Test yourself on multiplication of integers by a fraction and a mixed number!
\( 6\times\frac{3}{4}= \)
Incorrect
Correct Answer:
\( 4\frac{1}{2} \)
Practice more now
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
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
\( 2\times\frac{5}{7}= \)
Incorrect
Correct Answer:
\( 1\frac{3}{7} \)
Question 2
\( 4\times\frac{2}{3}= \)
Incorrect
Correct Answer:
\( 2\frac{2}{3} \)
Question 3
\( 3\times\frac{1}{2}= \)
Incorrect
Correct Answer:
\( 1\frac{1}{2} \)
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 1.
For example
Convert 3 into an imaginary fraction.
Solution:
We will write 3 in the numerator and 1 in the denominator.
We obtain: 13β
We will do this for any given number.
Examples:
172β=72
11β=1
15β=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 432β into an equivalent fraction.
Solution:
We will multiply the whole number 4 by the denominator 3 and then add 2.
We obtain: 4Γ3+2=14
14 will be the new numerator.
The denominator will remain the same: 3.
We obtain that:
432β=314β
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:
\( 7\times\frac{3}{8}= \)
Incorrect
Correct Answer:
\( 2\frac{5}{8} \)
Question 2
\( 8\times\frac{1}{2}= \)
Incorrect
Correct Answer:
\( 4 \)
Question 3
\( 3\times\frac{6}{7}= \)
Incorrect
Correct Answer:
\( 2\frac{4}{7} \)
Multiplication Practice with Integers, Fractions, and Mixed Numbers
Here is an exercise
4Γ31βΓ521β
Solution:
The first step
Convert whole numbers and mixed numbers into equivalent fractions and rewrite the exercise:
14β=4
521β=211β
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:
31βΓ211βΓ14β=
Now let's go to the second step:
Multiply both the numerators and the denominators separately.
We obtain:
3Γ2Γ11Γ11Γ4β=644β
We can convert the result we obtained 644β into a mixed number 262β.
Another exercise
792βΓ2Γ52β=
Solution:
First, we convert all whole and mixed numbers into equivalent fractions. We obtain:
792β=27Γ9+2β=965β
12β=2
Rewrite the exercise only with equivalent fractions:
965βΓ12βΓ52β=
Multiply both the numerators and the denominators separately and you get:
965βΓ12βΓ52β=45260β
We can convert the final result we obtained 45260β into a mixed number 54535β
Note: we can reduce the fraction even further and obtain: 54535β=5ββ97β
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 5.
Now remains the fraction 4535β which is identical in value to the fraction 97β
And therefore 45535β and 957β are identical in value.
Check your understanding
Question 1
\( 7\times\frac{2}{5}= \)
Incorrect
Correct Answer:
\( 2\frac{4}{5} \)
Question 2
\( 3\times1\frac{1}{2}= \)
Incorrect
Correct Answer:
\( 4\frac{1}{2} \)
Question 3
\( 8\times\frac{5}{9}= \)
Incorrect
Correct Answer:
\( 4\frac{4}{9} \)
Additional study
How to convert an equivalent fraction into a mixed number?
We will learn with the example
Convert the equivalent fraction 62241β 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=3β¦β¦.
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:
241β(3Γ62)=55
The result we receive will be written in the numerator.
The denominator will remain the same.
We obtain: 36255β
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
6Γ43β=
Video Solution
Step-by-Step Solution
To solve the problem 6Γ43β, we follow these steps:
Step 1: Express the integer 6 as a fraction: 16β.
Step 2: Multiply the fractions: 16βΓ43β.
Step 3: Multiply the numerators: 6Γ3=18.
Step 4: Multiply the denominators: 1Γ4=4.
Step 5: Form the resulting fraction: 418β.
Step 6: Simplify the fraction by dividing numerator and denominator by their greatest common divisor, which is 2: 418βΓ·22β=29β.
Step 7: Convert 29β to a mixed number: Divide 9 by 2 to get 4 with a remainder of 1, thus 29β=421β.
Therefore, the solution to the problem is 421β.
Answer
421β
Exercise #2
2Γ75β=
Video Solution
Step-by-Step Solution
To solve this problem, we'll multiply the whole number 2 by the fraction 75β:
Step 1: Multiply the numerator 5 by the whole number 2:
2Γ5=10
Step 2: Write the result over the original denominator 7:
710β
Step 3: Convert 710β to a mixed number:
Since 10 divided by 7 is 1 with a remainder of 3, we can express this as:
173β
Therefore, the solution to the problem is 173β.
Answer
173β
Exercise #3
4Γ32β=
Video Solution
Step-by-Step Solution
To solve this problem, we'll multiply the whole number 4 by the fraction 32β as follows:
Step 1: Convert the whole number 4 into a fraction. This can be written as 14β.
Step 2: Use fraction multiplication rules: multiply the numerators together and the denominators together.
Step 3: So, multiply the numerators: 4Γ2=8.
Step 4: Multiply the denominators: 1Γ3=3.
Step 5: The result is 38β.
Step 6: Since 38β is an improper fraction, convert it to a mixed number. 8Γ·3=2 with a remainder of 2.
Thus, 38β=232β.
Therefore, the solution to the problem is 232β.
Answer
232β
Exercise #4
3Γ21β=
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 21β, which is 1, by 3: 1Γ3=3.
Step 2: Write the result over the original denominator: 23β.
Step 3: Convert the improper fraction 23β to a mixed number:
Divide 3 by 2. This gives 1 as the quotient and 1 as the remainder, so: 23β=121β.
Therefore, the solution to the problem is 121β.
Answer
121β
Exercise #5
Solve:
7Γ83β=
Video Solution
Step-by-Step Solution
To solve this problem, we will start by multiplying the whole number 7 by the fraction 83β using the rule for multiplying a whole number by a fraction.
Calculate the product:
7Γ83β=87Γ3β
=821β
The fraction 821β 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: 285β
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 285β.
Answer
285β
Do you think you will be able to solve it?
Question 1
\( 3\times\frac{8}{12}= \)
Incorrect
Correct Answer:
\( 2 \)
Question 2
\( 10\times\frac{7}{9}= \)
Incorrect
Correct Answer:
\( 7\frac{7}{9} \)
Question 3
\( 7\times\frac{6}{8}= \)
Incorrect
Correct Answer:
\( 5\frac{1}{4} \)
Start practice
More Questions
Multiplication of Integers by a Fraction and a Mixed number