Unit 3: Subtraction
Topic F: Problem Solving
Why are you studying mathematics?
Some of you are taking math because you have to, but we hope you all want to have math skills to help you in your jobs, in job training, and in your everyday life. Numbers are an important part of our lives – we are surrounded by numbers.
Numbers are not often by themselves or set up neatly on a page for us to add or subtract. Numbers are usually in the middle of sentences and mixed in with other numbers. Sorting out the numbers you want and deciding what to do with those numbers is called problem-solving.
You are going to learn five problem-solving steps that will be useful in all your math work in courses, in jobs, and in your everyday life.
Problem Solving Steps
Step 1: Read or listen to the problem carefully. Understand the problem. Are there words that help you imagine what is happening? Can you draw a picture or diagram to show what is happening? Can you say the problem in your own words? What is the question? Underline it.
Step 2: What does the problem tell you? What do you know? Write down or circle the information you have. Often you have more information than you need. Think about the question you need to answer, and use only the information that will help you answer that question. What do you want to find out?
Step 3: What must you do with the information to answer the question? What arithmetic operation should you use addition, subtraction, multiplication or division? You will be learning keywords and patterns that will help you choose the correct operation. Write an equation for the problem, An equation is a number sentence such as
12 + 5 =
Step 4: Estimate the answer.
- Round the numbers so you can work with them quickly.
- Use the operation you chose in Step 3 and come to a quick answer.
- Does this estimated answer make sense? Does it answer the question in the problem? THINK about this before you do Step 5.
Step 5: Solve the problem using the actual numbers.
- Check your arithmetic calculations.
- Compare your result to your estimated answer.
- Reread the problem. Does your answer make sense?
- Write a sentence answer to the problem.
You must always say what the numbers are counting. He has 4, means nothing. We need to know 4 what… 4 children? 4 dogs? 4 dollars? These are called the units.
Some abbreviations used with numerals:
kilometre km
metre m
centimetre cm
kilogram kg
gram g
litre L
hour h
minute min
Now study the three example problems that show the five steps.
Example A
Jorge earned $165 last week and $142 this week in his job pumping gas at the service station. He spent $15 on his girlfriend’s gift. How much did he earn pumping gas?
Step 1: Read. Understand the problem. Find the question. Underline it.
How much did Jorge earn pumping gas?
Step 2: Find the Needed information. circle it.
Jorge earned $165 and $142.
The information about his girlfriend’s gift has nothing to do with finding out how much he earned.
Step 3: What Arithmetic operation to use?
We are putting together two amounts. That is addition.
The equation: $165 + $142 = what he earned.
Step 4: Estimate
$165 + $142
$170 + $140 = $310
$200 + $100 = $300
Is about $300 a reasonable answer to the question? Is it sensible to earn $300 for two weeks of pumping gas? Probably. $3 000 would not be sensible, and $30 would not be sensible.
Step 5: Solve, check, write a sentence answer.
$165 + $142 = $307
check by adding again
is $307 close to the estimate?
make sense?
Jorge earned $307 pumping gas.
Example B
The town of Gloryville had a population of 4 206 people before the mill had a big lay-off in May 2007. Since then 858 people have moved away. Find the population of Gloryville now.
Step 1: Read, understand the problem, find the question underline it.
Find the population of Gloryville now.
Step 2: Circle needed information.
4206 people before
858 people moved away
The date of the lay-off is not needed to answer the question.
Step 3: Operation
One amount is being taken away. That is subtraction.
Equation: 4206 – 858 = people in Gloryville now.
Step 4: estimate
4206 − 858
4000 − 1000 = 3000
4200 − 900= 3300
Step 5: solve, check, write sentence answer
Gloryville has a population now of 3 348 people.
Example C
Paul works at a lumber mill and is paid every two weeks. He has an account at the bank. Today he got a cheque for $845. He and his wife decided to deposit $600 in the account and keep the rest of the money out for a weekend trip. How much money did Paul and his wife keep out for the weekend trip?
Step 1: Question
how many did Paul and his wife keep for the weekend trip?
Step 2: needed information
Paul got a cheque for $845 for two weeks work.
He and his wife decided to put $600 in their account.
Step 3: Operation
One amount is being taken away. That is subtraction. Equation: $845 ─ $600 = money left over for weekend trip
Step 4: Estimate
$845 − $600 = $245
Step 5: check.
$250 + $600 = $845
close to estimate?
makes sense?
Paul and his wife have $245 for the weekend trip.
Addition Problems
The problems in this section all use the addition operation to find the solution (the answer to the problem). Addition problems give two or more amounts that must be put together (added). When you read the problems, pay special attention to key words and patterns that will help you to recognize other addition problems.
