Multiplication and Division of Fractions

1. Multiplying Fractions

Unlike adding and subtracting fractions, when multiplying or dividing fractions, you do not need a common denominator. Hooray! But why would you want to multiply a fraction? Remember, multiplication is just repeated addition. For example, if you want to make 4 times as many cookies for a bake sale, you don’t want to add the ingredients four times—just multiply and get your answer a lot faster. Additionally, multiplication and division involving fractions is useful when converting between measurements, which is common in many scientific fields, such as chemistry, physics, and nursing.

So how do you multiply fractions if you don’t need a common denominator? When you multiply fractions, you multiply the numerators together to find the numerator of your answer. Similarly, you multiply the denominators together to find the denominator of your answer. In other words, just multiply straight across!

EXAMPLE

Suppose you want to multiply 3/4 times 1/5.

	The expression
	Multiply the numerators, 3 and 1, to find the final numerator. Multiply the denominators, 4 and 5, to find the final denominator.
	The final numerator is 3 and the final denominator is 20. The solution is .

EXAMPLE

You are cooking a recipe that calls for ¾ cups of water. You want to triple the recipe for a party you plan to go to. How much water will you need to triple the recipe?

	We need 3 times ¾ cups of water.
	We can write the whole number 3 as a fraction by making the denominator 1.
	Now that we have 2 fractions, we can multiply across.
	The numerator is larger than the denominator so this is an improper fraction. We must simplify the fraction to determine the amount of water needed.
	When we divide 9 by 4, we get 2 with a remainder of 1.
	We will need to add 2 ¼ cups of water to triple the recipe.

try it

You are making 5 batches of cookies. Each batch of cookies uses ¼ teaspoon of salt.

How much salt will we need to bake all 5 batches of cookies?

	We need 5 times 1/4 teaspoon.
	We can write the whole number 5 as a fraction by making the denominator 1.
	Now that we have 2 fractions, we can multiply across.
	This is an improper fraction. We must simplify the fraction to determine the amount of water needed.
	When we divide 5 by 4, we get 1 with a remainder of 1.
	We will need to add 1 ¼ teaspoons of salt to make 5 batches of cookies.

2. Converting Mixed Fractions

So far, we’ve only converted whole numbers into improper fractions so we can multiply them, but what if we had a mixed fraction like 1 ½? Let’s think about it like a pizza again.

Here, we have 1 ½ pizzas….or 3/2. But how do we do this mathematically so we don’t have to draw pizzas each time? You multiply the denominator with the whole number and then add the numerator.

So, if we multiply the denominator (2) with the whole number (1), we get . Then, add the numerator (1), or . So, 1 ½ can be written as 3/2.

EXAMPLE

Convert to an improper fraction

	The fraction
	Multiply the denominator by the whole number.
	Add the numerator.
	Put this value as the numerator, and keep the original denominator

try it

You are combining 2 boxes of wild rice into a pot. Each box calls for 1 ¾ cups of water.

How much water will you need to make 2 boxes?

	We need 2 times 1 ¾ cups of water.
	We can write the whole number 2 as a fraction by making the denominator 1. But, we will need to convert 1 ¾ into an improper fraction so that we have 2 fractions that we can multiply.
	To convert 1 ¾ into an improper fraction, we multiply the whole number (1) times the denominator (4), or 1x4 = 4, then add the numerator of the mixed fraction (3). So, we have 4+3 = 7. Our mixed fraction 1 ¾ can be written as 7/4.
	Now we have 2 fractions and can multiply across the numerator and denominator.
	This is an improper fraction. We can simplify by converting this number back into a mixed number.
	We divided 14 by 4. We get 3 with a remainder of 2.
	The fraction portion of our mixed number is not in simplest terms because we can divide 2 into the numerator and denominator. To make 2 boxes of rice, we will need 3 ½ cups of water.

3. Dividing Fractions

Now that we know how to multiply fractions, dividing fractions is similar. Technically, you can’t divide fractions. No, that doesn't mean this lesson is done. When dividing fractions, you’ll need to multiply by the reciprocal. Reciprocal means “opposite,” so finding the reciprocal of a fraction means flipping it, or switching the numerator and the denominator.

EXAMPLE

Suppose you want to divide by .

	The expression
	First, find the reciprocal of by flipping it. This is . Change the sign to multiplication.
	Next, multiply the numerators, 3 and 2, and multiply the denominators, 8 and 1.
	Since the numerator is 6 and the denominator is 8, the answer is .
	Now simplify the fraction. Our common factor is 2 so divide your numerator and denominator by 2.

hint

It is important to note that you flipped your second fraction, and you changed the division sign to multiplication. Remember to only flip one fraction. If you flip both fractions, you can no longer multiply them to gain the correct answer.

EXAMPLE

Divide by .

	The expression
	First, find the reciprocal of by flipping it. This is . Change the sign to multiplication.
	Next, multiply the numerators, 3 and 4, and multiply the denominators, 4 and 1.
	Our solution is an improper fraction. We can convert this to a mixed fraction by dividing 12 by 4.
	This divides evenly 3 times, so our solution is 3 by converting our improper fraction into a whole number.

EXAMPLE

You have a 5-pound bag of sand. You want to pour 1 1/4 pounds of sand each into jars for table centerpieces. How many centerpieces can you make with the 5 pounds of sand?

	We want to divide 5 pounds by 1 ¼.
	We can write the whole number as a fraction by writing it over 1. We will also need to convert the mixed fraction 1 ¼ into an improper fraction. We multiply the whole number (1) by the denominator (4), 1x4 = 4, then we add this to the numerator of the mixed fraction 1 (4 + 1 = 5). 1 ¼ written as an improper fraction is 5/4.
	Next, find the reciprocal of 5/4 by flipping it. This is 4/5. Change the sign to multiplication.
	Multiply the numerators, 5 and 4, and the denominators, 1 and 5.
	We get the fraction This is an improper fraction and we can simplify this by converting it to a mixed number.
	We divide 20 by 5 and get a solution of 4. So we can divide the 5-pound bag of sand into 4 jars.

try it

You have a 7 ½-pound bag of candy. You want to divide it evenly between 5 children.

How much candy will each child receive (in pounds)?

	Our problem.
	We convert our mixed fraction (7 ½) into an improper fraction. We multiply (7x2 = 14) then add the numerator (1): 14 + 1 = 15. We can write our whole number 5 as the fraction .
	The reciprocal of is . We change the sign to multiplication.
	Then multiply the numerators, 15 and 1, and the denominators, 2 and 5.
	We get . We can simplify this by converting this to a mixed number.
	We divide 15 by 10 and get 1 with a remainder of 5, which turns into the fraction .
	The fraction portion 5/10 is not in simplest terms because 5 can divide evenly into the numerator and the denominator. When simplified, the solution is . Each of the 5 children will receive 1 ½ pounds of candy from the bag.

term to know

Reciprocal (of a fraction)

A fraction in which the numerator and denominator have been switched.