4.6: Applications Involving Fractions
- be able to solve missing product statements
- be able to solve 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).
- M = (factor 1) ⋅ (factor 2) Missing product statement.
- M ⋅ (factor 2) = product Missing factor statement.
- (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 \(M = 8 \cdot 4\) 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 \(\dfrac{3}{4}\) of \(\dfrac{8}{9}\). We are being asked the question, "What number is \(\dfrac{3}{4}\) of \(\dfrac{8}{9}\)?" We must translate from words to mathematical symbols.
\(M = \dfrac{\begin{array} {c} {^1} \\ {\cancel{3}} \end{array}}{\begin{array} {c} {\cancel{4}} \\ {^1} \end{array}} \cdot \dfrac{\begin{array} {c} {^2} \\ {\cancel{8}} \end{array}}{\begin{array} {c} {\cancel{9}} \\ {^3} \end{array}} = \dfrac{1 \cdot 2}{1 \cdot 3} = \dfrac{2}{3}\)
Thus, \(\dfrac{3}{4}\) of \(\dfrac{8}{9}\) is \(\dfrac{2}{3}\).
\(M = \dfrac{3}{\begin{array} {c} {\cancel{4}} \\ {^1} \end{array}} \cdot \dfrac{\begin{array} {c} {^6} \\ {\cancel{24}} \end{array}}{1} = \dfrac{3 \cdot 6}{1 \cdot 1} = \dfrac{18}{1} = 18\)
Thus, 18 is \(\dfrac{3}{4}\) of 24.
Practice Set A
Find \(\dfrac{3}{8}\) of \(\dfrac{16}{15}\).
- Answer
-
\(\dfrac{2}{5}\)
Practice Set A
What number is \(\dfrac{9}{10}\) of \(\dfrac{5}{6}\)?
- Answer
-
\(\dfrac{3}{4}\)
Practice Set A
\(\dfrac{11}{16}\) of \(\dfrac{8}{33}\) is what number?
- Answer
-
\(\dfrac{1}{6}\)
Missing Factor Statements
The equation \(8 \cdot M = 32\) is a missing factor statement. We can find the value of \(M\) that makes this statement true by dividing (since we know that \(32 \div 8 = 4\).
Finding the Missing Factor
To find the missing factor in a missing factor statement, divide the product by the known factor.
missing factor = (product) \(\div\) (known factor)
Missing factor statements can be used to answer such questions as
\(\dfrac{3}{8}\) of what number is \(\dfrac{9}{4}\)?
What part of \(1 \dfrac{2}{7}\) is \(1 \dfrac{13}{14}\)?
Sample Set B
Now, using
missing factor = (product) \(\div\) (known factor)
We get
\(\begin{array} {rcl} {M = \dfrac{9}{4} \div \dfrac{3}{8} = \dfrac{9}{4} \cdot \dfrac{8}{3}} & = & {\dfrac{\begin{array} {c} {^3} \\ {\cancel{9}} \end{array}}{\begin{array} {c} {\cancel{4}} \\ {^1} \end{array}} \cdot \dfrac{\begin{array} {c} {^2} \\ {\cancel{8}} \end{array}}{\begin{array} {c} {\cancel{3}} \\ {^1} \end{array}}} \\ {} & = & {\dfrac{3 \cdot 2}{1 \cdot 1}} \\ {} & = & {6} \end{array}\)
Thus, \(\dfrac{3}{8}\) of 6 is \(\dfrac{9}{4}\).
For convenience, let's convert the mixed numbers to improper fractions.
\(M \cdot \dfrac{9}{7} = \dfrac{27}{14}\)
Now, using
missing factor = (product) \(\div\) (known factor)
we get
\(\begin{array} {rcl} {M = \dfrac{27}{14} \div \dfrac{9}{7} = \dfrac{27}{14} \cdot \dfrac{7}{9}} & = & {\dfrac{\begin{array} {c} {^3} \\ {\cancel{27}} \end{array}}{\begin{array} {c} {\cancel{14}} \\ {^2} \end{array}} \cdot \dfrac{\begin{array} {c} {^1} \\ {\cancel{7}} \end{array}}{\begin{array} {c} {\cancel{9}} \\ {^1} \end{array}}} \\ {} & = & {\dfrac{3 \cdot 1}{2 \cdot 1}} \\ {} & = & {\dfrac{3}{2}} \end{array}\)
Thus, \(\dfrac{3}{2}\) of \(1 \dfrac{2}{7}\) is \(1 \dfrac{13}{14}\).