Key words that point to Addition
- sum
- combine
- total
- entire
- in all
- altogether
- complete
Exercise One
Do these problems by following the five problem solving steps. It is good practice to write down each step while you are learning this method. Check your work using the answer key at the end of the exercise.
- It was raining so Gita decided to bake several batches of cookies and freeze them. She made 75 chocolate chip cookies, 96 of her son’s favourite ginger snaps, and 42 fancy ―Birds’ nest‖ cookies for when she had company. How many cookies did Gita bake altogether?
- Step 1: What is the question? Underline it.
- Step 2: What information are you given that you need to solve the problem? Circle it.
- Step 3: What arithmetic operation should you use? addition Why?
- Step 4: Estimate the answer using rounded numbers.
- Step 5: Solve, check, and write a sentence answer.
- Levi wanted to paint his apartment and needed to buy some supplies. Brushes cost $10, sandpaper cost $4, a paint roller and tray cost $9 and the paint was $55. How much did it cost for all the paint supplies?
- Step 1: What is the question? Underline it.
- Step 2: What information are you given that you need to solve the problem? Circle it.
- Step 3: What arithmetic operation should you use? addition Why?
- Step 4: Estimate the answer using rounded numbers.
- Step 5: Solve, check, and write a sentence answer.
- Altogether, the college has 475 students in the Adult Basic Education department, 320 University Transfer students, 64 students in the Early Childhood Education program, 232 students in the Forestry department, and 125 students in trades courses. How many students are at the college?
- Step 1: What is the question? Underline it.
- Step 2: What information are you given that you need to solve the problem? Circle it.
- Step 3: What arithmetic operation should you use? addition Why?
- Step 4: Estimate the answer using rounded numbers.
- Step 5: Solve, check, and write a sentence answer.
- Zhou works part-time at the daycare centre. Last month she worked every week. The first week she worked 24 hours, 36 hours the second week, 29 hours the third week, and only 17 hours in the fourth week. Give the total number of hours that Zhou worked last month.
- Step 1: What is the question? Underline it.
- Step 2: What information are you given that you need to solve the problem? Circle it.
- Step 3: What arithmetic operation should you use? addition Why?
- Step 4: Estimate the answer using rounded numbers.
- Step 5: Solve, check, and write a sentence answer.
- The rest of the problems in this exercise just ask you for the estimate and the actual solution. You must still follow all five steps but you do not have to write everything down. Remember that the solution to problems must include the units (what is being counted) and should be written in a sentence answer.
- September is hard on the family budget! Amul figured they spent $275 for clothes and shoes for their two little daughters, $43 for school supplies, $24 for haircuts, and $130 to enroll them in the Figure Skating Club. How much has Amul spent getting his children ready for school and skating?
- Estimation:
- Actual Solution:
- The sign in the elevator says ―1200 kg maximum weight‖. Can the elevator hold all these large football players safely? Sean weighs 91 kg, Raja is 114 kg, Eyota is a heavyweight at 159 kg. Kiefer is even heavier at 168 kg, the two fullbacks weigh 135 kg and 148 kg, and the quarterback Juan is a muscular 87 kg. Find their combined weight to see if they are all safe in the elevator.
- Estimation:
- Actual Solution:
- On their holidays, the Matthews family drove to Saskatchewan from their home in Langley. They drove 620 km the first day, 810 km the second day, and only drove 350 km the next day because they went to Head Smashed-in Buffalo Jump Museum. On the fourth day, they drove a long 1 208 km. How many kilometres did they drive on their trip to Saskatchewan?
- Estimation:
- Actual Solution:
Answers to Exercise One
(The wording in the sentences will vary, but this is the idea)
- How many cookies altogether?
- she made 75, 96, and 42 cookies
- All the amounts have to be put together to find a total.
- 80+100+40= 220 cookies
- 75+96+42=213 cookies
- Gita baked 213 cookies altogether
- How much did it cost for all the paint supplies?
- He paid $ 10, $4, $9, and $55
- All the amounts have to be put together to find a total
- Rounding one digit numbers isn’t too helpful, but $10+$0+$10+$60=$80
- $10+$4+$9+$55=$78 Levi paid $78
- how many students at the college?
- There are 475, 320, 232, and 125 students.
- You must find a total.
- 500 + 300 + 100 + 200 + 100 = 1200 students
- 475 + 320 + 64 + 232 + 125 = 1216 students
- The college has 1 216 students
- How many hours did Zhou work last month?
- She worked 24, 36, 29, and 17 hours.
- You are looking for an amount altogether.
