How to Divide an Integer by Fraction

When you divide an integer by a fraction, you will find how many groups of fractions match that integer. The standard way to divide an integer by a fraction is to multiply the integer by the reciprocal of the given fraction. You can also draw diagrams to help you visualize the process.

Table of Contents

Multiply by the inverse

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Convert integers to fractions. To do this, put the integer in the numerator position of the fraction. The denominator will be 1. ^{[1] X Research Source}

For example, if you calculate $7$ $\div$ $3$ $4$ ${displaystyle 7div {frac {3}{4}}}$ ${displaystyle 7div {frac {3}{4}}}$ , first you need to change $7$ ${displaystyle 7}$ $7$ wall $7$ $first$ ${displaystyle {frac {7}{1}}}$ ${displaystyle {frac {7}{1}}}$ .

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Find the reciprocal of the divisor. The reciprocal of a number is the number that is reversed. To find the reciprocal of a fraction, invert the numerator and denominator. ^{[2] X Research Source}

For example, the reciprocal of the fraction $3$ $4$ ${displaystyle {frac {3}{4}}}$ ${frac {3}{4}}$ To be $4$ $3$ ${displaystyle {frac {4}{3}}}$ ${displaystyle {frac {4}{3}}}$ .

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Multiply two fractions. To multiply fractions, first multiply the numerators together. Then multiply the denominators together. The product of two fractions is equal to the quotient of the original division. ^{[3] X Research Sources}

For example, $7$ $first$ $\times$ $4$ $3$ $=$ $28$ $3$ ${displaystyle {frac {7}{1}}times {frac {4}{3}}={frac {28}{3}}}$ ${displaystyle {frac {7}{1}}times {frac {4}{3}}={frac {28}{3}}}$

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Shorten if necessary. If you have a fraction greater than 1 (a fraction with a larger numerator than the denominator), your teacher may ask you to change it to a mixed number. Often your teacher will ask you to reduce fractions to a minimum.

For example, $28$ $3$ ${displaystyle {frac {28}{3}}}$ ${displaystyle {frac {28}{3}}}$ shorten to mixed numbers $9$ $first$ $3$ ${displaystyle 9{frac {1}{3}}}$ ${displaystyle 9{frac {1}{3}}}$ .

Draw map

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Draw an integer representation. The drawing must be a shape that can be divided into equal groups, such as a square or a circle. Draw the shapes large enough that you can divide them into smaller parts.

For example, if you have the calculation $5$ $\div$ $3$ $4$ ${displaystyle 5div {frac {3}{4}}}$ ${displaystyle 5div {frac {3}{4}}}$ , you need to draw 5 circles.

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Divide each figure by the denominator of the fraction. The denominator of a fraction tells you how many parts an integer is divided into. Divide the form into equal proportions. ^{[4] X Research Sources}

For example, if you do the math $3$ $4$ ${displaystyle {frac {3}{4}}}$ ${displaystyle {frac {3}{4}}}$ , the number 4 in the denominator indicates that the figure is divided into four parts.

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Color the group representing the sections. Since you’re converting integers to fractions, you want to know how many groups of fractions are in the integer. ^{[5] X Research Resources} So first you need to create groups. Coloring each group a different color will help, as some groups will have parts at two different integers. Do not color the rest.

For example, if you divide 5 by $3$ $4$ ${displaystyle {frac {3}{4}}}$ ${frac {3}{4}}$ , you will color 3/4 of each group a different color. Note that many groups can contain 2 quarters of one integer and 1 quarter of another.

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Count the number of groups of integers. This will help you know the integer part of the answer.

For example, you should create 6 groups $3$ $4$ ${displaystyle {frac {3}{4}}}$ ${displaystyle {frac {3}{4}}}$ out of 5 circles.

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Explain the rest. Compare the number of remains with a complete group. The fraction of the group of remaining parts will indicate the fractional part of the answer. Make sure you don’t compare the number of parts with the whole figure as doing so will give incorrect fractions.

For example, after dividing 5 form groups $3$ $4$ ${displaystyle {frac {3}{4}}}$ ${displaystyle {frac {3}{4}}}$ , you have 2 quarters or $2$ $4$ ${displaystyle {frac {2}{4}}}$ ${displaystyle {frac {2}{4}}}$ remaining. Since the group has a total of 3 parts and you have 2 parts, your fraction is $2$ $3$ ${displaystyle {frac {2}{3}}}$ ${displaystyle {frac {2}{3}}}$ .

