Test yourself on multiplication of integers by a fraction and a mixed number!
\( 2\times\frac{5}{7}= \)
Incorrect
Correct Answer:
\( 1\frac{3}{7} \)
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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
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
\( 3\times\frac{1}{2}= \)
Incorrect
Correct Answer:
\( 1\frac{1}{2} \)
Question 2
\( 4\times\frac{2}{3}= \)
Incorrect
Correct Answer:
\( 2\frac{2}{3} \)
Question 3
\( 6\times\frac{3}{4}= \)
Incorrect
Correct Answer:
\( 4\frac{1}{2} \)
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: 13
We will do this for any given number.
Examples:
172=72
11=1
15=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 432 into an equivalent fraction.
Solution:
We will multiply the whole number 4 by the denominator 3 and then add 2.
We obtain: 4×3+2=14
14 will be the new numerator.
The denominator will remain the same: 3.
We obtain that:
432=314
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
\( 3\times\frac{6}{7}= \)
Incorrect
Correct Answer:
\( 2\frac{4}{7} \)
Question 3
\( 7\times\frac{2}{5}= \)
Incorrect
Correct Answer:
\( 2\frac{4}{5} \)
Multiplication Practice with Integers, Fractions, and Mixed Numbers
Here is an exercise
4×31×521
Solution:
The first step
Convert whole numbers and mixed numbers into equivalent fractions and rewrite the exercise:
14=4
521=211
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:
31×211×14=
Now let's go to the second step:
Multiply both the numerators and the denominators separately.
We obtain:
3×2×11×11×4=644
We can convert the result we obtained 644 into a mixed number 262.
Another exercise
792×2×52=
Solution:
First, we convert all whole and mixed numbers into equivalent fractions. We obtain:
792=27×9+2=965
12=2
Rewrite the exercise only with equivalent fractions:
965×12×52=
Multiply both the numerators and the denominators separately and you get:
965×12×52=45260
We can convert the final result we obtained 45260 into a mixed number 54535
Note: we can reduce the fraction even further and obtain: 54535=597
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 4535 which is identical in value to the fraction 97
And therefore 45535 and 957 are identical in value.
Check your understanding
Question 1
\( 8\times\frac{1}{2}= \)
Incorrect
Correct Answer:
\( 4 \)
Question 2
\( 10\times\frac{7}{9}= \)
Incorrect
Correct Answer:
\( 7\frac{7}{9} \)
Question 3
\( 2\times2\frac{1}{2}= \)
Incorrect
Correct Answer:
\( 5 \)
Additional study
How to convert an equivalent fraction into a mixed number?
We will learn with the example
Convert the equivalent fraction 62241 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…….
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:
241−(3×62)=55
The result we receive will be written in the numerator.
The denominator will remain the same.
We obtain: 36255
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
6×43=
Video Solution
Step-by-Step Solution
To solve the problem 6×43, we follow these steps:
Step 1: Express the integer 6 as a fraction: 16.
Step 2: Multiply the fractions: 16×43.
Step 3: Multiply the numerators: 6×3=18.
Step 4: Multiply the denominators: 1×4=4.
Step 5: Form the resulting fraction: 418.
Step 6: Simplify the fraction by dividing numerator and denominator by their greatest common divisor, which is 2: 418÷22=29.
Step 7: Convert 29 to a mixed number: Divide 9 by 2 to get 4 with a remainder of 1, thus 29=421.
Therefore, the solution to the problem is 421.
Answer
421
Exercise #2
3×21=
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 21, which is 1, by 3: 1×3=3.
Step 2: Write the result over the original denominator: 23.
Step 3: Convert the improper fraction 23 to a mixed number:
Divide 3 by 2. This gives 1 as the quotient and 1 as the remainder, so: 23=121.
Therefore, the solution to the problem is 121.
Answer
121
Exercise #3
2×75=
Video Solution
Step-by-Step Solution
To solve this problem, we'll multiply the whole number 2 by the fraction 75:
Step 1: Multiply the numerator 5 by the whole number 2:
2×5=10
Step 2: Write the result over the original denominator 7:
710
Step 3: Convert 710 to a mixed number:
Since 10 divided by 7 is 1 with a remainder of 3, we can express this as:
173
Therefore, the solution to the problem is 173.
Answer
173
Exercise #4
4×32=
Video Solution
Step-by-Step Solution
To solve this problem, we'll multiply the whole number 4 by the fraction 32 as follows:
Step 1: Convert the whole number 4 into a fraction. This can be written as 14.
Step 2: Use fraction multiplication rules: multiply the numerators together and the denominators together.
Step 3: So, multiply the numerators: 4×2=8.
Step 4: Multiply the denominators: 1×3=3.
Step 5: The result is 38.
Step 6: Since 38 is an improper fraction, convert it to a mixed number. 8÷3=2 with a remainder of 2.
Thus, 38=232.
Therefore, the solution to the problem is 232.
Answer
232
Exercise #5
7×52=
Video Solution
Step-by-Step Solution
To solve the problem 7×52, we will follow a structured approach:
Step 1: Multiply the whole number by the numerator of the fraction.
Step 2: Retain the denominator of the fraction.
Step 3: Simplify the resulting fraction, if possible.
Let's work through each step:
Step 1: Multiply the whole number by the numerator.
We have 7×2=14.
Step 2: Keep the denominator the same.
The resulting fraction is 514.
Step 3: Convert the improper fraction to a mixed number if possible.
Divide the numerator by the denominator: 14÷5=2 with a remainder of 4.
This results in the mixed number 254.
Therefore, the solution to the problem 7×52 is 254, which corresponds to choice 3 in the provided options.
Answer
254
Do you think you will be able to solve it?
Question 1
\( 2\times2\frac{1}{3}= \)
Incorrect
Correct Answer:
\( 4\frac{2}{3} \)
Question 2
\( 3\times1\frac{1}{2}= \)
Incorrect
Correct Answer:
\( 4\frac{1}{2} \)
Question 3
\( 3\times1\frac{1}{3}= \)
Incorrect
Correct Answer:
\( 4 \)
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More Questions
Multiplication of Integers by a Fraction and a Mixed number