- 20 + 40 + 30 + 20 = 110 hours
- 24 + 36 + 29 + 17 = 106 hours
- Zhou worked 106 hours last month
- $472 altogether
- 902 kg altogether; safe
- 2988 km
Subtraction Problems
These problems will give you a change to get the feel of subtraction problems.
Subtraction problems tell you an amount and then take something away from that amount. Money might be spent, saved, or deducted (taken off), people might move away, items might be sold or lost. These types of subtraction problems are quite easy to recognize.
A more difficult type of subtraction problem compares two amounts. You will be asked to find the difference between the amounts. Subtract to find the difference. These problems might ask you, how much more?, how much less?, how many fewer?, how much farther?, how much did it increase (go up)?, what is the decrease (amount it went down)? You might also have to find the age of something by comparing the dates.
Key Words that point to SUBTRACTION
- difference
- balance
- amount left
- the saving
- how much more (or greater, or farther)
- how much less (or fewer, or smaller)
- how old, find the age
Exercise Two
Use the five problem steps to solve these problems. Write down each step for the first three problems. Check your work using the answer key at the end of the exercise.
- Only 368 people went to the movie theatre on Friday night, but on Saturday 756 went to see the new comedy movie they were showing. How many more people went to the theatre on Saturday than on Friday?
- Step 1: What is the question? Underline it.
- Step 2: What information are you given that you need to solve the problem? Circle it.
- Step 3: What arithmetic operation should you use? subtraction Why?
- Step 4: Estimate the answer using rounded numbers.
- Step 5: Solve, check, and write a sentence answer.
- The highway construction started in 2004 and it was finished in 2010. How long did the construction take?
- Step 1: What is the question? Underline it.
- Step 2: What information are you given that you need to solve the problem? Circle it.
- Step 3: What arithmetic operation should you use? subtraction Why?
- Step 4: Estimate the answer using rounded numbers. In a question like this, an estimation using rounded numbers is not useful because the numbers are too similar and would round to the same number. Instead, think about the question carefully and figure out an approximate answer in your head.
- Step 5: Solve, check, and write a sentence answer.
- Aimee’s gross pay was $1656, but she had $331 of deductions. What is her net pay? (Gross pay is the amount we earn before anything is taken off. Net pay is the amount we take home after taxes, pension, employment insurance, etc. have been deducted.)
- Step 1: What is the question? Underline it.
- Step 2: What information are you given that you need to solve the problem? Circle it.
- Step 3: What arithmetic operation should you use? subtraction Why?
- Step 4: Estimate the answer using rounded numbers.
- Step 5: Solve, check, and write a sentence answer.
- Mike and Ann want to can 240 jars of fruit this year. They have already canned 165 jars. How many more jars do they need to do?
- Estimation:
- Actual Solution:
- Jian has purchased a used car for $3599. He has paid $450 so far. How much more money does he owe?
- Estimation:
- Actual Solution:
- In 1956 the population of the town was 10874. Many people left after the dam construction was finished. The population in 1989 was only 7 892 people. How much less was the population in 1989 than in 1956?
- Estimation:
- Actual Solution:
Answers to Exercise Two
- How many more people at the theatre on Saturday than on Friday?
- 368 people on Friday; 756 on Saturday
- You must find the difference between two amounts.
- 800 – 400 = 400 more people on Saturday
- 756 – 368 = 388 more people on Saturday
- How long did the construction take?
- Started in 2004; ended in 2010.
- Find the difference between the two dates.
- Think ―from 2004 to 2010 – about 5 years‖
- 2010 – 2004 = 6 years for the road construction
- What is Aimee’s net pay?
- Her gross pay was $1 656 and she had $331 taken off (deducted).
- Subtract to find how much is left.
- $1700 – $300 = $1400
- $1656 – $331 = $1325 net pay
- 75 jars
- $3149 still owed
- 2982 people less
Mixed Addition and Subtraction Problems
Exercise Three
Use the 5 problem solving steps. Look for key words and patterns to help you choose the correct operation. Estimate the answer using rounded numbers if the numbers have 2 digits or more. Check your work using the answer key at the end of the exercise.
- Enrico worked 37 hours one week and 26 hours the next week. How many hours did he work?
- Estimation:
- Actual Solution:
- Myung-Hee had $85. She spent $37 for groceries. How much did she have left?
- Estimation:
- Actual Solution:
- Ann bought 25 kg of potatoes. She used 13 kg the first week. How much did she have left?
- Estimation:
- Actual Solution:
- The sign in a furniture store read, $35 off all chairs. How much will a chair cost that was $125 before the sale?