Practice Set B
\(\dfrac{3}{5}\) of what number is \(\dfrac{9}{20}\)?
- Answer
-
\(\dfrac{3}{4}\)
Practice Set B
\(3 \dfrac{3}{4}\) of what number is \(2 \dfrac{2}{9}\)?
- Answer
-
\(\dfrac{16}{27}\)
Practice Set B
What part of \(\dfrac{3}{5}\) is \(\dfrac{9}{10}\)?
- Answer
-
\(1 \dfrac{1}{2}\)
Practice Set B
What part of \(1 \dfrac{1}{4}\) is \(1 \dfrac{7}{8}\)?
- Answer
-
\(1 \dfrac{1}{2}\)
Exercises
Exercise \(\PageIndex{1}\)
Find \(\dfrac{2}{3}\) of \(\dfrac{3}{4}\).
- Answer
-
\(\dfrac{1}{2}\)
Exercise \(\PageIndex{2}\)
Find \(\dfrac{5}{8}\) of \(\dfrac{1}{10}\).
Exercise \(\PageIndex{3}\)
Find \(\dfrac{12}{13}\) of \(\dfrac{13}{36}\).
- Answer
-
\(\dfrac{1}{3}\)
Exercise \(\PageIndex{4}\)
Find \(\dfrac{1}{4}\) of \(\dfrac{4}{7}\).
Exercise \(\PageIndex{5}\)
\(\dfrac{3}{10}\) of \(\dfrac{15}{4}\) is what number?
- Answer
-
\(\dfrac{9}{8}\) or \(1 \dfrac{1}{8}\)
Exercise \(\PageIndex{6}\)
\(\dfrac{14}{15}\) of \(\dfrac{20}{21}\) is what number?
Exercise \(\PageIndex{7}\)
\(\dfrac{3}{44}\) of \(\dfrac{11}{12}\) is what number?
- Answer
-
\(\dfrac{1}{16}\)
Exercise \(\PageIndex{8}\)
\(\dfrac{1}{3}\) of 2 is what number?
Exercise \(\PageIndex{9}\)
\(\dfrac{1}{4}\) of 3 is what number?
- Answer
-
\(\dfrac{3}{4}\)
Exercise \(\PageIndex{10}\)
\(\dfrac{1}{10}\) of \(\dfrac{1}{100}\) is what number?
Exercise \(\PageIndex{11}\)
\(\dfrac{1}{100}\) of \(\dfrac{1}{10}\) is what number?
- Answer
-
\(\dfrac{1}{1,000}\)
Exercise \(\PageIndex{12}\)
\(1 \dfrac{5}{9}\) of \(2 \dfrac{4}{7}\) is what number?
Exercise \(\PageIndex{13}\)
\(1 \dfrac{7}{18}\) of \(\dfrac{4}{15}\) is what number?
- Answer
-
\(\dfrac{10}{27}\)
Exercise \(\PageIndex{14}\)
\(1 \dfrac{1}{8}\) of \(1 \dfrac{11}{16}\) is what number?
Exercise \(\PageIndex{15}\)
Find \(\dfrac{2}{3}\) of \(\dfrac{1}{6}\) of \(\dfrac{9}{2}\).
- Answer
-
\(\dfrac{1}{2}\)
Exercise \(\PageIndex{16}\)
Find \(\dfrac{5}{8}\) of \(\dfrac{9}{20}\) of \(\dfrac{4}{9}\).
Exercise \(\PageIndex{17}\)
\(\dfrac{5}{12}\) of what number is \(\dfrac{5}{6}\)?
- Answer
-
2
Exercise \(\PageIndex{18}\)
\(\dfrac{3}{14}\) of what number is \(\dfrac{6}{7}\)?
Exercise \(\PageIndex{19}\)
\(\dfrac{10}{3}\) of what number is \(\dfrac{5}{9}\)?
- Answer
-
\(\dfrac{1}{6}\)
Exercise \(\PageIndex{20}\)
\(\dfrac{15}{7}\) of what number is \(\dfrac{20}{21}\)?
Exercise \(\PageIndex{21}\)
\(\dfrac{8}{3}\) of what number is \(1 \dfrac{7}{9}\)?
- Answer
-
\(\dfrac{2}{3}\)
Exercise \(\PageIndex{22}\)
\(\dfrac{1}{3}\) of what number is \(\dfrac{1}{3}\)?
