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

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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

6Γ—34= 6\times\frac{3}{4}=

Video Solution

Step-by-Step Solution

To solve the problem 6Γ—346 \times \frac{3}{4}, we follow these steps:

  • Step 1: Express the integer 6 as a fraction: 61 \frac{6}{1} .
  • Step 2: Multiply the fractions: 61Γ—34\frac{6}{1} \times \frac{3}{4} .
  • Step 3: Multiply the numerators: 6Γ—3=186 \times 3 = 18.
  • Step 4: Multiply the denominators: 1Γ—4=41 \times 4 = 4.
  • Step 5: Form the resulting fraction: 184\frac{18}{4}.
  • Step 6: Simplify the fraction by dividing numerator and denominator by their greatest common divisor, which is 2: 184Γ·22=92\frac{18}{4} \div \frac{2}{2} = \frac{9}{2}.
  • Step 7: Convert 92\frac{9}{2} to a mixed number: Divide 9 by 2 to get 4 with a remainder of 1, thus 92=412\frac{9}{2} = 4\frac{1}{2}.

Therefore, the solution to the problem is 412 4\frac{1}{2} .

Answer

412 4\frac{1}{2}

Exercise #2

2Γ—57= 2\times\frac{5}{7}=

Video Solution

Step-by-Step Solution

To solve this problem, we'll multiply the whole number 2 by the fraction 57\frac{5}{7}:

  • Step 1: Multiply the numerator 5 by the whole number 2:

2Γ—5=10 2 \times 5 = 10

  • Step 2: Write the result over the original denominator 7:

107 \frac{10}{7}

  • Step 3: Convert 107\frac{10}{7} to a mixed number:

Since 10 divided by 7 is 1 with a remainder of 3, we can express this as:

137 1\frac{3}{7}

Therefore, the solution to the problem is 137\textbf{1}\frac{\textbf{3}}{\textbf{7}}.

Answer

137 1\frac{3}{7}

Exercise #3

4Γ—23= 4\times\frac{2}{3}=

Video Solution

Step-by-Step Solution

To solve this problem, we'll multiply the whole number 4 by the fraction 23 \frac{2}{3} as follows:

  • Step 1: Convert the whole number 4 into a fraction. This can be written as 41 \frac{4}{1} .
  • Step 2: Use fraction multiplication rules: multiply the numerators together and the denominators together.
  • Step 3: So, multiply the numerators: 4Γ—2=8 4 \times 2 = 8 .
  • Step 4: Multiply the denominators: 1Γ—3=3 1 \times 3 = 3 .
  • Step 5: The result is 83 \frac{8}{3} .
  • Step 6: Since 83 \frac{8}{3} is an improper fraction, convert it to a mixed number.
        8Γ·3=2 8 \div 3 = 2 with a remainder of 2.
        Thus, 83=223 \frac{8}{3} = 2\frac{2}{3} .

Therefore, the solution to the problem is 223 2\frac{2}{3} .

Answer

223 2\frac{2}{3}

Exercise #4

3Γ—12= 3\times\frac{1}{2}=

Video Solution

Step-by-Step Solution

To solve this problem, we'll follow these steps:

  • Multiply the numerator of the fraction by the integer.
  • Keep the denominator unchanged.
  • Convert the resulting improper fraction to a mixed number, if necessary.

Now, let's work through each step:
Step 1: Multiply the numerator of 12 \frac{1}{2} , which is 1 1 , by 3 3 :
1Γ—3=3 1 \times 3 = 3 .

Step 2: Write the result over the original denominator:
32 \frac{3}{2} .

Step 3: Convert the improper fraction 32 \frac{3}{2} to a mixed number:
Divide 3 3 by 2 2 . This gives 1 1 as the quotient and 1 1 as the remainder, so:
32=112 \frac{3}{2} = 1\frac{1}{2} .

Therefore, the solution to the problem is 112 1\frac{1}{2} .

Answer

112 1\frac{1}{2}

Exercise #5

Solve:

7Γ—38= 7\times\frac{3}{8}=

Video Solution

Step-by-Step Solution

To solve this problem, we will start by multiplying the whole number 7 by the fraction 38 \frac{3}{8} using the rule for multiplying a whole number by a fraction.

Calculate the product:

  • 7Γ—38=7Γ—38 7 \times \frac{3}{8} = \frac{7 \times 3}{8}
  • =218 = \frac{21}{8}

The fraction 218 \frac{21}{8} is an improper fraction, meaning the numerator is greater than the denominator. To convert it to a mixed number, we divide 21 by 8:

  • 21 divided by 8 equals 2 with a remainder of 5.
  • This gives us the mixed number: 258 2\frac{5}{8}

The remainder becomes the numerator of the fraction part, and the denominator remains the same as in the original fraction.

Therefore, the solution to the problem is 258 2\frac{5}{8} .

Answer

258 2\frac{5}{8}

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