- Estimation:
- Actual Solution:
- Guillaume bought a pair of jeans for $29 at a sale. When he got home, he found the price tag on the jeans had been $48. How much did Guillaume save?
- Estimation:
- Actual Solution:
- British Columbia has an area of 947 800 square kilometres. The area of Alberta is 666 190 square kilometres. BC is how much larger than Alberta?
- Estimation:
- Actual Solution:
- Maxine paid $26 for an electric iron and $39 for an ironing board. How much did she pay for both?
- Estimation:
- Actual Solution:
- Ang bought a used TV set for $125. She made a down payment of $40. How much does she still owe on the set?
- Estimation:
- Actual Solution:
- Paulo had $325 in the bank. He wrote a cheque for $76. How much money did he have left in the bank?
- Estimation:
- Actual Solution:
- Mizu weighs 99 kg. Akula weighs 81 kg. How much heavier is Mizu than Akula?
- Estimation:
- Actual Solution:
- Kenji has three children. One weighs 25 kg, another weighs 20 kg, and the last weighs 17 kg. How much do they weigh together?
- Estimation:
- Actual Solution:
- Rafael bought a boat priced at $8 400. He was given $1 250 as a trade-in on his old boat. How much does he owe on the new boat?
- Estimation:
- Actual Solution:
- Last week Luis earned $212. The week before he earned $198. This week he earned $133. How much did he earn in all?
- Estimation:
- Actual Solution:
- Jakob went on a trip of 739 km. The first day he drove 561 km. How many kilometres did he have left to drive?
- Estimation:
- Actual Solution:
- In 2005 Jacques’ net income was $29675. In 2006 his net income was $30207. How much more did he earn in 2006?
- Estimation:
- Actual Solution:
Answers to Exercise Three
- 63 hours
- $48 left
- 12 kg of potatoes left
- $90 for the chair
- $19 saved
- 281610 square kilometres
- $65 in all
- $85 still owed
- $249 left in the bank
- 18 kg heavier
- 62 kg altogether
- $7150 still owed
- $543 in all
- 178 km left to drive
- $532 more
Two-Operation Questions
Sometimes you may need to use two operations to solve a question. We work from left to right when solving questions that involve two operations. If addition is first, you must do the addition first then the subtraction. If subtraction is first, you must do the subtraction first and then do the addition.
Example D: 342 + 325 − 146 =
Step 1: 342 + 325 = 667
Step 2: Use your answer and subtract 146
[latex]\begin{array}[t]{rr}& 667\\-&146\\ \hline &521\end{array}[/latex]
342 + 325 − 146 = 521
Example E: 475 − 284 + 362 =
Step 1: [latex]\begin{array}[t]{rr}& 475\\-&284\\ \hline &191\end{array}[/latex]
Step 2: use your answer and add 362
[latex]\begin{array}[t]{rr}& 191\\+&362\\ \hline &553\end{array}[/latex]
475 − 284 + 362 = 553
Exercise Four
Find the sum or difference for each question. Check your work using the answer key at the end of the exercise.
- 312 + 541 − 135 =
- 427 + 231 − 384 =
- 687 − 434 + 256 =
- 754 − 576 + 393 =
- 1 456 + 218 − 295 =
- 2 461 + 723 − 349 =
- 3 857 − 665 + 1 234 =
- 4 367 − 843 + 5 679 =
- 5 247 + 2 216 − 4 673 =
- 1 285 + 4 672 − 1 401 =
- 7 354 − 4 038 + 2 348 =
- 4 187 − 2 574 + 1 846 =
- 5 314 + 7 053 − 597 =
- 4 315 + 3 197 − 2 106 =
- 46 124 − 9 762 + 2 534 =
- 70 534 − 7 689 + 1 824 =
Answers to Exercise Four
- 718
- 274
- 509
- 571
- 1379
- 2835
- 4426
- 9203
- 2790
- 4556
- 5664
- 3459
- 11770
- 5406
- 38896
- 64669
Two-Operation Problems
Sometimes you may need to use more than one operation to solve a word problem or a real- life problem.
Example F
Janet bought a submarine sandwich for $5, a soft drink for $1, and some carrot cake for $3. She gave the cashier a twenty dollar bill. How much money did she get back as change?
Step 1: Question: How much change from $20?
Step 2: Information: Spent $5 and $1 and $3. Gave cashier $20.
Step 3: Operations
- Add the amounts she spent to find the total: $5 + $1 + $3 =
- Subtract the amount she spent from $20: $20 – total of what she spent = change
Step 4: Estimate
Numbers are only one digit so do not round them. But a quick add tells you that her change will be about $10.
