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4.6 Applications involving fractions

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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 ⋅ 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 3 4 size 12{ { {3} over {4} } } {} of 8 9 size 12{ { {8} over {9} } } {} . We are being asked the question, "What number is 3 4 size 12{ { {3} over {4} } } {} of 8 9 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 = 3 1 4 1 ⋅ 8 2 9 3 = 1 ⋅ 2 1 ⋅ 3 = 2 3

Thus, 3 4 of 8 9 is 2 3 .

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 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 3 4 size 12{ { {3} over {4} } } {} of 24.

Practice set a

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

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

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

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

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

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

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Missing factor statements

The equation 8 ⋅ M = 32 size 12{"8 " cdot " M "=" 32 "} {} is a missing factor statement. We can find the value of M size 12{M} {} that makes this statement true by dividing (since we know that 32 ÷ 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. 3 8 size 12{ { {3} over {8} } } {} of what number is 9 4 size 12{ { {9} over {4} } } {} ?
  2. What part of 1 2 7 size 12{1 { {2} over {7} } } {} is 1 13 14 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

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, 3 8 size 12{ { {3} over {8} } } {} of 6 is 9 4 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 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 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, 3 2 size 12{ { {3} over {2} } } {} of 1 2 7 size 12{1 { {2} over {7} } } {} is 1 13 14 size 12{1 { {"13"} over {"14"} } } {} .

Practice set b

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

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

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

16 27 size 12{ { {"16"} over {"27"} } } {}

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

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

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

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

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Exercises

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

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

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

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

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

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

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

9 8 size 12{ { {9} over {8} } } {} or 1 1 8 size 12{1 { {1} over {8} } } {}

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

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

1 16 size 12{ { {1} over {"16"} } } {}

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

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

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

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

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

1 1, 000 size 12{ { {1} over {1,"000"} } } {}

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

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

10 27 size 12{ { {"10"} over {"27"} } } {}

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

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

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

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

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

2

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

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

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

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

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

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

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

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

1

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

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

1

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

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

3 2 size 12{ { {3} over {2} } } {} or 1 2 3 size 12{1 { {2} over {3} } } {}

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

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

3

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

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

5 3 size 12{ { {5} over {3} } } {} or 1 2 3 size 12{1 { {2} over {3} } } {}

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

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

27 40 size 12{ { {"27"} over {"40"} } } {}

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

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

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

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

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What part of 24 is 9?

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

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What part of 42 is 26?

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

9 10 size 12{ { {9} over {"10"} } } {}

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

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

9 2 = 4 1 2 size 12{ { {9} over {2} } =4 { {1} over {2} } } {}

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

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

16 11 size 12{ { {"16"} over {"11"} } } {} or 1 5 11 size 12{1 { {5} over {"11"} } } {}

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

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

30 77 size 12{ { {"30"} over {"77"} } } {}

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Exercises for review

( [link] ) Use the numbers 2 and 7 to illustrate the commutative property of addition.

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( [link] ) Is 4 divisible by 0?

no

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( [link] ) Expand 3 7 size 12{3 rSup { size 8{7} } } {} . Do not find the actual value.

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( [link] ) Convert 3 5 12 size 12{3 { {5} over {"12"} } } {} to an improper fraction.

41 12 size 12{ { {"41"} over {"12"} } } {}

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( [link] ) Find the value of 3 8 ÷ 9 16 ⋅ 6 5 size 12{ { {3} over {8} } div { {9} over {"16"} } cdot { {6} over {5} } } {} .

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Source:  OpenStax, Fundamentals of mathematics. OpenStax CNX. Aug 18, 2010 Download for free at http://cnx.org/content/col10615/1.4
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