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Write the answer. Combine groups of integers with groups of fractions to find the quotient of the original division.

For example, $5$ $\div$ $3$ $4$ $=$ $6$ $2$ $3$ ${displaystyle 5div {frac {3}{4}}=6{frac {2}{3}}}$ ${displaystyle 5div {frac {3}{4}}=6{frac {2}{3}}}$ .

Sample problem solving

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Solve the following problem: Multiply

first

2

{displaystyle {frac {1}{2}}}

{frac {1}{2}}

with how much to get

8

{displaystyle 8}

{displaystyle 8}

?

Because the problem asks how many groups $first$ $2$ ${displaystyle {frac {1}{2}}}$ ${frac {1}{2}}$ in 8, the problem is division.
Convert 8 to a fraction with a denominator of 1: $8$ $=$ $8$ $first$ ${displaystyle 8={frac {8}{1}}}$ ${displaystyle 8={frac {8}{1}}}$ .
Find the reciprocal of a fraction by reversing the numerator and denominator: $first$ $2$ ${displaystyle {frac {1}{2}}}$ ${frac {1}{2}}$ wall $2$ $first$ ${displaystyle {frac {2}{1}}}$ ${displaystyle {frac {2}{1}}}$ .
Multiply two fractions together: $8$ $first$ $\times$ $2$ $first$ $=$ $16$ $first$ ${displaystyle {frac {8}{1}}times {frac {2}{1}}={frac {16}{1}}}$ ${displaystyle {frac {8}{1}}times {frac {2}{1}}={frac {16}{1}}}$ .
Shorten if needed: $16$ $first$ $=$ $16$ ${displaystyle {frac {16}{1}}=16}$ ${displaystyle {frac {16}{1}}=16}$ .

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Solve the following problem:

16

\div

5

8

{displaystyle 16div {frac {5}{8}}}

{displaystyle 16div {frac {5}{8}}}

.

Convert 16 to a fraction with a denominator of 1: $16$ $=$ $16$ $first$ ${displaystyle 16={frac {16}{1}}}$ ${displaystyle 16={frac {16}{1}}}$ .
Find the reciprocal of a fraction by reversing the numerator and denominator: $5$ $8$ ${displaystyle {frac {5}{8}}}$ ${displaystyle {frac {5}{8}}}$ wall $8$ $5$ ${displaystyle {frac {8}{5}}}$ ${displaystyle {frac {8}{5}}}$ .
Multiply two fractions together: $16$ $first$ $\times$ $8$ $5$ $=$ $128$ $5$ ${displaystyle {frac {16}{1}}times {frac {8}{5}}={frac {128}{5}}}$ ${displaystyle {frac {16}{1}}times {frac {8}{5}}={frac {128}{5}}}$ .
Shorten if needed: $128$ $5$ $=$ $25$ $3$ $5$ ${displaystyle {frac {128}{5}}=25{frac {3}{5}}}$ ${displaystyle {frac {128}{5}}=25{frac {3}{5}}}$ .

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Solve the following problem by drawing a diagram. Rufus has 9 boxes of food. Every day, she eats it all

2

3

{displaystyle {frac {2}{3}}}

{displaystyle {frac {2}{3}}}

box. How many days can the food last?

Draw 9 circles representing 9 food boxes.
Because every time she eats it all $2$ $3$ ${displaystyle {frac {2}{3}}}$ ${displaystyle {frac {2}{3}}}$ box, divide the circle into three parts.
Color the group of fractions $2$ $3$ ${displaystyle {frac {2}{3}}}$ ${displaystyle {frac {2}{3}}}$ .
Counting the number of groups that have been shaded, we get 13.
Explain the rest. There is 1 remaining part, which is $first$ $3$ ${displaystyle {frac {1}{3}}}$ ${frac {1}{3}}$ . Because the group is $2$ $3$ ${displaystyle {frac {2}{3}}}$ ${displaystyle {frac {2}{3}}}$ , you have half of the group left. So the fraction will be $first$ $2$ ${displaystyle {frac {1}{2}}}$ ${frac {1}{2}}$ .
Calculating the sum of groups of integers and groups of fractions, we have the answer: $9$ $\div$ $2$ $3$ $=$ $13$ $first$ $2$ ${displaystyle 9div {frac {2}{3}}=13{frac {1}{2}}}$ ${displaystyle 9div {frac {2}{3}}=13{frac {1}{2}}}$ .

Steps

Multiply by the inverse

Draw map

Sample problem solving

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