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Equivalent Fractions, Reducing Fractions to Lowest Terms, and Raising Fractions to Higher Terms

Module by: Wade Ellis, Denny Burzynski. E-mail the authorsEdited By: Math Editors

Summary: This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses equivalent fractions, reducing fractions to lowest terms, and raising fractions to higher terms. By the end of the module students should be able to recognize equivalent fractions, reduce a fraction to lowest terms and be able to raise a fraction to higher terms.

Section Overview

  • Equivalent Fractions
  • Reducing Fractions to Lowest Terms
  • Raising Fractions to Higher Terms

Equivalent Fractions

Let's examine the following two diagrams.

A rectangle divided equally into three parts, each marked one-third. The left two parts are shaded. To the right of the box is the caption, two-thirds of the whole is shaded. Below this is a rectangle equally divided into six part, with the leftmost four part shaded. to the right of this rectangle is the caption, four-sixths of the whole is shaded.

Notice that both 2323 size 12{ { {2} over {3} } } {} and 4646 size 12{ { {4} over {6} } } {} represent the same part of the whole, that is, they represent the same number.

Equivalent Fractions

Fractions that have the same value are called equivalent fractions. Equiva­lent fractions may look different, but they are still the same point on the number line.

There is an interesting property that equivalent fractions satisfy.

two-thirds and four-sixths, with an arrow from each denominator pointing to the numerator of the opposite fraction.

A Test for Equivalent Fractions Using the Cross Product

These pairs of products are called cross products.

Is two time six equal to three times four? Yes.

If the cross products are equal, the fractions are equivalent. If the cross products are not equal, the fractions are not equivalent.

Thus, 2323 size 12{ { {2} over {3} } } {} and 4646 size 12{ { {4} over {6} } } {} are equivalent, that is, 23=4623=46 size 12{ { {2} over {3} } = { {4} over {6} } } {}.

Sample Set A

Determine if the following pairs of fractions are equivalent.

Example 1

34and 6834and 68 size 12{ { {3} over {4} } `"and " { {6} over {8} } } {}. Test for equality of the cross products.

three-fourths and six-eigths, with an arrow from each denominator pointing to the numerator of the opposite fraction.

Is three times eight equal to six times four? yes. The cross products are equals.

The fractions 3434 and 6868 are equivalent, so 34=6834=68.

Example 2

38 and 91638 and 916 size 12{ { {3} over {8} } " and " { {9} over {"16"} } } {}. Test for equality of the cross products.

Three-eights and nine-sixteenths, with an arrow from each denominator pointing to the numerator of the opposite fraction.

is three times sixteen equal to nine times eight? No. forty-eight does not equal seventy-two. The cross products are not equal.

The fractions 3838 size 12{ { {3} over {8} } } {} and 916916 size 12{ { {9} over {"16"} } } {} are not equivalent.

Practice Set A

Determine if the pairs of fractions are equivalent.

Exercise 1

1212 size 12{ { {1} over {2} } } {}, 3636 size 12{ { {3} over {6} } } {}

Solution

Six equals six., yes

Exercise 2

4545 size 12{ { {4} over {5} } } {}, 12151215 size 12{ { {"12"} over {"15"} } } {}

Solution

Sixty equals sixty., yes

Exercise 3

2323 size 12{ { {2} over {3} } } {}, 815815 size 12{ { {8} over {"15"} } } {}

Solution

30≠2430≠24, no

Exercise 4

1818 size 12{ { {1} over {8} } } {}, 540540 size 12{ { {5} over {"40"} } } {}

Solution

Forty equals forty., yes

Exercise 5

312312 size 12{ { {3} over {"12"} } } {}, 1414 size 12{ { {1} over {4} } } {}

Solution

Twelve equals twelve., yes

Reducing Fractions to Lowest Terms

It is often very useful to convert one fraction to an equivalent fraction that has reduced values in the numerator and denominator. We can suggest a method for doing so by considering the equivalent fractions 915915 size 12{ { {9} over {"15"} } } {} and 3535 size 12{ { {3} over {5} } } {}. First, divide both the numerator and denominator of 915915 size 12{ { {9} over {"15"} } } {} by 3. The fractions 915915 size 12{ { {9} over {"15"} } } {} and 3535 size 12{ { {3} over {5} } } {} are equivalent.

