  Some number word problems ask us to find two or more numbers. It may be tempting to name them all with different variables, but so far we have only solved equations with one variable. In order to avoid using more than one variable, we will define the numbers in terms of the same variable. Be sure to read the problem carefully to discover how all the numbers relate to each other.<\/p>\n

                                  \n
                                  \n

                                  EXAMPLE 8<\/p>\n<\/header>\n

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                                  \n

                                  One number is five more than another. The sum of the numbers is 21. Find the numbers.<\/p>\n<\/div>\n

                                  \n
                                  <\/div>\n
                                  Solution<\/strong><\/div>\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
                                  Step 1. Read<\/strong> the problem.<\/td>\n<\/td>\n<\/tr>\n
                                  Step 2. Identify<\/strong> what we are looking for.<\/td>\nWe are looking for two numbers.<\/td>\n<\/tr>\n
                                  Step 3. Name.<\/strong> We have two numbers to name and need a name for each.<\/td>\n<\/td>\n<\/tr>\n
                                  Choose a variable to represent the first number.<\/td>\nLet \"n={1}^{\mathrm{st}}\" number.<\/td>\n<\/tr>\n
                                  What do we know about the second number?<\/td>\nOne number is five more than another.<\/td>\n<\/tr>\n
                                  <\/td>\n\"n+5={2}^{\mathrm{nd}}\" number<\/td>\n<\/tr>\n
                                  Step 4. Translate. <\/strong>Restate the problem as one sentence with all the important information.<\/td>\nThe sum of the 1st<\/sup> number and the 2nd<\/sup> number is 21.<\/td>\n<\/tr>\n
                                  Translate into an equation.<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
                                  Substitute the variable expressions.<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
                                  Step 5. Solve<\/strong> the equation.<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
                                  Combine like terms.<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
                                  Subtract 5 from both sides and simplify.<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
                                  Divide by 2 and simplify.<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
                                  Find the second number, too.<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
                                  <\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
                                  <\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
                                  Step 6. Check.<\/strong><\/td>\n<\/td>\n<\/tr>\n
                                  Do these numbers check in the problem?<\/td>\n<\/td>\n<\/tr>\n
                                  Is one number 5 more than the other?<\/td>\n\"\phantom{\rule{1.6em}{0ex}}13\stackrel{?}{=}8+5\"<\/td>\n<\/tr>\n
                                  Is thirteen 5 more than 8? Yes.<\/td>\n\"\phantom{\rule{1.6em}{0ex}}13=13✓\"<\/td>\n<\/tr>\n
                                  Is the sum of the two numbers 21?<\/td>\n\"8+13\stackrel{?}{=}21\"<\/td>\n<\/tr>\n
                                  <\/td>\n\"\phantom{\rule{1.6em}{0ex}}21=21✓\"<\/td>\n<\/tr>\n
                                  Step 7. Answer<\/strong> the question.<\/td>\nThe numbers are 8 and 13.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n
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                                  TRY IT 8<\/p>\n<\/header>\n

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                                  One number is six more than another. The sum of the numbers is twenty-four. Find the numbers.<\/p>\n<\/div>\n

                                  \n
                                  \nShow answer<\/summary>\n

                                  9, 15<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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                                  Now, we will use the problem solving strategy to solve some geometry problems.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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                                  \n

                                  EXAMPLE 9<\/p>\n<\/header>\n

                                  \n
                                  \n

                                  The length of a rectangle is \"32\" metres and the width is \"20\" metres. Find a) the perimeter, and b) the area.<\/p>\n<\/div>\n

