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Applications Involving Fractions

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 applications involving fractions. By the end of the module students should be able to solve missing product statements and solve missing factor statements.

Section Overview

  • Multiplication Statements
  • Missing Product Statements
  • Missing Factor Statements

Multiplication Statements

Statement, Multiplication Statement

A statement is a sentence that is either true or false. A mathematical statement of the form

product = (factor 1) ⋅ (factor 2)

is a multiplication statement. Depending on the numbers that are used, it can be either true or false.

Omitting exactly one of the three numbers in the statement will produce exactly one of the following three problems. For convenience, we'll represent the omitted (or missing) number with the letter M (M for Missing).

  1. M = (factor 1) ⋅ (factor 2) Missing product statement.
  2. M ⋅ (factor 2) = product Missing factor statement.
  3. (factor 1) ⋅ M = product Missing factor statement.

We are interested in developing and working with methods to determine the missing number that makes the statement true. Fundamental to these methods is the ability to translate two words to mathematical symbols. The word


 of translates to times
is translates to equals

Missing Products Statements

The equation M = 8 ⋅ 4M = 8 ⋅ 4 size 12{"M "=" 8 " cdot " 4"} {} is a missing product statement. We can find the value of M that makes this statement true by multiplying the known factors.

Missing product statements can be used to determine the answer to a question such as, "What number is fraction 1 of fraction 2?

Sample Set A

Find 3434 size 12{ { {3} over {4} } } {} of 8989 size 12{ { {8} over {9} } } {}. We are being asked the question, "What number is 3434 size 12{ { {3} over {4} } } {} of 8989 size 12{ { {8} over {9} } } {}?" We must translate from words to mathematical symbols.

Two statements in a row. Each element is aligned with something above it. First, what number is three-fourths of eight-ninths, becomes. Second, M equals three-fourths times eight-ninths, multiply. M is the missing product, and three-fourths and eight-ninths are known factors.

M = 31 4 1 ⋅ 8 2 93 = 1 ⋅ 2 1 ⋅ 3 = 2 3 M = 31 4 1 ⋅ 8 2 93 = 1 ⋅ 2 1 ⋅ 3 = 2 3

Thus, 3434 of 8989 is 2323.

Two statements in a row. Each element is aligned with something above it. First, what number is three-fourths of twenty-four. Second, M equals three-fourths times twenty-four. M is the missing product, and three-fourths and eight-ninths are known factors.

M = 3 4 1 ⋅ 24 6 1 = 3 ⋅ 6 1 ⋅ 1 = 18 1 = 18 M = 3 4 1 ⋅ 24 6 1 = 3 ⋅ 6 1 ⋅ 1 = 18 1 = 18 size 12{M= { {3} over { {4} cSub { size 8{1} } } } cdot { { {"24"} cSup { size 8{6} } } over {1} } = { {3 cdot 6} over {1 cdot 1} } = { {"18"} over {1} } ="18"} {}

Thus, 18 is 3434 size 12{ { {3} over {4} } } {} of 24.

Practice Set A

Exercise 1

Find 3838 size 12{ { {3} over {8} } } {} of 16151615 size 12{ { {"16"} over {"15"} } } {}.

[ Show Solution ]

Exercise 2

What number is 910910 size 12{ { {9} over {"10"} } } {} of 5656 size 12{ { {5} over {6} } } {} ?

[ Show Solution ]

Exercise 3

11161116 size 12{ { {"11"} over {"16"} } } {} of 833833 size 12{ { {8} over {"33"} } } {} is what number?

[ Show Solution ]

Missing Factor Statements

The equation 8 ⋅ M = 32 8 ⋅ M = 32 size 12{"8 " cdot " M "=" 32 "} {} is a missing factor statement. We can find the value of MM size 12{M} {} that makes this statement true by dividing (since we know that 32 ÷ 8 = 432 ÷ 8 = 4 size 12{"32 " div " 8 "=" 4"} {}).

The expression 8 times M equals 32 means that M equals thirty-two divided by eight. M is the missing factor, thirty-two is the product, and eight is the known factor.

Finding the Missing Factor

To find the missing factor in a missing factor statement, divide the product by the known factor.
missing factor = (product) ÷ (known factor)

Missing factor statements can be used to answer such questions as

  1. 3838 size 12{ { {3} over {8} } } {} of what number is 9494 size 12{ { {9} over {4} } } {}?
  2. What part of 127127 size 12{1 { {2} over {7} } } {} is 1131411314 size 12{1 { {"13"} over {"14"} } } {}?

Sample Set B

Three eighths of what number is nine fourths? This is the same as three eighths times M equals nine fourths. Three eighths is the known factor, M is the missing factor, and nine-fourths is the product.

Now, using

missing factor = (product) ÷ (known factor)

We get

M = 9 4 ÷ 3 8 = 9 4 ⋅ 8 3 = 9 3 4 1 ⋅ 8 2 3 1 = 3 ⋅ 2 1 ⋅ 1 = 6 M = 9 4 ÷ 3 8 = 9 4 ⋅ 8 3 = 9 3 4 1 ⋅ 8 2 3 1 = 3 ⋅ 2 1 ⋅ 1 = 6

Check the work: is three eighths times six equal to nine fourths? Cancel out the six and the eight by dividing each by two. This can be simplified to find out that yes, they are equal.