(Can you prove this?) So, 915=35915=35 size 12{ { {9} over {"15"} } = { {3} over {5} } } {}. We wish to convert 915915 size 12{ { {9} over {"15"} } } {} to 3535 size 12{ { {3} over {5} } } {}. Now divide the numerator and denominator of 915915 size 12{ { {9} over {"15"} } } {} by 3, and see what happens.

9 ÷ 3 15 ÷ 3 = 3 5 9 ÷ 3 15 ÷ 3 = 3 5 size 12{ { {9 div 3} over {"15" div 3} } = { {3} over {5} } } {}

The fraction 915915 size 12{ { {9} over {"15"} } } {} is converted to 3535 size 12{ { {3} over {5} } } {}.

A natural question is "Why did we choose to divide by 3?" Notice that

9 15 = 3 ⋅ 3 5 ⋅ 3 9 15 = 3 ⋅ 3 5 ⋅ 3 size 12{ { {9} over {"15"} } = { {3 cdot 3} over {5 cdot 3} } } {}

We can see that the factor 3 is common to both the numerator and denominator.

Reducing a Fraction

From these observations we can suggest the following method for converting one fraction to an equivalent fraction that has reduced values in the numerator and denominator. The method is called reducing a fraction.

A fraction can be reduced by dividing both the numerator and denominator by the same nonzero whole number.

Nine-twelfths is equal to nine divided by three, over nine divided by three, which is equal to three-fourths. Sixteen thirtieths is equal to sixteen divided by two, over thirty divided by 2, which is equal to eight-fifteenths. Notice that three over three and two over two are both equal to 1.

Consider the collection of equivalent fractions

520520 size 12{ { {5} over {"20"} } } {}, 416416 size 12{ { {4} over {"16"} } } {}, 312312 size 12{ { {3} over {"12"} } } {}, 2828 size 12{ { {2} over {8} } } {}, 1414 size 12{ { {1} over {4} } } {}

Reduced to Lowest Terms

Notice that each of the first four fractions can be reduced to the last fraction, 1414 size 12{ { {1} over {4} } } {}, by dividing both the numerator and denominator by, respectively, 5, 4, 3, and 2. When a fraction is converted to the fraction that has the smallest numerator and denomi­nator in its collection of equivalent fractions, it is said to be reduced to lowest terms. The fractions 1414 size 12{ { {1} over {4} } } {}, 3838 size 12{ { {3} over {8} } } {}, 2525 size 12{ { {2} over {5} } } {}, and 710710 size 12{ { {7} over {"10"} } } {} are all reduced to lowest terms.

Observe a very important property of a fraction that has been reduced to lowest terms. The only whole number that divides both the numerator and denominator without a remainder is the number 1. When 1 is the only whole number that divides two whole numbers, the two whole numbers are said to be relatively prime.

Relatively Prime

A fraction is reduced to lowest terms if its numerator and denominator are relatively prime.

Methods of Reducing Fractions to Lowest Terms

Method 1: Dividing Out Common Primes

  1. Write the numerator and denominator as a product of primes.
  2. Divide the numerator and denominator by each of the common prime factors. We often indicate this division by drawing a slanted line through each divided out factor. This process is also called cancelling common factors.
  3. The product of the remaining factors in the numerator and the product of remaining factors of the denominator are relatively prime, and this fraction is reduced to lowest terms.

Sample Set B

Reduce each fraction to lowest terms.

Example 3

618=21⋅3121⋅31⋅3=13618=21⋅3121⋅31⋅3=13 size 12{ { {6} over {"18"} } = { { { { {2}}} cSup { size 8{1} } cdot { { {3}}} cSup { size 8{1} } } over { { { {2}}} cSub { size 8{1} } cdot { { {3}}} cSub { size 8{1} } cdot 3} } = { {1} over {3} } } {} 1 and 3 are relatively prime.

