Multiplication of Integers by a Fraction and a Mixed Number

πŸ†Practice multiplication of integers by a fraction and a mixed number

Multiplying a whole number by a fraction and a mixed number is solved in the following steps:

The first step:

Convert each whole number and mixed number into a similar fraction and rewrite the problem.

The second stage:

Multiply the numerators and the denominators separately.

The multiplication of numerators will be written in the new numerator.

The multiplication of denominators will be written in the new denominator.

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\( 2\times\frac{5}{7}= \)

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Multiplication of integers by a fraction and a mixed number

In this article, we will learn how to multiply an integer with a fraction and a mixed number without any problem!

When it comes to multiplication exercises, there is no need to find a common denominator and all we have to do is convert the integers and mixed numbers into equivalent fractions.


Steps to Solve the Multiplication of Integers with a Fraction and a Mixed Number

The first step

Convert whole numbers and mixed numbers into equivalent fractions, with only the numerator and denominator and rewrite the exercise.

The second step

Multiply the numerators -> numerator by numerator by numerator

Multiply the denominators -> denominator by denominator by denominator

Product of the numerators -> will be the new numerator.

Product of the denominators -> will be the new denominator.


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How to convert a whole number into an equivalent fraction?

To convert an integer into an equivalent fraction, we will write the given number in the numerator and in the denominator we will write 11.

For example

Convert 33 into an imaginary fraction.

Solution:

We will write 33 in the numerator and 11 in the denominator.

We obtain: 313 \over 1

We will do this for any given number.

Examples:

721=72\frac{72}{1}=72

11=1\frac{1}{1}=1

51=5\frac{5}{1}=5


How to convert a mixed number into an equivalent fraction?

We will multiply the whole number by the denominator and add the numerator to the obtained result.

The final result we obtain will appear in the new numerator.

The denominator will remain the same.

For example

Convert the mixed number 4234 \frac{2}{3} into an equivalent fraction.

Solution:

We will multiply the whole number 44 by the denominator 33 and then add 22.

We obtain: 4Γ—3+2=144\times 3+2=14

1414 will be the new numerator.

The denominator will remain the same: 33.

We obtain that:

423=1434 \frac{2}{3}=\frac {14}{3}

Now that we know how to convert whole and mixed numbers into equivalent fractions, we can continue.


Do you know what the answer is?

Multiplication Practice with Integers, Fractions, and Mixed Numbers

Here is an exercise

4Γ—13Γ—5124\times \frac {1}{3}\times 5\frac {1}{2}

Solution:

The first step

Convert whole numbers and mixed numbers into equivalent fractions and rewrite the exercise:

41=4\frac{4}{1}=4

512=1125 \frac{1}{2}=\frac{11}{2}

Note that, since this is a multiplication exercise, we can apply the substitution property.

It doesn't matter in which position we place the equivalent fractions we received, the result will not change.

Rewrite the exercise:

1a - Convert whole numbers and mixed numbers into equivalent fractions

13Γ—112Γ—41= \frac{1}{3}\times\frac{11}{2}\times\frac{4}{1}=

Now let's go to the second step:

Multiply both the numerators and the denominators separately.

We obtain:

1Γ—11Γ—43Γ—2Γ—1=446\frac{1\times 11\times 4}{3\times 2\times 1}=\frac {44}{6}

We can convert the result we obtained 446\frac {44}{6} into a mixed number 2262 \frac {2}{6}.