Thus, 3838 size 12{ { {3} over {8} } } {} of 6 is 9494 size 12{ { {9} over {4} } } {}.

What part of 1 and two-sevenths is 1 and thirteen-fourteenths? This is equivalent to M times 1 and two-sevenths equals 1 and thirteen-fourteenths. M is the missing factor, 1 and two-sevenths is the known factor, and 1 and thirteen-fourteenths is the product.

For convenience, let's convert the mixed numbers to improper fractions.

M ⋅ 9 7 = 27 14 M ⋅ 9 7 = 27 14 size 12{M cdot { {9} over {7} } = { {"27"} over {"14"} } } {}

Now, using

missing factor = (product)÷(known factor)

we get

M = 27 14 ÷ 9 7 = 27 14 ⋅ 7 9 = 27 3 14 2 ⋅ 7 1 9 1 = 3 ⋅ 1 2 ⋅ 1 = 3 2 M = 27 14 ÷ 9 7 = 27 14 ⋅ 7 9 = 27 3 14 2 ⋅ 7 1 9 1 = 3 ⋅ 1 2 ⋅ 1 = 3 2 alignr { stack { size 12{M= { {"27"} over {"14"} } div { {9} over {7} } = { {"27"} over {"14"} } cdot { {7} over {9} } = { { {"27"} cSup { size 8{3} } } over { {"14"} cSub { size 8{2} } } } cdot { { {7} cSup { size 8{1} } } over { {9} cSub { size 8{1} } } } } {} # = { {3 cdot 1} over {2 cdot 1} } {} # = { {3} over {2} } {} } } {}

Check: is three halves times nine sevenths equal to twenty-seven fourteenths? Yes.

Thus, 3232 size 12{ { {3} over {2} } } {} of 127127 size 12{1 { {2} over {7} } } {} is 1131411314 size 12{1 { {"13"} over {"14"} } } {}.

Practice Set B

Exercise 4

3535 size 12{ { {3} over {5} } } {} of what number is 920920 size 12{ { {9} over {"20"} } } {}?

[ Show Solution ]

Exercise 5

334334 size 12{3 { {3} over {4} } } {} of what number is 229229 size 12{2 { {2} over {9} } } {}?

[ Show Solution ]

Exercise 6

What part of 3535 size 12{ { {3} over {5} } } {} is 910910 size 12{ { {9} over {"10"} } } {}?

[ Show Solution ]

Exercise 7

What part of 114114 size 12{1 { {1} over {4} } } {} is 178178 size 12{1 { {7} over {8} } } {}?

[ Show Solution ]

Exercises

Exercise 8

Find 2323 size 12{ { {2} over {3} } } {} of 3434 size 12{ { {3} over {4} } } {}.

[ Show Solution ]

Exercise 9

Find 5858 size 12{ { {5} over {8} } } {} of 110110 size 12{ { {1} over {"10"} } } {}.

Exercise 10

Find 12131213 size 12{ { {"12"} over {"13"} } } {} of 13361336 size 12{ { {"13"} over {"36"} } } {}.

[ Show Solution ]

Exercise 11

Find 1414 size 12{ { {1} over {4} } } {} of 4747 size 12{ { {4} over {7} } } {}.

Exercise 12

310310 size 12{ { {3} over {"10"} } } {} of 154154 size 12{ { {"15"} over {4} } } {} is what number?

[ Show Solution ]

Exercise 13

14151415 size 12{ { {"14"} over {"15"} } } {} of 20212021 size 12{ { {"20"} over {"21"} } } {} is what number?

Exercise 14

344344 size 12{ { {3} over {"44"} } } {} of 11121112 size 12{ { {"11"} over {"12"} } } {} is what number?

[ Show Solution ]

Exercise 15

1313 size 12{ { {1} over {3} } } {} of 2 is what number?

Exercise 16

1414 size 12{ { {1} over {4} } } {} of 3 is what number?

[ Show Solution ]

Exercise 17

110110 size 12{ { {1} over {"10"} } } {} of 11001100 size 12{ { {1} over {"100"} } } {} is what number?

Exercise 18

11001100 size 12{ { {1} over {"100"} } } {} of 110110 size 12{ { {1} over {"10"} } } {} is what number?

[ Show Solution ]

Exercise 19

159159 size 12{1 { {5} over {9} } } {} of 247247 size 12{2 { {4} over {7} } } {} is what number?

Exercise 20

17181718 size 12{1 { {7} over {"18"} } } {} of 415415 size 12{ { {4} over {"15"} } } {} is what number?

[ Show Solution ]

Exercise 21

118118 size 12{1 { {1} over {8} } } {} of 1111611116 size 12{1 { {"11"} over {"16"} } } {} is what number?

Exercise 22

Find 2323 size 12{ { {2} over {3} } } {} of 1616 size 12{ { {1} over {6} } } {} of 9292 size 12{ { {9} over {2} } } {}.