Example 4

1620=21⋅21⋅2⋅221⋅21⋅5=451620=21⋅21⋅2⋅221⋅21⋅5=45 size 12{ { {"16"} over {"20"} } = { { { { {2}}} cSup { size 8{1} } cdot { { {2}}} cSup { size 8{1} } cdot 2 cdot 2} over { { { {2}}} cSub { size 8{1} } cdot { { {2}}} cSub { size 8{1} } cdot 5} } = { {4} over {5} } } {} 4 and 5 are relatively prime.

Example 5

56104=21⋅21⋅21⋅721⋅21⋅21⋅13=71356104=21⋅21⋅21⋅721⋅21⋅21⋅13=713 size 12{ { {"56"} over {"104"} } = { { { { {2}}} cSup { size 8{1} } cdot { { {2}}} cSup { size 8{1} } cdot { { {2}}} cSup { size 8{1} } cdot 7} over { { { {2}}} cSub { size 8{1} } cdot { { {2}}} cSub { size 8{1} } cdot { { {2}}} cSub { size 8{1} } cdot "13"} } = { {7} over {"13"} } } {} 7 and 13 are relatively prime (and also truly prime)

Example 6

315 336 = 31⋅3⋅5⋅712⋅2⋅2⋅2⋅31⋅71⋅=1516 315 336 = 31⋅3⋅5⋅712⋅2⋅2⋅2⋅31⋅71⋅=1516 size 12{ { {"315"} over {"336"} } = { { { { {3}}} cSup { size 8{1} } cdot 3 cdot cdot 5 cdot { { {7}}} cSup { size 8{1} } } over {2 cdot 2 cdot 2 cdot 2 cdot { { {3}}} cSub { size 8{1} } cdot { { {7}}} cSub { size 8{1} } cdot } } = { {"15"} over {"16"} } } {} 15 and 16 are relatively prime.

Example 7

815=2⋅2⋅23⋅5815=2⋅2⋅23⋅5 size 12{ { {8} over {"15"} } = { {2 cdot 2 cdot 2} over {3 cdot 5} } } {} No common prime factors, so 8 and 15 are relatively prime.

The fraction 815815 size 12{ { {8} over {"15"} } } {} is reduced to lowest terms.

Practice Set B

Reduce each fraction to lowest terms.

Exercise 6

4848 size 12{ { {4} over {8} } } {}

Solution

1212 size 12{ { {1} over {2} } } {}

Exercise 7

615615 size 12{ { {6} over {"15"} } } {}

Solution

2525 size 12{ { {2} over {5} } } {}

Exercise 8

648648 size 12{ { {6} over {"48"} } } {}

Solution

1818 size 12{ { {1} over {8} } } {}

Exercise 9

21482148 size 12{ { {"21"} over {"48"} } } {}

Solution

716716 size 12{ { {7} over {"16"} } } {}

Exercise 10

72427242 size 12{ { {"72"} over {"42"} } } {}

Solution

127127 size 12{ { {"12"} over {7} } } {}

Exercise 11

135243135243 size 12{ { {"135"} over {"243"} } } {}

Solution

5959 size 12{ { {5} over {9} } } {}

Method 2: Dividing Out Common Factors

  1. Mentally divide the numerator and the denominator by a factor that is com­mon to each. Write the quotient above the original number.
  2. Continue this process until the numerator and denominator are relatively prime.

Sample Set C

Reduce each fraction to lowest terms.

Example 8

25302530 size 12{ { {"25"} over {"30"} } } {}. 5 divides into both 25 and 30.

25 5 30 6 = 5 6 25 5 30 6 = 5 6 size 12{ { { { { {2}} { {5}}} cSup { size 8{5} } } over { { { {3}} { {0}}} cSub { size 8{6} } } } = { {5} over {6} } } {} 5 and 6 are relatively prime.

Example 9

18241824 size 12{ { {"18"} over {"24"} } } {}. Both numbers are even so we can divide by 2.

