Chapter Notes: Multiply Fractions
Multiplying fractions helps us solve many everyday problems. When you need to find half of a third of a pizza, or figure out how much flour you need when you cut a recipe in half, you are multiplying fractions! At first, multiplying fractions might seem tricky, but it follows simple rules that make it easier than adding or subtracting fractions. You don't even need a common denominator! In this chapter, you will learn how to multiply fractions by whole numbers, fractions by fractions, and mixed numbers by other numbers. You will also learn how to simplify your answers to make them as clear as possible.
Understanding What It Means to Multiply Fractions
When you multiply a fraction by a whole number, you are finding that many groups of the fraction. For example, 3 × 1/4 means you have three groups of one-fourth. If you have three pieces that are each one-fourth of a pizza, you have 3/4 of a pizza total.
When you multiply a fraction by another fraction, you are finding a part of a part. For example, 1/2 × 1/3 means you are finding one-half of one-third. Imagine a candy bar cut into three equal pieces. One piece is one-third of the bar. If you take half of that one piece, you now have one-sixth of the whole candy bar.
Think of multiplying fractions like zooming in on a smaller piece. You start with a part, then take a part of that part, and you end up with an even smaller piece of the whole.
Multiplying a Whole Number by a Fraction
To multiply a whole number by a fraction, you can think of the whole number as a fraction by putting it over 1. Any whole number divided by 1 equals itself, so 3 is the same as 3/1.
Rule: Multiply the whole number by the numerator (top number) of the fraction. Keep the denominator (bottom number) the same.
Here are the steps:
- Write the whole number as a fraction with 1 as the denominator.
- Multiply the numerators together.
- Multiply the denominators together.
- Simplify the answer if possible.
Example: A recipe calls for 2/3 cup of sugar.
You want to make 4 batches of the recipe.How much sugar do you need in total?
Solution:
You need to find 4 × 2/3.
Write 4 as a fraction: 4 = 4/1
Multiply the numerators: 4 × 2 = 8
Multiply the denominators: 1 × 3 = 3
The answer is 8/3.
Since 8/3 is an improper fraction, we can write it as a mixed number: 8 ÷ 3 = 2 with a remainder of 2, so 8/3 = 2 2/3.
You need 2 2/3 cups of sugar.
Example: Juan runs 3/4 of a mile each day.
How many miles does he run in 5 days?Solution:
You need to find 5 × 3/4.
Write 5 as a fraction: 5 = 5/1
Multiply the numerators: 5 × 3 = 15
Multiply the denominators: 1 × 4 = 4
The answer is 15/4.
Convert to a mixed number: 15 ÷ 4 = 3 with a remainder of 3, so 15/4 = 3 3/4.
Juan runs 3 3/4 miles in 5 days.
Multiplying a Fraction by a Fraction
Multiplying two fractions is actually simpler than adding them because you don't need to find a common denominator. You just multiply straight across!
Rule: Multiply the numerators together to get the new numerator. Multiply the denominators together to get the new denominator.
Here are the steps:
- Multiply the top numbers (numerators) together.
- Multiply the bottom numbers (denominators) together.
- Simplify the answer if needed.
The formula looks like this:
\[ \frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d} \]Where \(a\), \(b\), \(c\), and \(d\) are numbers, and \(b\) and \(d\) are not zero.
Example: Multiply 1/2 × 1/3.
Solution:
Multiply the numerators: 1 × 1 = 1
Multiply the denominators: 2 × 3 = 6
The answer is 1/6.
The product is 1/6.
Example: Multiply 2/5 × 3/4.
Solution:
Multiply the numerators: 2 × 3 = 6
Multiply the denominators: 5 × 4 = 20
The answer is 6/20.
Simplify by finding the greatest common factor of 6 and 20, which is 2.
Divide both numerator and denominator by 2: 6 ÷ 2 = 3 and 20 ÷ 2 = 10.
The simplified answer is 3/10.
Example: Maria has 3/4 of a yard of ribbon.
She uses 2/3 of that ribbon to wrap a present.How much ribbon does she use?
Solution:
You need to find 2/3 of 3/4, which means 2/3 × 3/4.
Multiply the numerators: 2 × 3 = 6
Multiply the denominators: 3 × 4 = 12
The answer is 6/12.
Simplify by dividing both by 6: 6 ÷ 6 = 1 and 12 ÷ 6 = 2.
The simplified answer is 1/2.
