"

Chapter 1: Algebra Review

1.2 Fractions (Review)

Working with fractions is a very important foundational skill in algebra. This section will briefly review reducing, multiplying, dividing, adding, and subtracting fractions. As this is a review, concepts will not be explained in as much detail as they are in other lessons. Final answers of questions working with fractions tend to always be reduced. Reducing fractions is simply done by dividing both the numerator and denominator by the same number.

Example 1.2.1

Reduce 3684.3684.

3684Both the numerator and the denominator are divisible by 4.36÷484÷4=921Both the numerator and the denominator are divisible by 3.9÷321÷3=37Solution3684Both the numerator and the denominator are divisible by 4.36÷484÷4=921Both the numerator and the denominator are divisible by 3.9÷321÷3=37Solution

The previous example could have been done in one step by dividing both the numerator and the denominator by 12. Another solution could have been to divide by 2 twice and then by 3 once (in any order). It is not important which method is used as long as the fraction is reduced as much as possible.

The easiest operation to complete with fractions is multiplication. Fractions can be multiplied straight across, meaning all numerators and all denominators are multiplied together.

Example 1.2.2

Multiply 6735.6735.

6735Multiply numerators and denominators, respectively.1835Solution6735Multiply numerators and denominators, respectively.1835Solution

Before multiplying, fractions can be reduced. It is possible to reduce vertically within a single fraction, or diagonally within several fractions, as long as one number from the numerator and one number from the denominator are used.

Example 1.2.3

Multiply 25243255.25243255.
25 524 332 455 11Reduce 25 and 55 by dividing by 5, and reduce 32 and 24 by dividing by 8.54311Multiply numerators and denominators across.2033Solution25 524 332 455 11Reduce 25 and 55 by dividing by 5, and reduce 32 and 24 by dividing by 8.54311Multiply numerators and denominators across.2033Solution

Dividing fractions is very similar to multiplying, with one extra step. Dividing fractions necessitates first taking the reciprocal of the second fraction. Once this is done, multiply the fractions together. This multiplication problem solves just like the previous problem.

Example 1.2.4

Divide 2116÷286.2116÷286.

2116÷286Take the reciprocal of the second fraction and multiply it by the first.21 316 86 328 4Reduce 21 and 28 by dividing by 7, and reduce 6 and 16 by dividing by 2.3384Multiply numerators and denominators across.932Solution2116÷286Take the reciprocal of the second fraction and multiply it by the first.21 316 86 328 4Reduce 21 and 28 by dividing by 7, and reduce 6 and 16 by dividing by 2.3384Multiply numerators and denominators across.932Solution

To add and subtract fractions, it is necessary to first find the least common denominator (LCD). There are several ways to find the LCD. One way is to break the denominators into primes, write out the primes that make up the first denominator, and only add primes that are needed to make the other denominators.

Example 1.2.5

Find the LCD of 8 and 12.

Break 8 and 12 into primes:

8=2×2×212=2×2×38=2×2×212=2×2×3

The LCD will contain all the primes needed to make each number above.

LCD=82×2×22×2×2×312=4LCD=82×2×22×2×2×312=4

Adding and subtracting fractions is identical in process. If both fractions already have a common denominator, simply add or subtract the numerators and keep the denominator.

Example 1.2.6

Add 78+38.78+38.

78+38Same denominator, so add 7+3.108Reduce answer by dividing the numerator and denominator by 2.54Solution78+38Same denominator, so add 7+3.108Reduce answer by dividing the numerator and denominator by 2.54Solution

While 5454 can be written as the mixed number 114114, algebra almost never uses mixed numbers. For this reason, always use the improper fraction, not the mixed number.

Example 1.2.7

Subtract 13696.13696.

13696Same denominator, so subtract 139.46Reduce answer by dividing by 2.23Solution13696Same denominator, so subtract 139.46Reduce answer by dividing by 2.23Solution

If the denominators do not match, it is necessary to first identify the LCD and build up each fraction by multiplying the numerator and denominator by the same number so each denominator is built up to the LCD.

Example 1.2.8

Add 56+49.56+49.
62×32×3×39LCD is 18.3536+4292Multiply the first fraction by 3 and the second by 2.1518+818Same denominator, so add 15+8.2318Solution62×32×3×39LCD is 18.3536+4292Multiply the first fraction by 3 and the second by 2.1518+818Same denominator, so add 15+8.2318Solution

Example 1.2.9

Subtract 2316.2316.

2316LCD is 6.222316Multiply the first fraction by 2.4616Same denominator, so subtract 41.36Reduce answer by dividing by 3.12Solution2316LCD is 6.222316Multiply the first fraction by 2.4616Same denominator, so subtract 41.36Reduce answer by dividing by 3.12Solution

Questions

For questions 1 to 18, simplify each fraction. Leave your answer as an improper fraction.

  1. 42124212
  2. 25202520
  3. 35253525
  4. 248248
  5. 54365436
  6. 30243024
  7. 45364536
  8. 36273627
  9. 27182718
  10. 48184818
  11. 40164016
  12. 48424842
  13. 63186318
  14. 16121612
  15. 80608060
  16. 72487248
  17. 72607260
  18. 126108126108

For questions 19 to 36, find each product. Leave your answer as an improper fraction.

  1. (9)(89)(9)(89)
  2. (2)(56)(2)(56)
  3. (2)(29)(2)(29)
  4. (2)(13)(2)(13)
  5. (2)(138)(2)(138)
  6. (32)(12)(32)(12)
  7. (65)(118)(65)(118)
  8. (37)(118)(37)(118)
  9. (8)(12)(8)(12)
  10. (2)(97)(2)(97)
  11. (23)(34)(23)(34)
  12. (179)(35)(179)(35)
  13. (2)(32)(2)(32)
  14. (179)(35)(179)(35)
  15. (12)(75)(12)(75)
  16. (12)(57)(12)(57)
  17. (52)(05)(52)(05)
  18. (60)(67)(60)(67)

For questions 37 to 52, find each quotient. Leave your answer as an improper fraction.

  1. 2÷742÷74
  2. 127÷95127÷95
  3. 19÷1219÷12
  4. 2÷322÷32
  5. 32÷13732÷137
  6. 53÷7553÷75
  7. 1÷231÷23
  8. 109÷6109÷6
  9. 89÷1589÷15
  10. 16÷5316÷53
  11. 97÷1597÷15
  12. 138÷158138÷158
  13. 29÷3229÷32
  14. 45÷13845÷138
  15. 110÷32110÷32
  16. 53÷5353÷53

For questions 53 to 70, evaluate each expression. Leave your answer as an improper fraction.

  1. 13+(43)13+(43)
  2. 17+(117)17+(117)
  3. 37173717
  4. 13+5313+53
  5. 116+76116+76
  6. (2)+(158)(2)+(158)
  7. 35+5435+54
  8. (1)23(1)23
  9. 25+5425+54
  10. 1279712797
  11. 98+(27)
  12. (2)+56
  13. 1+(13)
  14. 12116
  15. (12)+32
  16. 118122
  17. 15+34
  18. 6585

Answer Key 1.2

License

Icon for the Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License

Intermediate Algebra Copyright © 2020 by Terrance Berg is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.

Share This Book