18 9 24 12 18 9 24 12 Now, both 9 and 12 are divisible by 3.

18 9 3 24 124 = 3 4 18 9 3 24 124 = 3 4 size 12{ { { { { {1}} { {8}}} cSup { size 8{ { { {9}}} cSup { size 6{3} } } } } over { { { {2}} { {4}}} cSub {"12"} } } size 12{ {}= { {3} over {4} } }} {} 3 and 4 are relatively prime.

Example 10

210217150155=75210217150155=75 size 12{ { { { { {2}} { {1}} { {0}}} cSup { size 8{ { { {2}} { {1}}} cSup { size 6{7} } } } } over { { { {1}} { {5}} { {0}}} cSub { { { {1}} { {5}}} cSub { size 6{5} } } } } size 12{ {}= { {7} over {5} } }} {}. 7 and 5 are relatively prime.

Example 11

3696=1848=924=383696=1848=924=38 size 12{ { {"36"} over {"96"} } = { {"18"} over {"48"} } = { {9} over {"24"} } = { {3} over {8} } } {}. 3 and 8 are relatively prime.

Practice Set C

Reduce each fraction to lowest terms.

Exercise 12

12161216 size 12{ { {"12"} over {"16"} } } {}

Solution

3434 size 12{ { {3} over {4} } } {}

Exercise 13

924924 size 12{ { {9} over {"24"} } } {}

Solution

3838 size 12{ { {3} over {8} } } {}

Exercise 14

21842184 size 12{ { {"21"} over {"84"} } } {}

Solution

1414 size 12{ { {1} over {4} } } {}

Exercise 15

48644864 size 12{ { {"48"} over {"64"} } } {}

Solution

3434 size 12{ { {3} over {4} } } {}

Exercise 16

63816381 size 12{ { {"63"} over {"81"} } } {}

Solution

7979 size 12{ { {7} over {9} } } {}

Exercise 17

150240150240 size 12{ { {"150"} over {"240"} } } {}

Solution

5858 size 12{ { {5} over {8} } } {}

Raising Fractions to Higher Terms

Equally as important as reducing fractions is raising fractions to higher terms. Raising a fraction to higher terms is the process of constructing an equivalent fraction that has higher values in the numerator and denominator than the original fraction.

The fractions 3535 size 12{ { {3} over {5} } } {} and 915915 size 12{ { {9} over {"15"} } } {} are equivalent, that is, 35=91535=915 size 12{ { {3} over {5} } = { {9} over {"15"} } } {}. Notice also,

3 ⋅ 3 5 ⋅ 3 = 9 15 3 ⋅ 3 5 ⋅ 3 = 9 15 size 12{ { {3 cdot 3} over {5 cdot 3} } = { {9} over {"15"} } } {}

Notice that 33=133=1 size 12{ { {3} over {3} } =1} {} and that 35⋅1=3535⋅1=35 size 12{ { {3} over {5} } cdot 1= { {3} over {5} } } {}. We are not changing the value of 3535 size 12{ { {3} over {5} } } {}.

From these observations we can suggest the following method for converting one fraction to an equivalent fraction that has higher values in the numerator and denominator. This method is called raising a fraction to higher terms.

Raising a Fraction to Higher Terms

A fraction can be raised to an equivalent fraction that has higher terms in the numerator and denominator by multiplying both the numerator and denominator by the same nonzero whole number.

The fraction 3434 size 12{ { {3} over {4} } } {} can be raised to 24322432 size 12{ { {"24"} over {"32"} } } {} by multiplying both the numerator and denominator by 8.

Three fourths equals three times eight, over four time eight, which is equal to twenty-four over thirty-two. Notice that eight over eight is equal to 1.

Most often, we will want to convert a given fraction to an equivalent fraction with a higher specified denominator. For example, we may wish to convert 5858 size 12{ { {5} over {8} } } {} to an equivalent fraction that has denominator 32, that is,

5 8 = ? 32 5 8 = ? 32 size 12{ { {5} over {8} } = { {?} over {"32"} } } {}

This is possible to do because we know the process. We must multiply both the numerator and denominator of 5858 size 12{ { {5} over {8} } } {} by the same nonzero whole number in order to 8 obtain an equivalent fraction.

