How to make multiplication and division of fractions?

The Multiplication and Division of Fractions are operations that, respectively, simplify the sum of numerators and represent the parts of a whole, that is, of an integer.

They can be done using two rules. Let's go to them!

It is important to remember that in fractions, the upper term is called the numerator while the lower term is called the denominator.

Fraction Multiplication

When multiplying fractions, simply multiply one numerator by another and then one denominator by another.

Example:

Multiplication is done this way regardless of the number of fractions.

Example:

How to do in the case below? Simple. You have at least three options:

1.ª 8 over 3 straight space x 6 over 1 equal to 48 over 3 equal to 16 over 1 equal to 16

2.ª 8 over 3 plus 8 over 3 plus 8 over 3 plus 8 over 3 plus 8 over 3 plus 8 over 3 equals 48 over 3 equals 16 over 1 equals 16

3.ª $numerator 8 straight space x space 6 over denominator 3 end of fraction equal to 48 over 3 equal to 16 over 1 equal to 16$

Check out this content in more detail at: Fraction Multiplication.

Division of Fractions

At division of fractions the rule is as follows:

1st The numerator of the first fraction multiplies the denominator of the second;
2. The denominator of the first fraction multiplies the numerator of the other fraction.

Example:

$10 over 5 divided by 2 over 8 equal to numerator 10 straight space x space 8 over denominator 5 straight space x space 2 end of fraction equal to 80 over 10 equal to 8 over 1 equal to 8$

As in multiplication, also in division the rule applies regardless of the number of fractions, ie:

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1st The numerator of the first fraction multiplies the denominator of the second and the remaining fractions;
2. The denominator of the first fraction multiplies the numerator of all other fractions.

Example:

$7 over 8 divided by 15 over 3 divided by 5 over 1 equal to numerator 7 straight space x space 3 straight space x space 1 over denominator 8 straight space x space 15 straight space x space 5 end of fraction equal to 21 over 600 equal to 7 over 200$

See also other operations with fractions: Addition and Subtraction of Fractions.

Solved exercises of multiplication and division of fractions

Now that you've learned how to multiply and divide fractions, test your knowledge:

question 1

Determine the result of the operations below.

The) 2 over 3 straight space x 3 over 2 space

B) 2 over 3 straight space x 3 over 7 space

ç) 3 over 5 space divided by 1 over 10

d) 1 bedroom space divided by space 2

See Answer

Correct answers: a) 1, b) 2/7 c) 6 and d) 1/8.

The) $2 over 3 straight space x space 3 over 2 space equal to numerator space 2 straight space x space 3 over denominator 3 straight space x space 2 end of fraction equals space 6 over 6 space equals space 1$
When the result of the multiplication of two fractions gives the result 1, it means that the fractions are inverse of each other, that is, the inverse fraction of 2/3 is 3/2.

So 2/3 times 3/2 is equal to 1.

B) $2 over 3 straight space x space 3 over 7 space equal to numerator space 2 straight space x space 3 over denominator 2 straight space x space 7 end of fraction space equal to space 6 to the power of divided by 3 end of exponential over 21 to the power of divided by 3 end of exponential space equal to space 2 about 7$

Another way to solve this multiplication is to cancel the similar term.

Note that fractions have the same factor in the numerator and denominator. In this case, we can cancel them by dividing both by the number itself, ie 3.

$2 over 3 space straight x space 3 over 7 space equal to space numerator 2 over diagonal denominator up risk 3 end of fraction straight space x space diagonal numerator up risk 3 over denominator 7 end of fraction space equal to space 2 over 7$

So 2/3 times 3/7 is equal to 2/7.

c) In the division operation, we must multiply the first fraction by the inverse of the second fraction, that is, multiply the numerator of the first by the denominator of the second and multiply the denominator of the first by the numerator of the Monday.

3 over 5 space divided by 1 over 10 space equal to space 3 over 5 straight space x space 10 over 1 space equal to space 30 over 5 space equal to space 6

So 3/5 divided by 1/10 equals 6.

d) In this example we have the division of a fraction by a natural number. To solve it, we must multiply the first by the inverse of the second.

Note that the number 2 does not have the denominator written, that is, we have the number 1 as the denominator and we can invert the fraction as follows: the inverse of 2 is 1/2.

We then solved the operation.

1 room space divided by space 2 space equal to space 1 room space straight x space 1 half space equal to space 1 over 8

So the 1/4 half is 1/8.

question 2

If a pot contains 3/4 kilogram of chocolate milk, how many kg of chocolate milk would have 8 pots equal to this?

a) 4 kg
b) 6 kg
c) 2 kg

See Answer

Correct answer: b) 6 kg.

In this situation we have the multiplication of a fraction by a natural number.

To solve it we must multiply the natural number by the numerator of the fraction and repeat the denominator.

8 space. space 3 over 4 space equal to space 24 over 4 space equal to space 6

If each pot has 3/4 kg of chocolate milk, 8 pots would have a total of 6 kg.

question 3

In the pantry at her house, Maria noticed that she had four packages with half a kg of rice and 6 packages with a quarter of a kilo of noodles. What was in the greatest amount?

a) Rice
b) Pasta
c) In the pantry there was the same amount of both

See Answer

Correct answer: a) Rice.

First, let's calculate the amount of rice. Remember that a pound is 1/2, because 1 divided by 2 is 0.5.

$4 space. numerator space 1 space over denominator 2 end of fraction equals space 4 over 2 equals space 2$

Now, we calculate the amount of noodles.

6 space. 1 bedroom space equal to 6 over 4 space

Since the division of 6 by 2 is not an exact number, we can simplify the numerator and denominator by 2.

6 to the power of divided by 2 end of exponential over 4 to power of divided by 2 end of exponential space equal to space 3 over 2

As the division of 3 by 2 results in 1.5 we concluded that rice is in greater quantity, as it has 2 kg.

question 4

In a classroom 2/3 of the students are girls. Among girls, 3/4 have brown hair. What fraction of the students in the class have brown hair?

a) 3/2
b) 1/2
c) 1/3

See Answer

Correct answer: b) 1/2.

If in a class 2/3 of the total are girls and in that number 3/4 have brown hair, then we must calculate the product of two fractions.

2 over 3 straight space x 3 over 4 space

We solve the multiplication of fractions by writing in the numerator the product of 2 by 3 and in the denominator the product of 3 by 4.

$2 over 3 straight space x space 3 over 4 space equal to numerator 2 straight space x space 3 over denominator 3 straight space x space 4 end of fraction space equal to space 6 over 12$

Note that 12 is double 6. We can simplify this fraction by dividing the numerator and denominator by 6.

6 to the power of divided by 2 end of exponential over 12 to power of divided by 2 end of exponential space equal to space 1 half

Thus, 1/2, that is, half have brown hair.

For more questions, check outFraction Exercises.

question 5

When he got home, João found an open chocolate package on the table. There was 1/3 of the chocolate bar and he ate half that amount. How much chocolate did John eat?

a) 1/4
b) 1/5
c) 1/6

See Answer

Correct answer: c) 1/6.

In the statement we have the information that João ate half of 1/3, that is, he divided 1/3 into two parts and ate only one. Therefore, the operation that must be performed is 1/3: 2.

To solve this question we must multiply the first fraction (1/3) by the inverse of the second fraction (2), that is, 1/3 multiplied by 1/2.

$1 third space divided by space 2 space equal to space 1 third straight space x space 1 half equal space numerator 1 straight space x space 1 over denominator 3 straight space x space 2 end of fraction space equal to space 1 about 6$