                                  \n
                                  <\/div>\n
                                  Solution<\/strong><\/div>\n\n\n\n\n\n\n\n\n\n\n
                                  a)<\/td>\n<\/td>\n<\/tr>\n
                                  Step 1. Read<\/strong> the problem. Draw the figure and label it with the given information.<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
                                  Step 2. Identify<\/strong> what you are looking for.<\/td>\nthe perimeter of a rectangle<\/td>\n<\/tr>\n
                                  Step 3. Name.<\/strong> Choose a variable to represent it.<\/td>\nLet P<\/em> = the perimeter<\/td>\n<\/tr>\n
                                  Step 4. Translate.<\/strong>
                                  \nWrite the appropriate formula.
                                  \nSubstitute.<\/td>\n                                  		\".\"<\/span><\/td>\n<\/tr>\n
                                  Step 5. Solve<\/strong> the equation.<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
                                  Step 6. Check:<\/strong><\/td>\n\".\"<\/td>\n<\/tr>\n
                                  Step 7. Answer<\/strong> the question.<\/td>\nThe perimeter of the rectangle is 104 metres.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\n\n\n\n\n\n\n\n\n\n
                                  b)<\/td>\n<\/td>\n<\/tr>\n
                                  Step 1. Read<\/strong> the problem. Draw the figure and label it with the given information.<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
                                  Step 2. Identify<\/strong> what you are looking for.<\/td>\nthe area of a rectangle<\/td>\n<\/tr>\n
                                  Step 3. Name.<\/strong> Choose a variable to represent it.<\/td>\nLet A<\/em> = the area<\/td>\n<\/tr>\n
                                  Step 4. Translate.<\/strong>
                                  \nWrite the appropriate formula.
                                  \nSubstitute.<\/td>\n                                  		\".\"<\/span><\/td>\n<\/tr>\n
                                  Step 5. Solve<\/strong> the equation.<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
                                  Step 6. Check:<\/strong><\/td>\n\".\"<\/td>\n<\/tr>\n
                                  Step 7. Answer<\/strong> the question.<\/td>\nThe area of the rectangle is 60 square metres.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n
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                                  TRY IT 9<\/p>\n<\/header>\n

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                                  \n

                                  The length of a rectangle is \"120\" yards and the width is \"50\" yards. Find a) the perimeter and b) the area.<\/p>\n<\/div>\n

                                  \n
                                  \nShow answer<\/summary>\n

                                  a) 340 yd<\/p>\n

                                  b) 6000 sq. yd<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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                                  EXAMPLE 10<\/p>\n<\/header>\n

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                                  \n

                                  Find the length of a rectangle with perimeter \"50\" inches and width \"10\" inches.<\/p>\n<\/div>\n

                                  \n
                                  <\/div>\n
                                  Solution<\/strong><\/div>\n\n\n\n\n\n\n\n\n\n
                                  Step 1. Read<\/strong> the problem. Draw the figure and label it with the given information.<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
                                  Step 2. Identify<\/strong> what you are looking for.<\/td>\nthe length of the rectangle<\/td>\n<\/tr>\n
                                  Step 3. Name.<\/strong> Choose a variable to represent it.<\/td>\nLet L<\/em> = the length<\/td>\n<\/tr>\n
                                  Step 4. Translate.<\/strong>
                                  \nWrite the appropriate formula.
                                  \nSubstitute.<\/td>\n                                  		\".\"<\/span><\/td>\n<\/tr>\n
                                  Step 5. Solve<\/strong> the equation.<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
                                  Step 6. Check:<\/strong><\/td>\n\".\"<\/td>\n<\/tr>\n
                                  Step 7. Answer<\/strong> the question.<\/td>\nThe length is 15 inches.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n
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                                  TRY IT 10<\/p>\n<\/header>\n

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                                  Find the length of a rectangle with a perimeter of \"80\" inches and width of \"25\" inches.<\/p>\n<\/div>\n

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                                  \nShow answer<\/summary>\n

                                  15 in.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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                                  EXAMPLE 11<\/p>\n<\/header>\n

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                                  \n

                                  The area of a rectangular room is \"168\" square feet. The length is \"14\" feet. What is the width?<\/p>\n<\/div>\n

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                                  <\/div>\n
                                  Solution<\/strong><\/div>\n\n\n\n\n\n\n\n\n\n
                                  Step 1. Read<\/strong> the problem.<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
                                  Step 2. Identify<\/strong> what you are looking for.<\/td>\nthe width of a rectangular room<\/td>\n<\/tr>\n
                                  Step 3. Name.<\/strong> Choose a variable to represent it.<\/td>\nLet W<\/em> = width<\/td>\n<\/tr>\n
                                  Step 4.Translate.<\/strong>
                                  \nWrite the appropriate formula and substitute in the given information.<\/td>\n                                  		\".\"<\/span><\/td>\n<\/tr>\n
                                  Step 5. Solve<\/strong> the equation.<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
                                  Step 6. Check:<\/strong><\/td>\n\".\"<\/td>\n<\/tr>\n
                                  Step 7. Answer<\/strong> the question.<\/td>\nThe width of the room is 12 feet.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n
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                                  TRY IT 11<\/p>\n<\/header>\n