We have some information. The denominator 8 was raised to 32 by multiplying it by some nonzero whole number. Division will give us the proper factor. Divide the original denominator into the new denominator.

32 ÷ 8 = 4 32 ÷ 8 = 4 size 12{"32 " div " 8 "=" 4"} {}

Now, multiply the numerator 5 by 4.

5 ⋅ 4 = 20 5 ⋅ 4 = 20 size 12{"5 " cdot "4 "=" 20"} {}

Thus,

5 8 = 5 ⋅ 4 8 ⋅ 4 = 20 32 5 8 = 5 ⋅ 4 8 ⋅ 4 = 20 32 size 12{ { {5} over {8} } = { {5 cdot 4} over {8 cdot 4} } = { {"20"} over {"32"} } } {}

So,

5 8 = 20 32 5 8 = 20 32 size 12{ { {5} over {8} } = { {"20"} over {"32"} } } {}

Sample Set D

Determine the missing numerator or denominator.

Example 12

37=?3537=?35 size 12{ { {3} over {7} } = { {?} over {"35"} } } {}. Divide the original denominator into the new denominator.

35 ÷ 7 = 5 35 ÷ 7 = 5 size 12{"35"¸7=5} {} The quotient is 5. Multiply the original numerator by 5.

3 7 = 3 ⋅ 5 7 ⋅ 5 = 15 35 3 7 = 3 ⋅ 5 7 ⋅ 5 = 15 35 size 12{ { {3} over {7} } = { {3 cdot 5} over {7 cdot 5} } = { {"15"} over {"35"} } } {} The missing numerator is 15.

Example 13

56=45?56=45? size 12{ { {5} over {6} } = { {"45"} over {?} } } {}. Divide the original numerator into the new numerator.

45 ÷ 5 = 9 45 ÷ 5 = 9 size 12{"45"¸5=9} {} The quotient is 9. Multiply the original denominator by 9.

5 6 = 5 ⋅ 9 6 ⋅ 9 = 45 54 5 6 = 5 ⋅ 9 6 ⋅ 9 = 45 54 size 12{ { {5} over {6} } = { {5 cdot 9} over {6 cdot 9} } = { {"45"} over {"54"} } } {} The missing denominator is 45.

Practice Set D

Determine the missing numerator or denominator.

Exercise 18

45=?4045=?40 size 12{ { {4} over {5} } = { {?} over {"40"} } } {}

Solution

32

Exercise 19

37=?2837=?28 size 12{ { {3} over {7} } = { {?} over {"28"} } } {}

Solution

12

Exercise 20

16=?2416=?24 size 12{ { {1} over {6} } = { {?} over {"24"} } } {}

Solution

4

Exercise 21

310=45?310=45? size 12{ { {3} over {"10"} } = { {"45"} over {?} } } {}

Solution

150

Exercise 22

815=?165815=?165 size 12{ { {8} over {"15"} } = { {?} over {"165"} } } {}

Solution

88

Exercises

For the following problems, determine if the pairs of fractions are equivalent.

