# Simplify fractions before multiplying them - a free lesson for 5th grade

In this 5th grade lesson, I explain how to simplify fractions before multiplying them. This is an extremely useful technique. It makes the fraction multiplication much easier because the numbers to be multiplied are smaller after the simplification.

 A new notation We will start using a new way to indicate simplifying fractions. When a numerator or a denominator gets simplified, we will cross it out with a slash and write the new numerator or denominator next to it (either above it or below it). The number you divide by (the 4) does not get indicated in any way! You only think about it in your mind: “I divide 12 by 4, and get 3. I divide 20 by 4, and get 5.” You may not see any advantage over the “old” method yet, but this shortcut will come in handy soon.

1. Simplify the fractions. Write the simplified numerator and denominator above and below the old ones.

 a. 14 16
 b. 33 27
 c. 12 26
 d. 9 33

Before you multiply, you can write another equivalent,
simpler fraction in the place of a fraction.

In the first example on the right, 3/6 is simplified to 1/2
before multiplying. We write a tiny “1” above the “3”

 1 3 6 2 × 5 8 = 5 16

 3 7 × 2  4  10 5 = 6 35

and a tiny “2” below the “6”. In the other example, 4/10 is simplified to 2/5 before multiplying.

Why does this work? Obviously, we can write 1/2 instead of 3/6, or 2/5 instead of 4/10, since they are equivalent.

2. Simplify one or both fractions before multiplying. Use equivalent fractions. Look at the example.

 a. 3 6 10 5 × 1 2 14 7 = 3 × 1 5 × 7 = 3 35
 b. 2 4 × 3 15 =
 c. 8 12 × 1 2 =
 d. 8 32 × 14 21 =
 e. 6 15 × 6 9 =
 f. 27 45 × 21 49 =

You can also simplify “criss-cross.” Look at this example: →

We simplify 3 and 6, writing 1 and 2 in their place. Think of it as the fraction
3/6 being simplified into 1/2, but the 3 and 6 are across from each other.

Why are we allowed to simplify in such a manner?

 Compare the above problem to this one: 7 9 × 3 6 . (It is almost the same, isn’t it?) Surely

you can see that in this problem, we could simplify 3/6 to 1/2 before multiplying.

And, these two multiplication problems are essentially the same problem, because they

 both lead to the same expression and the same answer: the first one becomes 7 × 3 6 × 9 = 21 54 ,
 and the second one becomes 7 × 3 9 × 6 = 21 54 (without simplifying). Therefore, since you can

simplify 3/6 into 1/2 in the one problem, you can do the same in the other also.

3. Simplify “criss-cross” before you multiply.

 a. 8 9 × 6 11
 b. 3 10 × 2 5
 c. 4 7 × 1 12
 d. 7 4 × 3 21
 e. 3 16 × 8 5
 f. 3 8 × 12 11

 You can even simplify criss-cross several times before multiplying. First, simplify 3 and 6 into 1 and 2. Then simplify 5 and 15 into 1 and 3.

4. Simplify before you multiply.

 a. 7 8 × 2 7
 b. 3 5 × 5 6
 c. 5 12 × 4 10
 d. 9 15 × 3 18
 e. 8 11 × 3 4
 f. 12 100 × 4 15

 Simplify 27 45 × 45 .  You can think of this problem in two manners:

1) Think of the fraction line as division. The problem is therefore the same as 27 ÷ 45 × 45.
Whenever you multiply and divide by the same number, you have essentially done nothing.
So, you can cross out both 45s in the original problem, and the answer is simply 27.

 2) First change the whole number 45 into the fraction 45/1. The problem is now 27 45 × 45 1 .
 Now you can simplify criss-cross, and multiply: =  27.

5. Simplify and multiply.

 a. 82 77 × 77  =
 b.   13 × 49 13 =
 c. 14 × 16 14 =
 d. 5 6 × 24  =
 e.   54 × 2 9 =
 f. 16 × 5 8 =

6. A toy block is 3/8 in. tall. How tall is a stack of 8 of them?

A stack of 20 of them?

7. Sandra buys 3/4 kg of meat every week. How much meat does she buy in a year?

8. The morning after his birthday, there is 12/20 of Sam's birthday cake left. He eats 2/3 of what is left.
When you multiply those two fractions, what does your answer mean or tell you?

To multiply three or more fractions, the same principles apply. You multiply all the numerators and all the denominators to get the numerator and the denominator for the answer.

 Example. We can do a lot of simplifying before multiplying with this problem: 14 25 × 10 9 × 5 6
 1. Simplify 10 and 25 into     2 and 5 (dividing by 5).
 2. Simplify 14 and 6      into 7 and 3.
 = 14 27 3. Lastly, simplify 5 and 5,     leaving 1 and 1.

9. Multiply three fractions. Simplify before multiplying.

 a. 4 5 × 3 4 × 2 3 =
 b. 11 8 × 6 8 × 2 3 =
 c. 9 10 × 5 2 × 2 7 =
 d. 3 5 × 6 12 × 5 3 =
 e. 4 5 × 9 8 × 10 24 =
 f. 7 12 × 3 5 × 6 7 =

10. a. Draw a bar model for this situation. Matthew pays 1/5 of his salary in taxes.
Of what is left, he uses 1/4 to purchase groceries.

b. Suppose Matthew’s salary is \$2,450.
Calculate how much he uses for groceries.

Epilogue: What happens if you don’t simplify before
multiplying?
Compare the two problems on the right →

Jack did all of the simplifying before multiplying.
Tina simplified after multiplying. Both of them got
the right answer. Simplifying before multiplying does
NOT change the final answer—it just makes it easier
to multiply
because the numbers are smaller!

 7 35 × 6 8 = 42 280 = 21 140 = 3 20

Tina multiplies first to get 42/280.
Lastly, she simplifies her answer in two
steps, first to 21/140, and then to 3/20.

 1 7 35 5 × 3 6 8 4 = 3 20

Jack simplifies
before multiplying.

This lesson is taken from my book Math Mammoth Fractions 2.

#### Math Mammoth Fractions 2

A self-teaching worktext that teaches fractions using visual models, a sequel to Math Mammoth Fractions 1. The book covers simplifying fractions, multiplication and division of fractions and mixed numbers, converting fractions to decimals, and ratios.