Dividing Fractions: Reciprocal Numbers
In this lesson we study reciprocal numbers and how to use them in fraction divisions. We also try to make sense of the reciprocal numbers and division of fractions visually.
The video below explains the reciprocal numbers and the "rule" for dividing fractions conceptually.
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First, let’s review a little.
How many times does one number go into another?
From this situation, you can always write a division. Yes — EVEN if the numbers are fractions! Ask:
“How many times does the divisor go
into
the dividend?” |
How many times does  go into  ?
| Three times. We write the division: 2 ÷ |
2
3 |
= 3. |
| Then check the division: 3 × |
2
3 |
= |
6
3 |
= 2. |
|
1. Solve. Write a
division. Then write a multiplication that checks your division.
| 3 ÷ |
1
3 |
= _____ |
| Check: ____ × |
1
3 |
= |
|
| ÷ |
 |
= _____ |
| Check: ____ × |
|
= |
|
| ÷ |
|
= _____ |
Check: |
| ÷ |
|
= _____ |
Check:
|
2. Solve. Think how many times the divisor
goes into the dividend. Can you find a pattern or a shortcut?
| a. 3 ÷ |
1
6 |
= |
|
| b. 4 ÷ |
1
5 |
= |
|
| c. 3 ÷ |
1
10 |
= |
|
| d. 5 ÷ |
1
10 |
= |
|
| e. 7 ÷ |
1
4 |
= |
|
| f. 4 ÷ |
1
8 |
= |
|
| g. 4 ÷ |
1
10 |
= |
|
| h. 9 ÷ |
1
8 |
= |
|
| The shortcut is this: |
|
| 5 |
÷ |
1
4 |
|
| |
↓ |
↓ |
|
| 5 |
× |
4 |
= 20 |
|
|
| 3 |
÷ |
1
8 |
|
| |
↓ |
↓ |
|
| 3 |
× |
8 |
= 24 |
|
|
| 9 |
÷ |
1
7 |
|
| |
↓ |
↓ |
|
| 9 |
× |
7 |
= 63 |
|
That is, multiply
the number by the reciprocal of the divisor. Notice that 1/4
upside down,
or inverted,
is 4/1 or just 4. We call 1/4 and 4
reciprocal
numbers, or just reciprocals.
Does the shortcut make sense to you? For example, consider the problem 5 ÷ (1/4).
Since 1/4
goes into 1
exactly four times, it must go into 5 exactly 5 × 4 = 20
times. |
| Two numbers are reciprocal
numbers (or reciprocals) of each other if, when multiplied, they make 1. |
|
3
4 |
is a reciprocal of |
4
3 |
, because |
3
4 |
× |
4
3 |
= |
12
12 |
= 1. |
|
|
1
7 |
is a reciprocal of 7, because |
1
7 |
× |
7 |
= |
7
7 |
= 1. |
|
| You can find the reciprocal of a fraction |
m
n |
by flipping the numerator and denominator:
|
n
m |
. |
| This works, because |
m
n |
× |
n
m |
= |
n × m
m × n |
= 1. |
|
|
To find the reciprocal of a mixed number, first write
it as a fraction,
then “flip” it.
| Since 2 |
3
4 |
= |
11
4 |
, its reciprocal number is |
4
11
|
. |
|
3. Find the reciprocal numbers.
Then, write a multiplication with the given number and its reciprocal.
|
a. |
5
8 |
|
|
5
8 |
× |
|
= 1 |
|
|
b. |
1
9 |
|
|
|
× |
|
= 1 |
|
|
c. 1 |
7
8 |
|
|
|
× |
|
= 1 |
|
d. 32
|
|
32 |
× |
|
= 1 |
|
|
e. 2 |
1
8 |
|
|
|
× |
|
= 1 |
|
4. Write a division sentence to match with each multiplication above.
|
a. |
1 |
÷ |
|
= |
|
|
|
b. |
1 |
÷ |
|
= |
|
|
|
c. |
1 |
÷ |
|
= |
|
|
|
d. |
___ |
÷ |
|
= |
|
|
|
e. |
___ |
÷ |
|
= |
|
|
Read, and try to understand.
This is important!
|
Let’s now try to make sense of the reciprocal numbers and division of
fractions visually.
|
|
Thinking of the division
problem 1 ÷ (2/5), we ask, first of all, how many times does 2/5
fit into 1?
| Using pictures: How many times does |
 |
go into |
 |
? |
|
| From the picture we can see that |
|
goes into |
|
two times, and then we have 1/5 left over. |
| But, how many times does |
2
5 |
fit into the leftover piece, |
1
5 |
? How many times does |
|
go into |
 |
? |
|
|
That is like trying to fit a
TWO-part piece into a hole that holds just ONE part.