Step 5: Solve
- $5 + $1 + $3 = $9 total spent
- $20 – $9 = $11
Janet will get $11 in change.
Exercise Five
Use the 5 problem solving steps. Look for key words and patterns to help you choose the correct operation. Estimate the answer using rounded numbers if the numbers have 2 digits or more. Show all your work. Check your work using the answer key at the end of the exercise.
- Maureen weighed 72 kg and decided to go on a diet for her New Year’s Resolution. She lost 3 kg in January, 2 kg in February, and 4 kg in March. How much did she weigh after her three month diet?
- Estimation:
- Actual Solution:
- The local Girl Guides and Brownies had a goal to sell 2 850 boxes of Girl Guide cookies. In the first week the Brownies sold 975 boxes and the Guides sold 1 138 boxes. How many more boxes do they need to sell to reach their goal?
- Estimation:
- Actual Solution:
- Pat is ready to start first year college; she received a Passport to Education award from the provincial government which was $625. She got a Rotary Club Scholarship of $250 and a science scholarship of $400. Her first year’s tuition and books are going to cost $2000. Pat will use all her awards and scholarships. How much more money will she need to pay?
- Estimation:
- Actual Solution:
- The elementary school had 83 girls and 95 boys enrolled in September. Five of the girls and three of the boys moved away in September. How many children were still enrolled in the school at the end of September?
- Estimation:
- Actual Solution:
- Franco is on a 1200 calorie-a-day diet. He had 320 calories at breakfast and 468 calories at lunch. How many calories does he have left for dinner?
- Lilo had a total of 150 hats in four boxes. In box one there were 72 hats. In box two, there were 28 hats. In box three, there were 47 hats. How many hats were in box four?
- Miguel wanted to buy a Blue ray player for $225. He got $65 for his birthday. He won $75. How much more money does Miguel need?
- Kehara and Omar decided to visit their grandmother who lives 160 kilometres away. They travelled 50 kilometres and stopped for gas. They travelled another 30 kilometres and stopped for lunch. How much farther is it to their grandmother’s house?
- Kuen had $7 342 in his bank account. He decided to buy a new television for $1 139. Kuen was able to save another $697. How much does Kuen have in his bank account?
- Giles wishes to buy three gifts that cost $15, $9 and $12. He has $11 of the money he needs. How much more money does he need to earn in order to buy the gifts?
- Colette bought items costing $34, $19, $65 and $129. She used a coupon worth $75. How much money does she still owe?
- Sahale had 25 metres of fencing. He wanted to fence his garden that was 53 metres long and 38 metres wide. How much more fencing does Sahale need to buy? (Hint: To put a fence around means the perimetre. Draw a picture before you begin.)
Answers to Exercise Five
- 63 kg
- 737 boxes of cookies more
- $725 more
- 170 children still enrolled
- 412 calories
- 3 hats
- $85 more
- 80 kilometres
- $6900
- $25 more
- $172
- 157 metres
Topic F: Self-Test
Mark /14 Aim 12/14
Solve these problems. Show all your work. Give yourself one mark for the correct method and one mark for the correct answer. (14 marks)
- Alice weighed 86 kg. She went on a diet. Now she weighs 69 kg. How much did she lose?
- Estimation:
- Actual Solution:
- Jacques spent $49 on a pair of jeans, $18 for a shirt, $12 for a belt, and $3 for socks. How much did he spend altogether?
- Estimation:
- Actual Solution:
- Bookshelf had 94 books on the top shelf, 86 on the middle shelf, and 79 on the bottom shelf. How many books are there on the three shelves?
- Estimation:
- Actual Solution:
- Mahad bought a new car for $9989. He traded in his old car for $1785. How much more was the new one than the value of his trade-in?
- Estimation:
- Actual Solution:
- Kian and Toran picked apples for their uncle. Kian picked 509 kg and Toran picked 436 kg. (4 marks)
- How many more kilograms of apples did Kian pick than Toran?
- Estimation:
- Actual Solution:
- How many kilograms of apples did they pick together?
- Estimation:
- Actual Solution:
- How many more kilograms of apples did Kian pick than Toran?
- During an election, Dominique counted 4 721 votes and 8 956 votes. The number of spoiled ballots was 1 639. How many were good votes? (This question is worth 4 marks).
Answers to Topic F Self-Test
- 86 kg − 69 kg = 17 kg
- $49 + $18 + $12 + $3 = $82
- 94 + 86 + 79 = 259 books
- $9989 – $1785 = $8204
-
- 509 kg − 436 kg = 73 kg more
- 509 kg + 436 kg = 945 kg altogether
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