Maria uses 1/2 yard of ribbon.
Simplifying Before You Multiply
Sometimes you can make multiplying fractions easier by simplifying before you multiply. This is called canceling or cross-canceling. When you simplify first, you work with smaller numbers, which makes the multiplication easier and your answer is already simplified!
You can cancel when a numerator and a denominator (from either fraction) share a common factor. Divide both by their greatest common factor before you multiply.
Here are the steps:
- Look at the numerators and denominators of both fractions.
- Find any common factors between a numerator and a denominator.
- Divide both the numerator and denominator by the common factor.
- Multiply the simplified numerators together.
- Multiply the simplified denominators together.
Example: Multiply 4/5 × 5/8 using simplification first.
Solution:
Notice that 5 appears in the numerator of the second fraction and the denominator of the first fraction.
Divide both 5s by 5: 5 ÷ 5 = 1.
Notice that 4 and 8 share a common factor of 4.
Divide both by 4: 4 ÷ 4 = 1 and 8 ÷ 4 = 2.
Now multiply: 1/1 × 1/2 = 1/2.
The answer is 1/2.
Example: Multiply 6/7 × 14/15 using simplification first.
Solution:
Look for common factors. Notice that 6 and 15 share a common factor of 3.
Divide both by 3: 6 ÷ 3 = 2 and 15 ÷ 3 = 5.
Notice that 7 and 14 share a common factor of 7.
Divide both by 7: 7 ÷ 7 = 1 and 14 ÷ 7 = 2.
Now multiply: 2/1 × 2/5 = 4/5.
The answer is 4/5.
Multiplying Mixed Numbers
A mixed number is a whole number and a fraction together, like 2 1/3. To multiply with mixed numbers, you must first change them into improper fractions. An improper fraction is a fraction where the numerator is larger than or equal to the denominator, like 7/3.
To convert a mixed number to an improper fraction:
- Multiply the whole number by the denominator.
- Add the numerator to that product.
- Write the sum over the original denominator.
For example, to convert 2 1/3:
- Multiply: 2 × 3 = 6
- Add: 6 + 1 = 7
- Write as fraction: 7/3
Once all mixed numbers are converted to improper fractions, multiply them just like you multiply any other fractions.
Example: Multiply 2 1/2 × 3.
Solution:
Convert 2 1/2 to an improper fraction.
Multiply: 2 × 2 = 4, then add: 4 + 1 = 5, so 2 1/2 = 5/2.
Write 3 as a fraction: 3 = 3/1.
Multiply: 5/2 × 3/1 = (5 × 3)/(2 × 1) = 15/2.
Convert 15/2 to a mixed number: 15 ÷ 2 = 7 with a remainder of 1, so 15/2 = 7 1/2.
The answer is 7 1/2.
Example: Multiply 1 1/3 × 2 1/4.
Solution:
Convert 1 1/3 to an improper fraction: 1 × 3 = 3, then 3 + 1 = 4, so 1 1/3 = 4/3.
Convert 2 1/4 to an improper fraction: 2 × 4 = 8, then 8 + 1 = 9, so 2 1/4 = 9/4.
Multiply: 4/3 × 9/4.
Simplify first: 4 in the numerator and 4 in the denominator both divide by 4 to get 1.
We also see that 3 and 9 share a common factor of 3: 3 ÷ 3 = 1 and 9 ÷ 3 = 3.
Now multiply: 1/1 × 3/1 = 3/1 = 3.
The answer is 3.
Example: A rectangular garden is 4 1/2 feet wide and 2 2/3 feet long.
What is the area of the garden?
Solution:
Area of a rectangle = length × width, so we need 4 1/2 × 2 2/3.
Convert 4 1/2: 4 × 2 = 8, then 8 + 1 = 9, so 4 1/2 = 9/2.
Convert 2 2/3: 2 × 3 = 6, then 6 + 2 = 8, so 2 2/3 = 8/3.
Multiply: 9/2 × 8/3.
Simplify first: 9 and 3 share a factor of 3, so 9 ÷ 3 = 3 and 3 ÷ 3 = 1.
Also, 2 and 8 share a factor of 2, so 2 ÷ 2 = 1 and 8 ÷ 2 = 4.
Now multiply: 3/1 × 4/1 = 12/1 = 12.
The area of the garden is 12 square feet.