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                                  The area of a rectangle is \"598\" square feet. The length is \"23\" feet. What is the width?<\/p>\n<\/div>\n

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                                  \nShow answer<\/summary>\n

                                  26 ft<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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                                  EXAMPLE 12<\/p>\n<\/header>\n

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                                  \n

                                  The perimeter of a rectangular swimming pool is \"150\" feet. The length is \"15\" feet more than the width. Find the length and width.<\/p>\n<\/div>\n

                                  \n
                                  <\/div>\n
                                  Solution<\/strong><\/div>\n\n\n\n\n\n\n\n\n\n
                                  Step 1. Read<\/strong> the problem. Draw the figure and label it with the given information.<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
                                  Step 2. Identify<\/strong> what you are looking for.<\/td>\nthe length and width of the pool<\/td>\n<\/tr>\n
                                  Step 3. Name.<\/strong> Choose a variable to represent it.
                                  \nThe length is 15 feet more than the width.<\/td>\n                                  		Let \"W=\text{width}\"
                                  \n\"W+15=\text{length}\"<\/td>\n<\/tr>\n
                                  Step 4.Translate.<\/strong>
                                  \nWrite the appropriate formula and substitute.<\/td>\n                                  		\".\"<\/span><\/td>\n<\/tr>\n
                                  Step 5. Solve<\/strong> the equation.<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
                                  Step 6. Check:<\/strong><\/td>\n\".\"<\/td>\n<\/tr>\n
                                  Step 7. Answer<\/strong> the question.<\/td>\nThe length of the pool is 45 feet and the width is 30 feet.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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                                  TRY IT 12<\/p>\n<\/header>\n

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                                  The perimeter of a rectangular swimming pool is \"200\" feet. The length is \"40\" feet more than the width. Find the length and width.<\/p>\n<\/div>\n

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                                  \nShow answer<\/summary>\n

                                  30 ft, 70 ft<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                                  \n
                                  \n

                                  The formula for the area of a triangle is \"A=\frac{1}{2}bh\", where \"b\" is the base and \"h\" is the height.<\/p>\n

                                  To find the area of the triangle, you need to know its base and height.<\/p>\n

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                                  EXAMPLE 13<\/p>\n<\/header>\n

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                                  \n

                                  Find the area of a triangle whose base is \"11\" inches and whose height is \"8\" inches.<\/p>\n<\/div>\n

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                                  <\/div>\n
                                  Solution<\/strong><\/div>\n\n\n\n\n\n\n\n\n\n
                                  Step 1. Read<\/strong> the problem. Draw the figure and label it with the given information.<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
                                  Step 2. Identify<\/strong> what you are looking for.<\/td>\nthe area of the triangle<\/td>\n<\/tr>\n
                                  Step 3. Name.<\/strong> Choose a variable to represent it.<\/td>\nlet A<\/em> = area of the triangle<\/td>\n<\/tr>\n
                                  Step 4.Translate.<\/strong>
                                  \nWrite the appropriate formula.
                                  \nSubstitute.<\/td>\n                                  		\".\"<\/span><\/td>\n<\/tr>\n
                                  Step 5. Solve<\/strong> the equation.<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
                                  Step 6. Check:<\/strong><\/td>\n\".\"<\/td>\n<\/tr>\n
                                  Step 7. Answer<\/strong> the question.<\/td>\nThe area is 44 square inches.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n
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                                  TRY IT 13<\/p>\n<\/header>\n

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                                  Find the area of a triangle with base \"13\" inches and height \"2\" inches.<\/p>\n<\/div>\n

                                  \n
                                  \nShow answer<\/summary>\n

                                  13 sq. in.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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                                  EXAMPLE 14<\/p>\n<\/header>\n

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                                  \n

                                  The perimeter of a triangular garden is \"24\" feet. The lengths of two sides are \"4\" feet and \"9\" feet. How long is the third side?<\/p>\n<\/div>\n

