Multiplication of Integers by a Fraction and a Mixed Number

Multiplication of Integers by a Fraction and a Mixed number - Examples, Exercises and Solutions
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Multiplication of Integers by a Fraction and a Mixed number
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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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Test your knowledge
Question 1

\( 3\times\frac{1}{2}= \)

Question 2

\( 4\times\frac{2}{3}= \)

Question 3

\( 6\times\frac{3}{4}= \)

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: 4×3+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?
Question 1

Solve:

\( 7\times\frac{3}{8}= \)

Question 2

\( 3\times\frac{6}{7}= \)

Question 3

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

Multiplication Practice with Integers, Fractions, and Mixed Numbers

Here is an exercise

4×13×5124\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

13×112×41= \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:

1×11×43×2×1=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

729×2×25=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=7×9+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:

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

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

659×21×25=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
Question 1

\( 8\times\frac{1}{2}= \)

Question 2

\( 10\times\frac{7}{9}= \)

Question 3

\( 2\times2\frac{1}{2}= \)

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

241−(3×62)=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

3×214= 3\times2\frac{1}{4}=

Video Solution

Step-by-Step Solution

We will use the distributive property of multiplication and separate the fraction into an addition exercise between fractions. This allows us to work with smaller numbers and simplify the operation

Reminder - The distributive property of multiplication allows us to break down the larger term in the multiplication exercise into a sum or difference of smaller numbers, which makes the multiplication operation easier and gives us the ability to solve the exercise without a calculator

3×(2+14)= 3\times(2+\frac{1}{4})=

We will use the distributive property formula a(b+c)=ab+ac a(b+c)=ab+ac

(3×2)−(3×14)= (3\times2)-(3\times\frac{1}{4})=

Let's solve what's in the left parentheses:

3×2=6 3\times2=6

Let's solve what's in the right parentheses:

3=31 3=\frac{3}{1}

31×14=3×11×4=34 \frac{3}{1}\times\frac{1}{4}=\frac{3\times1}{1\times4}=\frac{3}{4}

And we get the exercise:

6+34=634 6+\frac{3}{4}=6\frac{3}{4}

And now let's see the solution centered:

3×214=3×(2+14)=(3×2)+(3×14)=6+34=634 3\times2\frac{1}{4}=3\times(2+\frac{1}{4})=(3\times2)+(3\times\frac{1}{4})=6+\frac{3}{4}=6\frac{3}{4}

Answer

634 6\frac{3}{4}

Exercise #2

5×313= 5\times3\frac{1}{3}=

Video Solution

Step-by-Step Solution

We will use the distributive property of multiplication and separate the fraction into an addition exercise between fractions. This allows us to work with smaller numbers and simplify the operation

Reminder - The distributive property of multiplication actually allows us to separate the larger term in the multiplication exercise into a sum or difference of smaller numbers, which makes the multiplication operation easier and gives us the ability to solve the exercise without a calculator

5×(3+13)= 5\times(3+\frac{1}{3})=

We will use the distributive property formula a(b+c)=ab+ac a(b+c)=ab+ac

(5×3)+(5×13)= (5\times3)+(5\times\frac{1}{3})=

Let's solve what's in the left parentheses:

5×3=15 5\times3=15

Let's solve what's in the right parentheses:

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

51×13=5×11×3=53 \frac{5}{1}\times\frac{1}{3}=\frac{5\times1}{1\times3}=\frac{5}{3}

And we get the exercise:

15+53=15+123=1623 15+\frac{5}{3}=15+1\frac{2}{3}=16\frac{2}{3}

And now let's see the solution centered:

5×313=5×(3+13)=(5×3)+(5×13)=15+53=15+123=1623 5\times3\frac{1}{3}=5\times(3+\frac{1}{3})=(5\times3)+(5\times\frac{1}{3})=15+\frac{5}{3}=15+1\frac{2}{3}=16\frac{2}{3}

Answer

1623 16\frac{2}{3}

Exercise #3

9×389= 9\times3\frac{8}{9}=

Video Solution

Step-by-Step Solution

We will use the distributive property of multiplication and break down the fraction into a subtraction exercise between a whole number and a fraction. This allows us to work with smaller numbers and simplify the operation

Reminder - The distributive property of multiplication allows us to break down the larger term in a multiplication problem into a sum or difference of smaller numbers, which makes multiplication easier and gives us the ability to solve the problem even without a calculator

9×(4−19)= 9\times(4-\frac{1}{9})=

We will use the distributive property formula a(b+c)=ab+ac a(b+c)=ab+ac

(9×4)−(9×19)= (9\times4)-(9\times\frac{1}{9})=

Let's solve what's in the left parentheses:

9×4=36 9\times4=36

Note that in the right parentheses we can reduce 9 by 9 as follows:

9=91 9=\frac{9}{1}

91×19=9×11×9=99=11=1 \frac{9}{1}\times\frac{1}{9}=\frac{9\times1}{1\times9}=\frac{9}{9}=\frac{1}{1}=1

And we get the exercise:

36−1=35 36-1=35

And now let's see the solution centered:

9×389=9×(4−19)=(9×4)−(9×19)=36−1=35 9\times3\frac{8}{9}=9\times(4-\frac{1}{9})=(9\times4)-(9\times\frac{1}{9})=36-1=35

Answer

35 35

Exercise #4

5⋅(212+116+34)= 5\cdot\big(2\frac{1}{2}+1\frac{1}{6}+\frac{3}{4}\big)=

Video Solution

Step-by-Step Solution

Let's simplify this expression while following the order of operations which states that exponents come before multiplication and division, which come before addition and subtraction, and that parentheses come before all of these,

We'll start by simplifying the expression inside the parentheses.