Exercise 23

12,51012,510 size 12{ { {1} over {2} } , { {5} over {"10"} } } {}

Solution

equivalent

Exercise 24

23,81223,812 size 12{ { {2} over {3} } , { {8} over {"12"} } } {}

Exercise 25

512,1024512,1024 size 12{ { {5} over {"12"} } , { {"10"} over {"24"} } } {}

Solution

equivalent

Exercise 26

12,3612,36 size 12{ { {1} over {2} } , { {3} over {6} } } {}

Exercise 27

35,121535,1215 size 12{ { {3} over {5} } , { {"12"} over {"15"} } } {}

Solution

not equivalent

Exercise 28

16,74216,742 size 12{ { {1} over {6} } , { {7} over {"42"} } } {}

Exercise 29

1625,49751625,4975 size 12{ { {"16"} over {"25"} } , { {"49"} over {"75"} } } {}

Solution

not equivalent

Exercise 30

528,20112528,20112 size 12{ { {5} over {"28"} } , { {"20"} over {"112"} } } {}

Exercise 31

310,36110310,36110 size 12{ { {3} over {"10"} } , { {"36"} over {"110"} } } {}

Solution

not equivalent

Exercise 32

610,1832610,1832 size 12{ { {6} over {"10"} } , { {"18"} over {"32"} } } {}

Exercise 33

58,152458,1524 size 12{ { {5} over {8} } , { {"15"} over {"24"} } } {}

Solution

equivalent

Exercise 34

1016,15241016,1524 size 12{ { {"10"} over {"16"} } , { {"15"} over {"24"} } } {}

Exercise 35

45,3445,34 size 12{ { {4} over {5} } , { {3} over {4} } } {}

Solution

not equivalent

Exercise 36

57,152157,1521 size 12{ { {5} over {7} } , { {"15"} over {"21"} } } {}

Exercise 37

911,119911,119 size 12{ { {9} over {"11"} } , { {"11"} over {9} } } {}

Solution

not equivalent

For the following problems, determine the missing numerator or denominator.

Exercise 38

13=?1213=?12 size 12{ { {1} over {3} } = { {?} over {"12"} } } {}

Exercise 39

15=?3015=?30 size 12{ { {1} over {5} } = { {?} over {"30"} } } {}

Solution

6

Exercise 40

23=?923=?9 size 12{ { {2} over {3} } = { {?} over {9} } } {}

Exercise 41

15=?3015=?30 size 12{ { {1} over {5} } = { {?} over {"30"} } } {}

Solution

12

Exercise 42

2 3 = ? 9 2 3 = ? 9

Exercise 43

3 4 = ? 16 3 4 = ? 16

Solution

12

Exercise 44

56=?1856=?18 size 12{ { {5} over {6} } = { {?} over {"18"} } } {}

Exercise 45

45=?2545=?25 size 12{ { {4} over {5} } = { {?} over {"25"} } } {}

Solution

20

Exercise 46

12=4?12=4? size 12{ { {1} over {2} } = { {4} over {?} } } {}

Exercise 47

925=27?925=27? size 12{ { {9} over {"25"} } = { {"27"} over {?} } } {}

Solution

75

Exercise 48

32=18?32=18? size 12{ { {3} over {2} } = { {"18"} over {?} } } {}

Exercise 49

53=80?53=80? size 12{ { {5} over {3} } = { {"80"} over {?} } } {}

Solution

48

Exercise 50

18=3?18=3? size 12{ { {1} over {8} } = { {3} over {?} } } {}

Exercise 51

45=?10045=?100 size 12{ { {4} over {5} } = { {?} over {"100"} } } {}

Solution

80

Exercise 52

12=25?12=25? size 12{ { {1} over {2} } = { {"25"} over {?} } } {}

Exercise 53

316=?96316=?96 size 12{ { {3} over {"16"} } = { {?} over {"96"} } } {}

Solution

18

Exercise 54

1516=225?1516=225? size 12{ { {"15"} over {"16"} } = { {"225"} over {?} } } {}

Exercise 55

1112=?1681112=?168 size 12{ { {"11"} over {"12"} } = { {?} over {"168"} } } {}

Solution

154

Exercise 56

913=?286913=?286 size 12{ { {9} over {"13"} } = { {?} over {"286"} } } {}

Exercise 57

3233=?15183233=?1518 size 12{ { {"32"} over {"33"} } = { {?} over {"1518"} } } {}

Solution

1,472

Exercise 58

1920=1045?1920=1045? size 12{ { {"19"} over {"20"} } = { {"1045"} over {?} } } {}

Exercise 59

3750=1369?3750=1369? size 12{ { {"37"} over {"50"} } = { {"1369"} over {?} } } {}

Solution

1,850

For the following problems, reduce, if possible, each of the fractions to lowest terms.