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                                  <\/div>\n
                                  Solution<\/strong><\/div>\n\n\n\n\n\n\n\n\n\n
                                  Step 1. Read<\/strong> the problem. Draw the figure and label it with the given information.<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
                                  Step 2. Identify<\/strong> what you are looking for.<\/td>\nlength of the third side of a triangle<\/td>\n<\/tr>\n
                                  Step 3. Name.<\/strong> Choose a variable to represent it.<\/td>\nLet c<\/em> = the third side<\/td>\n<\/tr>\n
                                  Step 4.Translate.<\/strong>
                                  \nWrite the appropriate formula.
                                  \nSubstitute in the given information.<\/td>\n                                  		\".\"<\/span><\/td>\n<\/tr>\n
                                  Step 5. Solve<\/strong> the equation.<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
                                  Step 6. Check:<\/strong><\/td>\n\".\"<\/td>\n<\/tr>\n
                                  Step 7. Answer<\/strong> the question.<\/td>\nThe third side is 11 feet long.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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                                  TRY IT 14<\/p>\n<\/header>\n

                                  \n
                                  \n

                                  The perimeter of a triangular garden is \"24\" feet. The lengths of two sides are \"18\" feet and \"22\" feet. How long is the third side?<\/p>\n<\/div>\n

                                  \n
                                  \nShow answer<\/summary>\n

                                  8 ft<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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                                  \n

                                  EXAMPLE 15<\/p>\n<\/header>\n

                                  \n
                                  \n

                                  The area of a triangular church window is \"90\" square metres. The base of the window is \"15\" metres. What is the window\u2019s height?<\/p>\n<\/div>\n

                                  \n
                                  <\/div>\n
                                  Solution<\/strong><\/div>\n\n\n\n\n\n\n\n\n\n
                                  Step 1. Read<\/strong> the problem. Draw the figure and label it with the given information.<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
                                  Step 2. Identify<\/strong> what you are looking for.<\/td>\nheight of a triangle<\/td>\n<\/tr>\n
                                  Step 3. Name.<\/strong> Choose a variable to represent it.<\/td>\nLet h<\/em> = the height<\/td>\n<\/tr>\n
                                  Step 4.Translate.<\/strong>
                                  \nWrite the appropriate formula.
                                  \nSubstitute in the given information.<\/td>\n                                  		\".\"<\/span><\/td>\n<\/tr>\n
                                  Step 5. Solve<\/strong> the equation.<\/td>\n\".\"<\/span><\/td>\n<\/tr>\n
                                  Step 6. Check:<\/strong><\/td>\n\".\"<\/td>\n<\/tr>\n
                                  Step 7. Answer<\/strong> the question.<\/td>\nThe height of the triangle is 12 metres.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
                                  \n
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                                  \n
                                  \n

                                  TRY IT 15<\/p>\n<\/header>\n

                                  \n
                                  \n

                                  The area of a triangular painting is \"126\" square inches. The base is \"18\" inches. What is the height?<\/p>\n<\/div>\n

                                  \n
                                  \nShow answer<\/summary>\n

                                  14 in.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                                  Key Concepts<\/h1>\n<\/div>\n<\/div>\n<\/div>\n
                                    \n
                                  • To translate a sentence to an equation<\/strong>\n
                                      \n
                                    1. Locate the \u201cequals\u201d word(s). Translate to an equal sign (=).<\/li>\n
                                    2. Translate the words to the left of the \u201cequals\u201d word(s) into an algebraic expression.<\/li>\n
                                    3. Translate the words to the right of the \u201cequals\u201d word(s) into an algebraic expression.<\/li>\n<\/ol>\n<\/li>\n
                                    4. To solve an application<\/strong>\n
                                        \n
                                      1. Read the problem. Make sure all the words and ideas are understood.<\/li>\n
                                      2. Identify what we are looking for.<\/li>\n
                                      3. Name what we are looking for. Choose a variable to represent that quantity.<\/li>\n
                                      4. Translate into an equation. It may be helpful to restate the problem in one sentence with the important information.<\/li>\n
                                      5. Solve the equation using good algebra techniques.<\/li>\n
                                      6. Check the answer in the problem and make sure it makes sense.<\/li>\n
                                      7. Answer the question with a complete sentence.<\/li>\n<\/ol>\n<\/li>\n<\/ul>\n

                                        2.3 Exercise Set<\/h1>\n

                                        In the following exercises, translate to an equation and then solve it.<\/p>\n

                                          \n
                                        1. Nine more than \"x\" is equal to 52.<\/li>\n
                                        2. Ten less than m is \"-14\".<\/li>\n
                                        3. The sum of y and \"-30\" is 40.<\/li>\n
                                        4. The difference of \"n\" and \"\frac{1}{6}\" is \"\frac{1}{2}\".<\/li>\n
                                        5. The sum of \"-4n\" and \"5n\" is \"-82\".<\/li>\n
                                        6. 133 is the product of \"-19\" and n<\/em>.<\/li>\n
                                        7. The quotient of \"b\" and \"-6\" is 18.<\/li>\n
                                        8. Three-tenths of x<\/em> is 15.<\/li>\n
                                        9. The sum of two-fifths and f<\/em> is one-half.<\/li>\n
                                        10. The difference of q<\/em> and one-eighth is three-fourths<\/li>\n<\/ol>\n

