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.
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.
Download ($5.75). Also available as a printed copy.