In this expression, there are addition operations between mixed fractions, so in the first step we'll convert all mixed fractions in this expression to improper fractions.

We'll do this by multiplying the whole number by the denominator of the fraction, and adding the result to the numerator.

In the fraction's denominator (which is the divisor) - nothing will change of course.

We'll do this in the following way:

212=(2×2)+12=4+12=52 2\frac{1}{2}=\frac{(2\times2)+1}{2}=\frac{4+1}{2}=\frac{5}{2}

116=(1×6)+16=6+16=76 1\frac{1}{6}=\frac{(1\times6)+1}{6}=\frac{6+1}{6}=\frac{7}{6}

Now we'll get the exercise:

5⋅(52+76+34) 5\cdot\big(\frac{5}{2}+\frac{7}{6}+\frac{3}{4}\big)

We'll continue and perform the addition of fractions in the expression inside the parentheses.

First, we'll expand each fraction to the common denominator, which is 12 (since it is the least common multiple of all denominators in the expression), we'll do this by multiplying the numerator of the fraction by the number that answers the question: "By how much did we multiply the current denominator to get the common denominator?"

Then we'll perform the addition operations between the expanded numerators:

5⋅(52+76+34)=5⋅5⋅6+7⋅2+3⋅312=5⋅30+14+912=5⋅5312= 5\cdot\big(\frac{5}{2}+\frac{7}{6}+\frac{3}{4}\big) =\\ 5\cdot\frac{5\cdot6+7\cdot2+3\cdot3}{12} =\\ 5\cdot\frac{30+14+9}{12} =\\ 5\cdot\frac{53}{12} =\\ We performed the addition operation between the numerators above, after expanding the fractions mentioned.

Note that since multiplication comes before addition, we first performed the multiplications in the fraction's numerator and only then the addition operations,

We'll continue and simplify the expression we got in the last step, meaning - we'll perform the multiplication we got, while remembering that multiplying a fraction means multiplying the fraction's numerator.

In the next step, we'll write the result as a mixed fraction, we'll do this by finding the whole numbers (the answer to the question "How many complete times does the denominator go into the numerator?") and adding the remainder divided by the divisor:

5⋅5312=5⋅5312=26512=22112 5\cdot\frac{53}{12}=\\ \frac{5\cdot53}{12}=\\ \frac{265}{12}=\\ 22\frac{1}{12}

Let's summarize the steps of simplifying the given expression:

5⋅(212+116+34)=5⋅(52+76+34)=5⋅5⋅6+7⋅2+3⋅312=5⋅5312=22112 5\cdot\big(2\frac{1}{2}+1\frac{1}{6}+\frac{3}{4}\big)= \\ 5\cdot\big(\frac{5}{2}+\frac{7}{6}+\frac{3}{4}\big)=\\ 5\cdot\frac{5\cdot6+7\cdot2+3\cdot3}{12} =\\ 5\cdot\frac{53}{12} =\\ 22\frac{1}{12}

Therefore the correct answer is answer B.

Answer

22112 22\frac{1}{12}

Exercise #5

15×78×223= \frac{1}{5}\times\frac{7}{8}\times2\frac{2}{3}=

Video Solution

Step-by-Step Solution

First, let's convert the mixed fraction to an improper fraction as follows:

15×78×3×2+23= \frac{1}{5}\times\frac{7}{8}\times\frac{3\times2+2}{3}=

Let's solve the equation in the numerator:

15×78×6+23= \frac{1}{5}\times\frac{7}{8}\times\frac{6+2}{3}=

15×78×83= \frac{1}{5}\times\frac{7}{8}\times\frac{8}{3}=

Since the only operation in the equation is multiplication, we'll combine everything into one equation:

1×7×85×8×3= \frac{1\times7\times8}{5\times8\times3}=

Let's simplify the 8 in the numerator and denominator of the fraction:

1×75×3= \frac{1\times7}{5\times3}=

Let's solve the equations in the numerator and denominator and we get:

715 \frac{7}{15}

Answer

715 \frac{7}{15}

Do you think you will be able to solve it?
Question 1

\( 2\times2\frac{1}{3}= \)

Question 2

\( 3\times1\frac{1}{2}= \)

Question 3

\( 3\times1\frac{1}{3}= \)

Start practice

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