                                          In the following exercises, translate into an equation and solve.<\/p>\n

                                            \n
                                          1. Avril rode her bike a total of 18 miles, from home to the library and then to the beach. The distance from Avril\u2019s house to the library is 7 miles. What is the distance from the library to the beach?<\/li>\n
                                          2. Eva\u2019s daughter is 15 years younger than her son. Eva\u2019s son is 22 years old. How old is her daughter?<\/li>\n
                                          3. For a family birthday dinner, Celeste bought a turkey that weighed 5 pounds less than the one she bought for Thanksgiving. The birthday turkey weighed 16 pounds. How\u00a0much did the Thanksgiving turkey weigh?<\/li>\n
                                          4. Arjun\u2019s temperature was 0.7 degrees higher this morning than it had been last night. His temperature this morning was 101.2 degrees. What was his temperature last night?<\/li>\n
                                          5. \u00a0Ron\u2019s paycheck this week was $17.43 less than his paycheck last week. His paycheck this week was $103.76. How much was Ron\u2019s paycheck last week?<\/li>\n<\/ol>\n

                                            In the following exercises, solve each number word problem<\/p>\n

                                              \n
                                            1. The sum of a number and eight is 12. Find the number.<\/li>\n
                                            2. The difference of twice a number and seven is 17. Find the number.<\/li>\n
                                            3. Three times the sum of a number and nine is 12. Find the number.<\/li>\n
                                            4. One number is six more than the other. Their sum is 42. Find the numbers.<\/li>\n
                                            5. The sum of two numbers is \"-45.\" One number is nine more than the other. Find the numbers.<\/li>\n
                                            6. One number is 14 less than another. If their sum is increased by seven, the result is 85. Find the numbers.<\/li>\n
                                            7. One number is one more than twice another. Their sum is \"-5.\" Find the numbers.<\/li>\n<\/ol>\n

                                              In the following exercises, find the a) perimeter and b) area of each rectangle.<\/p>\n

                                                \n
                                              1. The length of a rectangle is \"85\" feet and the width is \"45\" feet.<\/li>\n
                                              2. A rectangular room is \"15\" feet wide by \"14\" feet long.<\/li>\n<\/ol>\n

                                                In the following exercises, solve.<\/p>\n

                                                  \n
                                                1. Find the length of a rectangle with perimeter \"124\" inches and width \"38\" inches.<\/li>\n
                                                2. Find the width of a rectangle with perimeter \"92\" metres and length \"19\" metres.<\/li>\n
                                                3. The area of a rectangle is \"414\" square metres. The length is \"18\" metres. What is the width?<\/li>\n
                                                4. The length of a rectangle is \"9\" inches more than the width. The perimeter is \"46\" inches. Find the length and the width.<\/li>\n
                                                5. The perimeter of a rectangle is \"58\" metres. The width of the rectangle is \"5\" metres less than the length. Find the length and the width of the rectangle.<\/li>\n
                                                6. The width of the rectangle is \"0.7\" metres less than the length. The perimeter of a rectangle is \"52.6\" metres. Find the dimensions of the rectangle.<\/li>\n
                                                7. The perimeter of a rectangle is \"150\" feet. The length of the rectangle is twice the width. Find the length and width of the rectangle.<\/li>\n
                                                8. The length of a rectangle is \"3\" metres less than twice the width. The perimeter is \"36\" metres. Find the length and width.<\/li>\n
                                                9. The width of a rectangular window is \"24\" inches. The area is \"624\" square inches. What is the length?<\/li>\n
                                                10. The area of a rectangular roof is \"2310\" square metres. The length is \"42\" metres. What is the width?<\/li>\n
                                                11. The perimeter of a rectangular courtyard is \"160\" feet. The length is \"10\" feet more than the width. Find the length and the width.<\/li>\n
                                                12. The width of a rectangular window is \"40\" inches less than the height. The perimeter of the doorway is \"224\" inches. Find the length and the width.<\/li>\n<\/ol>\n

