3E Formulas with Multiplying and Dividing Mixed Numbers

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A recipe calls for $5 and 3 fourths$ cups of flour, but Julie wants to make only half of the recipe. We learned in the last section that to take a fraction of a number, we need to multiply the fraction by the number. So it stands to reason that Julie will need to multiply 1/2 times the mixed number $5 and 3 fourths$ . We see that to figure out how much flour Julie needs, we will need to review multiplying and dividing mixed numbers.

To multiply or divide mixed numbers, we need to first convert them to improper fractions. Multiply or divide the fractions, simplify, and then if needed convert your answer back into a mixed number.

Look at the problem $5 and 5 eighths divided by 6 and 2 thirds$ . We first convert the mixed numbers into fractions. For these positive mixed numbers, remember we multiply the whole part times the denominator and then add the numerator. So $5 \frac{5}{8} = \frac{5 \times 8 + 5}{8} = \frac{45}{8}$ and $6 \frac{2}{3} = \frac{6 \times 3 + 2}{3} = \frac{20}{3}$ . Hence $5 \frac{5}{8} \div 6 \frac{2}{3} = \frac{45}{8} \div \frac{20}{3}$ .

Now we multiply by the reciprocal of the divisor and simplify. $\frac{45}{8} \div \frac{20}{3} = \frac{45}{8} \times \frac{3}{20} = \frac{5^{1} \times 9 \times 3}{8 \times 5 \times 4} = \frac{27}{32}$ .

Try the following examples with your instructor.

Practice Problems Section 3E

Multiply or divide the following mixed numbers. Write answers as a whole number, a simplified proper fraction, or a mixed number.

$6 and 1 eighth times 2 and 2 sevenths$

Check answer

14
$3 and 1 sixth divided by 6 and 1 third$
$10 \frac{1}{2} \times 3 \frac{5}{14}$

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$35 and 1 fourth$
$8 and 2 fifths divided by 2 and 1 tenth$
$6 and 1 fourth divided by 9 and 3 eighths$

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$\frac{2}{3}$
$1 and 5 sixteenths divided by 4 and 1 twelfth$
$10 and 2 fifths times 3 and 1 tenth$

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$32 \frac{6}{25}$
$7 \frac{3}{14} \div 1 \frac{1}{21}$
$8 \frac{1}{15} \times 2 \frac{2}{9}$

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$17 \frac{25}{27}$
$2 and 5 eighths times 7 and 1 third$
$1 and 7 ninths divided by 4 and 2 thirds$

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$\frac{8}{21}$
$7 and 1 fifth times 6 and 1 ninth$
$3 and 3 fifths divided by 2 and 1 tenth$

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$1 and 5 sevenths$
$5 and 5 sixths divided by 12 and 2 thirds$
$3 \frac{3}{4} \div 5 \frac{1}{15}$

Check answer

$\frac{225}{304}$
$4 and 3 eighths times 2 and 2 sevenths$
$9 and 1 sixth times 11 and 2 fifths$

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$104 and 1 half$
$7 \frac{3}{5} \times 2 \frac{2}{19}$
$3 \frac{1}{4} \times 1 \frac{3}{13}$

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4
$5 and 2 elevenths divided by 4 and 3 fourths$
$11 and 1 fourth divided by 6 and 2 thirds$

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$1 and 11 sixteenths$
$6 and 1 eighth times 12 and 4 sevenths$
$6 and 1 ninth divided by 7 and 1 third$

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$\frac{5}{6}$
$9 and 5 eighths divided by 2 and 3 fourths$
$2 and 5 eighths times 13 and 1 third$

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35
$3 and 1 twelfth divided by 5 and 1 sixth$
$10 and 1 eighth divided by 18$

Check answer

$\frac{9}{16}$

Are you ready to help Julie? Remember Julie needs to figure out half of the amounts in her recipe. Multiply all of the recipe amounts by 1/2.

Help Julie find $\frac{1}{2}$ of $5 and 3 fourths$ cups of flour.
Help Julie find $\frac{1}{2}$ of $2 and 1 fourth$ cups of sugar.

Check answer

$1 and 1 eighth$ cup sugar
Help Julie find $\frac{1}{2}$ of $3 and 1 fourth$ teaspoons of nutmeg.

Tyra owns a cupcake shop. Tyra has a great recipe for double chocolate cupcakes. Tyra needs to make 4 times the number of cupcakes that the recipe makes. Her cupcake shop is busy. Help Tyra by multiplying all of the recipe amounts by 4.

Help Tyra by finding 4 times $2 and 2 thirds$ cups of sugar.

Check answer

$10 and 2 thirds$ cups sugar
Help Tyra by finding 4 times $4 and 1 fifth$ cups of flour.
Help Tyra by finding 4 times $1 and 3 fourths$ cups of cocoa powder.

Check answer

7 cups cocoa
The distance around a circle is called the circumference and is given by the formula $C equals 2 π r$ . Juan raises horses and is building a fence around a circular region on his property with a radius of $30 and 5 eighths$ meters. Find the distance around the circle. Use the approximation $π \approx \frac{22}{7}$ . Write your answer as a mixed number.
The volume of a square based pyramid is given by the formula $V = \frac{1}{3} s^{2} h$ where s is the side of the base and h is the height of the pyramid. Find the volume of a pyramid if one of the sides is 100 feet long and the pyramid is 36 feet tall.

Check answer

120,000 cu ft

Adapted from Preparing for Algebra and Statistics, Third Edition, by Matt Teachout. License: CC BY 4.0. Copyright and sources.