Exercise 60

6868 size 12{ { {6} over {8} } } {}

Exercise 61

810810 size 12{ { {8} over {"10"} } } {}

Solution

4545 size 12{ { {4} over {5} } } {}

Exercise 62

510510 size 12{ { {5} over {"10"} } } {}

Exercise 63

614614 size 12{ { {6} over {"14"} } } {}

Solution

3737 size 12{ { {3} over {7} } } {}

Exercise 64

312312 size 12{ { {3} over {"12"} } } {}

Exercise 65

414414 size 12{ { {4} over {"14"} } } {}

Solution

2727 size 12{ { {2} over {7} } } {}

Exercise 66

1616 size 12{ { {1} over {6} } } {}

Exercise 67

4646 size 12{ { {4} over {6} } } {}

Solution

2323 size 12{ { {2} over {3} } } {}

Exercise 68

18141814 size 12{ { {"18"} over {"14"} } } {}

Exercise 69

208208 size 12{ { {"20"} over {8} } } {}

Solution

5252 size 12{ { {5} over {2} } } {}

Exercise 70

4646 size 12{ { {4} over {6} } } {}

Exercise 71

106106 size 12{ { {"10"} over {6} } } {}

Solution

5353 size 12{ { {5} over {3} } } {}

Exercise 72

614614 size 12{ { {6} over {"14"} } } {}

Exercise 73

146146 size 12{ { {"14"} over {6} } } {}

Solution

7373 size 12{ { {7} over {3} } } {}

Exercise 74

10121012 size 12{ { {"10"} over {"12"} } } {}

Exercise 75

16701670 size 12{ { {"16"} over {"70"} } } {}

Solution

835835 size 12{ { {8} over {"35"} } } {}

Exercise 76

40604060 size 12{ { {"40"} over {"60"} } } {}

Exercise 77

20122012 size 12{ { {"20"} over {"12"} } } {}

Solution

5353 size 12{ { {5} over {3} } } {}

Exercise 78

32283228 size 12{ { {"32"} over {"28"} } } {}

Exercise 79

36103610 size 12{ { {"36"} over {"10"} } } {}

Solution

185185 size 12{ { {"18"} over {5} } } {}

Exercise 80

36603660 size 12{ { {"36"} over {"60"} } } {}

Exercise 81

12181218 size 12{ { {"12"} over {"18"} } } {}

Solution

2323 size 12{ { {2} over {3} } } {}

Exercise 82

18271827 size 12{ { {"18"} over {"27"} } } {}

Exercise 83

18241824 size 12{ { {"18"} over {"24"} } } {}

Solution

3434 size 12{ { {3} over {4} } } {}

Exercise 84

32403240 size 12{ { {"32"} over {"40"} } } {}

Exercise 85

11221122 size 12{ { {"11"} over {"22"} } } {}

Solution

1212 size 12{ { {1} over {2} } } {}

Exercise 86

27812781 size 12{ { {"27"} over {"81"} } } {}

Exercise 87

17511751 size 12{ { {"17"} over {"51"} } } {}

Solution

1313 size 12{ { {1} over {3} } } {}

Exercise 88

16421642 size 12{ { {"16"} over {"42"} } } {}

Exercise 89

39133913 size 12{ { {"39"} over {"13"} } } {}

Solution

3

Exercise 90

44114411 size 12{ { {"44"} over {"11"} } } {}

Exercise 91

66336633 size 12{ { {"66"} over {"33"} } } {}

Solution

2

Exercise 92

151151 size 12{ { {"15"} over {1} } } {}

Exercise 93

15161516 size 12{ { {"15"} over {"16"} } } {}

Solution

already reduced

Exercise 94

15401540 size 12{ { {"15"} over {"40"} } } {}

Exercise 95

3610036100 size 12{ { {"36"} over {"100"} } } {}

Solution

925925 size 12{ { {9} over {"25"} } } {}

Exercise 96

45324532 size 12{ { {"45"} over {"32"} } } {}

Exercise 97

30753075 size 12{ { {"30"} over {"75"} } } {}

Solution

2525 size 12{ { {2} over {5} } } {}

Exercise 98

121132121132 size 12{ { {"121"} over {"132"} } } {}