                                                  In the following exercises, solve using the properties of triangles.<\/p>\n

                                                    \n
                                                  1. A triangular flag has base of \"8\" foot and height of \"1.5\" feet. What is its area?<\/li>\n
                                                  2. What is the base of a triangle with an area of \"207\" square inches and height of \"18\" inches?<\/li>\n
                                                  3. The perimeter of a triangular reflecting pool is \"36\" yards. The lengths of two sides are \"10\" yards and \"15\" yards. How long is the third side?<\/li>\n
                                                  4. The perimeter of a triangle is \"39\" feet. One side of the triangle is \"1\" foot longer than the second side. The third side is \"2\" feet longer than the second side. Find the length of each side.<\/li>\n<\/ol>\n

                                                    Answers<\/h1>\n
                                                      \n
                                                    1. \"x+9=52;x=43\"<\/li>\n
                                                    2. \"m-10=-14;m=-4\"<\/li>\n
                                                    3. \"y+\left(-30\right)=40;y=70\"<\/li>\n
                                                    4. \"n-\frac{1}{6}=\frac{1}{2};\frac{2}{3}\"<\/li>\n
                                                    5. \"-4n+5n=-82;-82\"<\/li>\n
                                                    6. 133=-19n; n=-7<\/li>\n
                                                    7. \"\frac{b}{-6}=18;b=-108\"<\/li>\n
                                                    8. 3\/10x=15; x=50<\/li>\n
                                                    9. \"\frac{2}{5}+f=\frac{1}{2};f=\frac{1}{10}\"<\/li>\n
                                                    10. \"q-\frac{1}{8}=\frac{3}{4};q=\frac{7}{8}\"<\/li>\n
                                                    11. \"11\" miles<\/li>\n
                                                    12. \"7\" years old<\/li>\n
                                                    13. \"21\" pounds<\/li>\n
                                                    14. \"100.5\" degrees<\/li>\n
                                                    15. \"\$121.19\"<\/li>\n
                                                    16. 4<\/li>\n
                                                    17. 12<\/li>\n
                                                    18. -5<\/li>\n
                                                    19. 18,24<\/li>\n
                                                    20. \"-18,-27\"<\/li>\n
                                                    21. 32,46<\/li>\n
                                                    22. \"-2,-3\"<\/li>\n
                                                    23. \n
                                                        \n
                                                      1. 260 ft<\/li>\n
                                                      2. 3825 sq. ft<\/li>\n<\/ol>\n<\/li>\n
                                                      3. \n
                                                          \n
                                                        1. 58 ft<\/li>\n
                                                        2. 210 sq. ft<\/li>\n<\/ol>\n<\/li>\n
                                                        3. 24 inches<\/li>\n
                                                        4. 27 metres<\/li>\n
                                                        5. 23 m<\/li>\n
                                                        6. 7 in., 16 in.<\/li>\n
                                                        7. 17 m, 12 m<\/li>\n
                                                        8. 13.5 m, 12.8 m<\/li>\n
                                                        9. 25 ft, 50 ft<\/li>\n
                                                        10. 7 m, 11 m<\/li>\n
                                                        11. 26 in.<\/li>\n
                                                        12. 55 m<\/li>\n
                                                        13. 35 ft, 45 ft<\/li>\n
                                                        14. 76 in., 36 in.<\/li>\n
                                                        15. 6 sq. ft<\/li>\n
                                                        16. 23 in.<\/li>\n
                                                        17. 11 ft<\/li>\n
                                                        18. 12 ft, 13 ft, 14 ft<\/li>\n<\/ol>\n","protected":false},"author":125,"menu_order":3,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-556","chapter","type-chapter","status-publish","hentry"],"part":272,"_links":{"self":[{"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/pressbooks\/v2\/chapters\/556","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/wp\/v2\/users\/125"}],"version-history":[{"count":1,"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/pressbooks\/v2\/chapters\/556\/revisions"}],"predecessor-version":[{"id":557,"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/pressbooks\/v2\/chapters\/556\/revisions\/557"}],"part":[{"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/pressbooks\/v2\/parts\/272"}],"metadata":[{"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/pressbooks\/v2\/chapters\/556\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/wp\/v2\/media?parent=556"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/pressbooks\/v2\/chapter-type?post=556"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/wp\/v2\/contributor?post=556"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/wp\/v2\/license?post=556"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}