CHAPTER
6

Fractions

In This Chapter

• Learning how to add and subtract fractions
• Learning how to multiply and divide fractions
• Applying the vertically and crosswise method to fractions

You may be one of the many people who doesn’t enjoy math using fractions. However, once you have a grasp of the many quick and easy ways you can work fraction problems, you’ll gain a comfort level even with the most complex-looking fraction problems.

In this chapter, you learn ways to add, subtract, multiply, and divide fractions, as well as how you can use the vertically and crosswise method when working fractions.

Adding Fractions

Adding fractions can be tricky to master, particularly if the denominators don’t match. In the following sections, I provide the fastest, simplest ways to solve these problems.

Fractions with the Same Denominator

Whenever two or more fractions have the same denominator, you add the numerators and simplify the answer as much as possible.

Example 1

Solve the problem .

Because the denominators are same, you simply add the numerators.

Solution: The answer is .

Example 2

Solve the problem .

Here, too, the denominators are same, so you just add the numerators and simplify the fraction to get your answer.

Solution: The answer is .

Fractions Where One Denominator Is a Factor of the Other

To add fractions that don’t have matching denominators, but where one is a factor of the other, you multiply the numerator and denominator by that factor and solve.

Example 1

Solve the problem .

Step 1: Because 5 is a factor of 20, multiply the numerator and denominator of by 4 to get the fraction you can use.

Step 2: Add the numerators of the fractions.

Solution: The answer is .

Example 2

Solve the problem .

Step 1: Because 15 is a factor of 30, multiply the numerator and denominator of by 2 to get the fraction you can use.

Step 2: Add the numerators of the fractions.

Solution: The answer is .

Addition with the Vertically and Crosswise Method

To add fractions without a common denominator or one that’s a factor of the other, you can use the vertically and crosswise method.

Example 1

Solve the problem .

Step 1: Multiply the fractions crosswise and add the products: (4 × 3) + (5 × 1) = 17. This is the numerator.

Step 2: Multiply the denominators: 5 × 4 = 20. This is the denominator.

Solution: The answer is .

Example 2

Solve the problem .

Step 1: Multiply the fractions crosswise and add the products: (9 × 2) + (11 × 7) = 18 + 77 = 95. This is the numerator.

Step 2: Multiply the denominators: 11 × 9 = 99. This is the denominator.

Solution: The answer is .

Subtracting Fractions

Subtracting fractions is very similar to what you learned for adding fractions—the main difference is that a minus sign is involved. The following show you different ways you can subtract, based on the setup of the denominators.

Fractions with the Same Denominator

This is the easiest type of subtraction for fractions. Whenever you spot that the denominators are same, you can just subtract the numerators and write your answer.

Example 1

Solve the problem .

Because the denominators are the same, you simply subtract the numerators and simplify the fraction.

Solution: The answer is .

SPEED BUMP

If you can simplify the fraction, do so; the fraction should be in the simplest form possible. Otherwise, you’ll have an incomplete answer.

Example 2

Solve the problem .

Because the denominators are the same, subtract the numerators and simplify.

Solution: The answer is .

Fractions Where One Denominator Is a Factor of the Other

As you learned in the addition section, fractions where one denominator is a factor of the other require you to multiply the numerator and denominator by that factor before you can solve the problem.

Example 1

Solve the problem .

Step 1: Because 2 is a factor of 24, multiply the numerator and denominator of by 12 to get the fraction you can use.

Step 2: Subtract the numerators of the fractions.

Solution: The answer is .

Example 2

Solve the problem .

Step 1: Because 3 is a factor of 9, multiply the numerator and denominator of by 3 to get the fraction you can use.

Step 2: Subtract the numerators of the fractions.

Solution: The answer is .

Subtraction with the Vertically and Crosswise Method

Just like in addition, the vertically and crosswise method plays an important role in subtraction. You can subtract any fraction from any other fraction, no matter the denominator, using this method.

Example 1

Solve the problem .

Step 1: Multiply the fractions crosswise and add the products: (6 × 2) − (7 × 1) = 12 − 7 = 5. This is the numerator.

Step 2: Multiply the denominators: 7 × 2 = 14. This is the denominator.

Step 3: Subtract the fractions.

Solution: The answer is .

Example 2

Solve the problem .

Step 1: Multiply the fractions crosswise and add the products: (12 × 50) − (25 × 3) = 600 − 75 = 525. This is the numerator.

Step 2: Multiply the denominators: 25 × 50 = 1,250. This is the denominator.

Step 3: Subtract the fractions.

Step 4: The numerator and denominator are multiples of 25, so reduce the fraction by 25.

Solution: The answer is .

Multiplying Fractions

If you need to multiply two fractions, all you need to do is multiply the numerators and then multiply the denominators.

Example 1

Solve the problem .

Step 1: Multiply the numerators: 4 × 6 = 24.

Step 2: Multiply the denominators: 11 × 19 = 209.

Solution: The answer is .

Example 2

Solve the problem .

Step 1: Make the whole numbers part of the fractions. For , integrate the 3 by multiplying it by the denominator and adding it to the numerator: 3 × 7 = 21 + 4 = 25. This makes the first fraction . For , integrate the 6 by multiplying it by denominator and adding it to the numerator: 6 × 3 = 18 + 2 = 20. This makes the second fraction .

Step 2: Multiply the numerators and denominators. For the numerators: 25 × 20 = 500. For the denominators: 7 × 3 = 21.

Step 3: Convert it to the mixed number form again: 500 ÷ 21 = 23, remainder 17. The 17 is the numerator.

Solution: The answer is .

Dividing Fractions

Dividing fractions can be a headache. However, it doesn’t have to be challenging—you can get the correct answer by multiplying the first fraction by the reciprocal of the second fraction.

Example 1

Solve the problem .

Step 1: Change the division sign to a multiplication sign and reverse the second fraction so 72 is on top and 32 is on the bottom.

Step 2: Simplify the fractions crosswise and multiply. In this case, 32 goes into 16 twice, so 16 becomes 1 and 32 becomes 2; 72 goes into 9 eight times, so 9 becomes 1 and 72 becomes 8. You can then further reduce when you get the answer.

Solution: The answer is .

QUICK TIP

Whether it’s on paper or in your head, right after you change the sign from division to multiplication, flip the numerator and denominator on the second fraction. That way, you’ll keep those two processes connected.

Example 2

Solve the problem .

Step 1: Change the division sign to a multiplication sign and reverse the second fraction so 7 is on top and 90 is on the bottom.

Step 2: Simplify the fractions crosswise and multiply. In this case, 90 goes into 45 twice, so 45 becomes 1 and 90 becomes 2; 18 and 7 can’t reduce.

Step 3: Because the numerator is larger than the denominator, convert it to a mixed number: 36 ÷ 7 = 5, remainder 1. The 1 is the numerator.

Solution: The answer is .

The Least You Need to Know

• If the fractions in the problem have the same denominator, you can simply solve and reduce if necessary—no extra steps are involved.
• For fractions in which one denominator is a factor of the other, you multiply the numerator and denominator by that factor to get the denominators the same.
• The vertically and crosswise method allows you to solve problems involving any fractions.
• To convert a mixed number into a fraction, multiply the whole number by the denominator and add it to the numerator.
• The easiest way to divide fractions is to reverse the second fraction and multiply it by the first fraction.