Real-World Applications of Multiplying Fractions
Multiplying fractions appears in many everyday situations. Here are some common examples:
- Cooking and baking: When you adjust recipes, you multiply fractions. If a recipe calls for 2/3 cup of milk and you want to make half the recipe, you multiply 1/2 × 2/3.
- Measurement: If you need to find part of a length, you multiply. For example, finding 3/4 of 2 1/2 yards of fabric.
- Money: If an item costs $12 and it's on sale for 1/4 off, you multiply 1/4 × 12 to find the discount.
- Time: If you can complete 2/3 of a project in one hour, and you work for 3 1/2 hours, you multiply to find how much you complete.
Example: A brownie recipe calls for 3/4 cup of cocoa powder.
You want to make only 1/2 of the recipe.How much cocoa powder do you need?
Solution:
You need to find 1/2 of 3/4, which means 1/2 × 3/4.
Multiply the numerators: 1 × 3 = 3.
Multiply the denominators: 2 × 4 = 8.
The answer is 3/8.
You need 3/8 cup of cocoa powder.
Important Properties of Multiplying Fractions
Understanding these properties helps you work with fractions more easily:
Commutative Property
You can multiply fractions in any order and get the same answer. This is called the commutative property.
For example: 1/2 × 3/4 = 3/4 × 1/2. Both equal 3/8.
Associative Property
When multiplying three or more fractions, you can group them in any way. This is called the associative property.
For example: (1/2 × 1/3) × 1/4 = 1/2 × (1/3 × 1/4). Both equal 1/24.
Identity Property
When you multiply any fraction by 1, the answer is the original fraction. This is called the identity property.
For example: 3/5 × 1 = 3/5.
Remember that any fraction where the numerator and denominator are the same equals 1, such as 4/4 or 7/7.
Zero Property
When you multiply any fraction by 0, the answer is always 0.
For example: 3/5 × 0 = 0.
Common Mistakes to Avoid
When learning to multiply fractions, watch out for these common errors:
- Adding instead of multiplying: Don't add the numerators and denominators. You must multiply them. For example, 1/2 × 1/3 is NOT 2/5. It is 1/6.
- Forgetting to simplify: Always check if your answer can be simplified. The fraction 6/8 should be simplified to 3/4.
- Not converting mixed numbers: You cannot multiply mixed numbers directly. Always convert them to improper fractions first.
- Mixing up numerators and denominators: Remember to multiply numerators together and denominators together, not across diagonally.
Comparing the Size of Products
When you multiply fractions, something interesting happens with the size of your answer:
- When you multiply a number by a fraction less than 1, the product is smaller than the original number.
- When you multiply a number by a fraction equal to 1, the product is the same as the original number.
- When you multiply a number by a fraction greater than 1 (an improper fraction or mixed number), the product is larger than the original number.
Think of it this way: multiplying by a fraction less than 1 is like taking only part of something, so you end up with less. Multiplying by a number greater than 1 is like having more than one whole group, so you end up with more.
Example: Compare 8 × 1/4 and 8 × 5/4.
Solution:
First calculate 8 × 1/4:
8/1 × 1/4 = 8/4 = 2. This is smaller than 8 because 1/4 is less than 1.
Now calculate 8 × 5/4:
8/1 × 5/4 = 40/4 = 10. This is larger than 8 because 5/4 is greater than 1.
We can see that 8 × 1/4 = 2 (smaller than 8) and 8 × 5/4 = 10 (larger than 8).
Step-by-Step Strategy for Any Fraction Multiplication Problem
Follow these steps every time you multiply fractions, and you'll get the right answer:
- Convert all mixed numbers to improper fractions. This is the most important first step.
- Write any whole numbers as fractions. Put them over 1.
- Look for opportunities to simplify. Cancel common factors between any numerator and any denominator before multiplying.
- Multiply the numerators together. This gives you the numerator of your answer.
- Multiply the denominators together. This gives you the denominator of your answer.
- Simplify your answer if possible. Find the greatest common factor and divide.
- Convert improper fractions to mixed numbers if needed. This often makes answers easier to understand.
- Check your work. Does your answer make sense? If you multiplied by a fraction less than 1, your answer should be smaller than what you started with.
With practice, multiplying fractions becomes quick and automatic. Remember that the most important rule is to multiply straight across-numerator times numerator, denominator times denominator. Unlike adding and subtracting fractions, you don't need a common denominator, which makes multiplication much simpler once you learn the basic steps!