Exercise 99

72647264 size 12{ { {"72"} over {"64"} } } {}

Solution

9898 size 12{ { {9} over {8} } } {}

Exercise 100

3010530105 size 12{ { {"30"} over {"105"} } } {}

Exercise 101

46604660 size 12{ { {"46"} over {"60"} } } {}

Solution

23302330 size 12{ { {"23"} over {"30"} } } {}

Exercise 102

75457545 size 12{ { {"75"} over {"45"} } } {}

Exercise 103

40184018 size 12{ { {"40"} over {"18"} } } {}

Solution

209209 size 12{ { {"20"} over {9} } } {}

Exercise 104

1087610876 size 12{ { {"108"} over {"76"} } } {}

Exercise 105

721721 size 12{ { {7} over {"21"} } } {}

Solution

1313 size 12{ { {1} over {3} } } {}

Exercise 106

651651 size 12{ { {6} over {"51"} } } {}

Exercise 107

51125112 size 12{ { {"51"} over {"12"} } } {}

Solution

174174 size 12{ { {"17"} over {4} } } {}

Exercise 108

81008100 size 12{ { {8} over {"100"} } } {}

Exercise 109

51545154 size 12{ { {"51"} over {"54"} } } {}

Solution

17181718 size 12{ { {"17"} over {"18"} } } {}

Exercise 110

A ream of paper contains 500 sheets. What frac­tion of a ream of paper is 200 sheets? Be sure to reduce.

Exercise 111

There are 24 hours in a day. What fraction of a day is 14 hours?

Solution

712712 size 12{ { {7} over {"12"} } } {}

Exercise 112

A full box contains 80 calculators. How many calculators are in 1414 size 12{ { {1} over {4} } } {} of a box?

Exercise 113

There are 48 plants per flat. How many plants are there in 1313 size 12{ { {1} over {3} } } {} of a flat?

Solution

16

Exercise 114

A person making $18,000 per year must pay $3,960 in income tax. What fraction of this per­son's yearly salary goes to the IRS?

For the following problems, find the mistake.

Exercise 115

324=33⋅8=08=0324=33⋅8=08=0 size 12{ { {3} over {"24"} } = { { { {3}}} over { { {3}} cdot 8} } = { {0} over {8} } =0} {}

Solution

Should be 1818 size 12{ { {1} over {8} } } {}; the cancellation is division, so the numerator should be 1.

Exercise 116

810=2+62+8=68=34810=2+62+8=68=34 size 12{ { {8} over {"10"} } = { { { {2}}+6} over { { {2}}+8} } = { {6} over {8} } = { {3} over {4} } } {}

Exercise 117

715=77+8=18715=77+8=18 size 12{ { {7} over {"15"} } = { { { {7}}} over { { {7}}+8} } = { {1} over {8} } } {}

Solution

Cancel factors only, not addends; 715715 size 12{ { {7} over {"15"} } } {} is already reduced.

Exercise 118

67=5+15+2=1267=5+15+2=12 size 12{ { {6} over {7} } = { { { {5}}+1} over { { {5}}+2} } = { {1} over {2} } } {}

Exercise 119

99=00=099=00=0 size 12{ { { { {9}}} over { { {9}}} } = { {0} over {0} } =0} {}

Solution

Same as Exercise 115; answer is 1111 size 12{ { {1} over {1} } } {} or 1.

Exercises for Review

Exercise 120

((Reference)) Round 816 to the nearest thousand.

Exercise 121

((Reference)) Perform the division: 0÷60÷6 size 12{0 div 6} {}.

Solution

0

Exercise 122

((Reference)) Find all the factors of 24.

Exercise 123

((Reference)) Find the greatest common factor of 12 and 18.

Solution

6

Exercise 124

((Reference)) Convert 158158 size 12{ { {"15"} over {8} } } {} to a mixed number.

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