Another exercise

729Γ—2Γ—25=7 \frac {2}{9}\times 2\times \frac {2}{5}=

Solution:

First, we convert all whole and mixed numbers into equivalent fractions. We obtain:

729=7Γ—9+22=6597 \frac {2}{9}=\frac {7\times 9+2}{2}=\frac {65}{9}

21=2\frac{2}{1}=2

Rewrite the exercise only with equivalent fractions:

659Γ—21Γ—25=\frac {65}{9}\times \frac {2}{1}\times \frac {2}{5}=

Multiply both the numerators and the denominators separately and you get:

659Γ—21Γ—25=26045\frac {65}{9}\times \frac {2}{1}\times \frac {2}{5}=\frac {260}{45}

We can convert the final result we obtained 26045\frac {260}{45} into a mixed number 535455 \frac {35}{45}

Note: we can reduce the fraction even further and obtain: 53545=5​​795\frac{35}{45}=5​​\frac{7}{9}


Why didn't we have to convert the fraction into an equivalent number to reduce?

Think of it this way:

The number is composed of integers and another fraction. We don't touch the integers and leave 55.

Now remains the fraction 3545\frac {35}{45} which is identical in value to the fraction 79\frac {7}{9}

And therefore 53545\frac {535}{45} and 579\frac {57}{9} are identical in value.


Check your understanding

Additional study

How to convert an equivalent fraction into a mixed number?

We will learn with the example

Convert the equivalent fraction 24162\frac {241}{62} into a mixed number.

Solution:

To convert an equivalent fraction into a mixed number, we will divide the numerator by the denominator and refer only to the whole number we receive (ignore the remainder).

241:62=3…….241:62=3…….

This will be the whole number.

Then, we subtract from the given numerator the result of multiplying the whole number by the denominator to see how much is left to "complete" it.

That is:

A1 - Multiplication of the whole number by the denominator

241βˆ’(3Γ—62)=55 241-(3\times 62)=55

The result we receive will be written in the numerator.

The denominator will remain the same.

We obtain: 355623 \frac {55}{62}

You can always test yourself and see if you return to the same equivalent fraction.


Examples and exercises with solutions for multiplying integers by a fraction and a mixed number

Exercise #1

4.1β‹…1.6β‹…3.2+4.7=? 4.1\cdot1.6\cdot3.2+4.7=\text{?}

Step-by-Step Solution

We convert decimal numbers into mixed fractions:

4110Γ—1610Γ—3210+4710= 4\frac{1}{10}\times1\frac{6}{10}\times3\frac{2}{10}+4\frac{7}{10}=

Now, we convert mixed fractions into simple fractions:

4110Γ—1610Γ—3210+4710= \frac{41}{10}\times\frac{16}{10}\times\frac{32}{10}+\frac{47}{10}=

We solve the exercise from left to right:

41Γ—1610Γ—10=656100 \frac{41\times16}{10\times10}=\frac{656}{100}

Now we get the exercise:

656100Γ—3210+4710= \frac{656}{100}\times\frac{32}{10}+\frac{47}{10}=

We solve the multiplication exercise:

656Γ—32100Γ—10=20,9921,000 \frac{656\times32}{100\times10}=\frac{20,992}{1,000}

Now we get the exercise:

20,9921,000+4710= \frac{20,992}{1,000}+\frac{47}{10}=

We multiply the fraction on the right so that its denominator is also 1000:

47Γ—10010Γ—100=4,7001,000 \frac{47\times100}{10\times100}=\frac{4,700}{1,000}

We get the exercise:

20,9921,000+4,7001,000=20,992+4,7001,000=25,6921,000 \frac{20,992}{1,000}+\frac{4,700}{1,000}=\frac{20,992+4,700}{1,000}=\frac{25,692}{1,000}

We convert the simple fraction into a decimal number:

25,6921,000=25.692 \frac{25,692}{1,000}=25.692

Answer

25.692

Exercise #2

2Γ—57= 2\times\frac{5}{7}=

Video Solution

Answer

137 1\frac{3}{7}

Exercise #3

3Γ—12= 3\times\frac{1}{2}=

Video Solution

Answer

112 1\frac{1}{2}

Exercise #4

4Γ—23= 4\times\frac{2}{3}=

Video Solution

Answer

223 2\frac{2}{3}

Exercise #5

6Γ—34= 6\times\frac{3}{4}=

Video Solution

Answer

412 4\frac{1}{2}

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