[ Show Solution ]

Exercise 23

Find 5858 size 12{ { {5} over {8} } } {} of 920920 size 12{ { {9} over {"20"} } } {} of 4949 size 12{ { {4} over {9} } } {}.

Exercise 24

512512 size 12{ { {5} over {"12"} } } {} of what number is 5656 size 12{ { {5} over {6} } } {}?

[ Show Solution ]

Exercise 25

314314 size 12{ { {3} over {"14"} } } {} of what number is 6767 size 12{ { {6} over {7} } } {}?

Exercise 26

103103 size 12{ { {"10"} over {3} } } {} of what number is 5959 size 12{ { {5} over {9} } } {}?

[ Show Solution ]

Exercise 27

157157 size 12{ { {"15"} over {7} } } {} of what number is 20212021 size 12{ { {"20"} over {"21"} } } {}?

Exercise 28

8383 size 12{ { {8} over {3} } } {} of what number is 179179 size 12{1 { {7} over {9} } } {}?

[ Show Solution ]

Exercise 29

1313 size 12{ { {1} over {3} } } {} of what number is 1313 size 12{ { {1} over {3} } } {}?

Exercise 30

1616 size 12{ { {1} over {6} } } {} of what number is 1616 size 12{ { {1} over {6} } } {}?

[ Show Solution ]

Exercise 31

3434 size 12{ { {3} over {4} } } {} of what number is 3434 size 12{ { {3} over {4} } } {}?

Exercise 32

811811 size 12{ { {8} over {"11"} } } {} of what number is 811811 size 12{ { {8} over {"11"} } } {}?

[ Show Solution ]

Exercise 33

3838 size 12{ { {3} over {8} } } {} of what number is 0?

Exercise 34

2323 size 12{ { {2} over {3} } } {} of what number is 1?

[ Show Solution ]

Exercise 35

315315 size 12{3 { {1} over {5} } } {} of what number is 1?

Exercise 36

19121912 size 12{1 { {9} over {"12"} } } {} of what number is 514514 size 12{5 { {1} over {4} } } {}?

[ Show Solution ]

Exercise 37

31253125 size 12{3 { {1} over {"25"} } } {} of what number is 28152815 size 12{2 { {8} over {"15"} } } {}?

Exercise 38

What part of 2323 size 12{ { {2} over {3} } } {} is 119119 size 12{1 { {1} over {9} } } {}?

[ Show Solution ]

Exercise 39

What part of 910910 size 12{ { {9} over {"10"} } } {} is 335335 size 12{3 { {3} over {5} } } {}?

Exercise 40

What part of 8989 size 12{ { {8} over {9} } } {} is 3535 size 12{ { {3} over {5} } } {}?

[ Show Solution ]

Exercise 41

What part of 14151415 size 12{ { {"14"} over {"15"} } } {} is 730730 size 12{ { {7} over {"30"} } } {}?

Exercise 42

What part of 3 is 1515 size 12{ { {1} over {5} } } {}?

[ Show Solution ]

Exercise 43

What part of 8 is 2323 size 12{ { {2} over {3} } } {}?

Exercise 44

What part of 24 is 9?

[ Show Solution ]

Exercise 45

What part of 42 is 26?

Exercise 46

Find 12131213 size 12{ { {"12"} over {"13"} } } {} of 39403940 size 12{ { {"39"} over {"40"} } } {}.

[ Show Solution ]

Exercise 47

14151415 size 12{ { {"14"} over {"15"} } } {} of 12211221 size 12{ { {"12"} over {"21"} } } {} is what number?

Exercise 48

815815 size 12{ { {8} over {"15"} } } {} of what number is 225225 size 12{2 { {2} over {5} } } {}?

[ Show Solution ]

Exercise 49

11151115 size 12{ { {"11"} over {"15"} } } {} of what number is 22352235 size 12{ { {"22"} over {"35"} } } {}?

Exercise 50

11161116 size 12{ { {"11"} over {"16"} } } {} of what number is 1?

[ Show Solution ]

Exercise 51

What part of 23402340 size 12{ { {"23"} over {"40"} } } {} is 39203920 size 12{3 { {9} over {"20"} } } {}?

Exercise 52

435435 size 12{ { {4} over {"35"} } } {} of 39223922 size 12{3 { {9} over {"22"} } } {} is what number?

[ Show Solution ]

Exercises for Review

Exercise 53

((Reference)) Use the numbers 2 and 7 to illustrate the commutative property of addition.

Exercise 54

((Reference)) Is 4 divisible by 0?

[ Show Solution ]

Exercise 55

((Reference)) Expand 3737 size 12{3 rSup { size 8{7} } } {}. Do not find the actual value.

Exercise 56

((Reference)) Convert 35123512 size 12{3 { {5} over {"12"} } } {} to an improper fraction.

[ Show Solution ]

Exercise 57

((Reference)) Find the value of 38÷916⋅6538÷916⋅65 size 12{ { {3} over {8} } div { {9} over {"16"} } cdot { {6} over {5} } } {}.

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