Exercise \(\PageIndex{23}\)
\(\dfrac{1}{6}\) of what number is \(\dfrac{1}{6}\)?
- Answer
-
1
Exercise \(\PageIndex{24}\)
\(\dfrac{3}{4}\) of what number is \(\dfrac{3}{4}\)?
Exercise \(\PageIndex{25}\)
\(\dfrac{8}{11}\) of what number is \(\dfrac{8}{11}\)?
- Answer
-
1
Exercise \(\PageIndex{26}\)
\(\dfrac{3}{8}\) of what number is 0?
Exercise \(\PageIndex{27}\)
\(\dfrac{2}{3}\) of what number is 1?
- Answer
-
\(\dfrac{3}{2}\) or \(1 \dfrac{1}{2}\)
Exercise \(\PageIndex{28}\)
\(3 \dfrac{1}{5}\) of what number is 1?
Exercise \(\PageIndex{29}\)
\(1 \dfrac{9}{12}\) of what number is \(5 \dfrac{1}{4}\)?
- Answer
-
3
Exercise \(\PageIndex{30}\)
\(3 \dfrac{1}{25}\) of what number is \(2 \dfrac{8}{15}\)?
Exercise \(\PageIndex{31}\)
What part of \(\dfrac{2}{3}\) is \(1 \dfrac{1}{9}\)?
- Answer
-
\(\dfrac{5}{3}\) or \(1 \dfrac{2}{3}\)
Exercise \(\PageIndex{32}\)
What part of \(\dfrac{9}{10}\) is \(3 \dfrac{3}{5}\)?
Exercise \(\PageIndex{33}\)
What part of \(\dfrac{8}{9}\) is \(\dfrac{3}{5}\)?
- Answer
-
\(\dfrac{27}{40}\)
Exercise \(\PageIndex{34}\)
What part of \(\dfrac{14}{15}\) is \(\dfrac{7}{30}\)?
Exercise \(\PageIndex{35}\)
What part of 3 is \(\dfrac{1}{5}\)?
- Answer
-
\(\dfrac{1}{15}\)
Exercise \(\PageIndex{36}\)
What part of 8 is \(\dfrac{2}{3}\)?
Exercise \(\PageIndex{37}\)
What part of 24 is 9?
- Answer
-
\(\dfrac{3}{8}\)
Exercise \(\PageIndex{38}\)
What part of 42 is 26?
Exercise \(\PageIndex{39}\)
Find \(\dfrac{12}{13}\) of \(\dfrac{39}{40}\).
- Answer
-
\(\dfrac{9}{10}\)
Exercise \(\PageIndex{40}\)
\(\dfrac{14}{15}\) of \(\dfrac{12}{21}\) is what number?
Exercise \(\PageIndex{41}\)
\(\dfrac{8}{15}\) of what number is \(2 \dfrac{2}{5}\)?
- Answer
-
\(\dfrac{9}{2} = 4 \dfrac{1}{2}\)
Exercise \(\PageIndex{42}\)
\(\dfrac{11}{15}\) of what number is \(\dfrac{22}{35}\)?
Exercise \(\PageIndex{43}\)
\(\dfrac{11}{16}\) of what number is 1?
- Answer
-
\(\dfrac{16}{11}\) or \(1 \dfrac{5}{11}\)
Exercise \(\PageIndex{44}\)
What part of \(\dfrac{23}{40}\) is \(3 \dfrac{9}{20}\)?
Exercise \(\PageIndex{45}\)
\(\dfrac{4}{35}\) of \(3 \dfrac{9}{22}\) is what number?
- Answer
-
\(\dfrac{30}{77}\)
Exercises for Review
Exercise \(\PageIndex{46}\)
Use the numbers 2 and 7 to illustrate the commutative property of addition.
Exercise \(\PageIndex{47}\)
Is 4 divisible by 0?
- Answer
-
no
Exercise \(\PageIndex{48}\)
Expand \(3^7\). Do not find the actual value.
Exercise \(\PageIndex{49}\)
Convert \(3 \dfrac{5}{12}\) to an improper fraction.
- Answer
-
\(\dfrac{41}{12}\)
Exercise \(\PageIndex{50}\)
Find the value of \(\dfrac{3}{8}\) \div \dfrac{9}{16} \cdot \dfrac{6}{5}\).