Only 1/2
of the two-part piece fits! So, 2/5
fits into 1/5 exactly half a time.
|
| In total, we find that
2/5 fits into
one exactly 2 1/2 times. We can write the division 1 ÷ |
2
5 |
= 2 |
1
2 |
or |
5
2 |
. |
| Notice, we got 1 ÷ |
2
5 |
= |
5
2 |
. Checking that with multiplication, we get |
5
2 |
× |
2
5 |
= 1. Reciprocals again! |
|
|
One more example. Thinking of the division problem
1 ÷ (5/7), we ask how many times does 5/7 fit into
1?
| Using pictures: How many times does |
 |
go into |
 |
? |
|
| From the picture we can see that |
|
goes into |
|
just once, and then we have 2/7 left over. |
| But, how many times does |
5
7 |
fit into the leftover piece, |
2
7 |
? How many times does |
|
go into |
 |
? |
|
|
The FIVE-part piece fits into a hole that is only big enough for two parts just 2/5 of
the way.
|
| So in total, 5/7 fits into
one exactly 1 2/5 times. The division is 1 ÷ |
5
7 |
= 1 |
2
5 |
or 1 ÷ |
5
7 |
= |
7
5 |
. |
|
5. Solve. Think how many
times the given fraction fits into one whole. Write a
division.
| a.
How many times does |
|
go into |
|
? |
|
1 |
÷ |
|
= |
|
| b.
How many times does |
|
go into |
 |
? |
|
1 |
÷ |
|
= |
|
|
c.
How many times does |
|
go into |
 |
? |
|
1 |
÷ |
|
= |
|
| d.
How many times does |
|
go into |
|
? |
|
1 |
÷ |
|
= |
|
| e.
How many times does |
|
go into |
 |
? |
|
1 |
÷ |
|
= |
|
|
f.
How many times does |
|
go into |
 |
? |
|
1 |
÷ |
|
= |
|
6. Solve. Think how many
times the given fraction fits into the other number. Write a
division.
|
|
2 |
÷ |
|
= |
|
 |
÷ |
|
= |
|
|
|
3 |
÷ |
|
= |
|
|
d.
How many times does |
|
go into |
 |
? |
|
|
÷ |
|
= |
|
SHORTCUT: instead of dividing, multiply by the reciprocal of
the divisor.
Study the
examples to see how this works. |
|
How many times
does
 go
into  ?
|
3
4 |
÷ |
1
3 |
|
|
|
|
| |
↓ |
↓ |
|
|
3
4 |
× |
3 |
= |
9
4 |
= 2 |
1
4 |
Answer: 2 1/4 times.
Does it make sense?
Yes,
fits into
a little
more
than two times. |
How many times
does
 go
into 
?
|
7
4 |
÷ |
2
5 |
|
|
|
|
| |
↓ |
↓ |
|
|
7
4 |
× |
5
2 |
= |
35
8
|
= 4
|
3
8
|
Answer: 4 3/8 times.
Does it make sense? Yes.
goes into 1 3/4
over four times. |
How many times
does
 go
into
 ?
|
2
9 |
÷ |
2
7 |
= |
|
| |
↓ |
↓ |
|
|
9 |
× |
7
|
= |
7
9 |
Answer: 7/9 of a time.
Does it make sense? Yes, because
does not
go
into
even one full time! |
|
Remember: There are
TWO changes in each calculation:
- You need to change the division into
multiplication.
- You need to use the reciprocal of the divisor.
|
7. Solve these division problems
using the shortcut.
Remember to check if your answer makes sense.
|
a. |
3
4 |
÷ |
5 |
|
| |
|
↓ |
↓ |
|
|
|
3
4 |
× |
1
5 |
= |
|
|
b. |
2
3 |
÷ |
6
7 |
|
|
c. |
4
7 |
÷ |
3
7 |
|
|
d. |
2
3 |
÷ |
3
5 |
|
|
e. 4 ÷ |
2
5 |
|
|
f. |
13
3 |
÷ |
1
5 |
|
8. a. Write a division to match the
situation on the right.
b. Check your division by
multiplication.
|
| How many times does |
|
fit into |
? |
We have 8/5, which is eight pieces, trying to fit
into five
pieces... so they fit 5/8 of the way. |
|
9. Fill in.
|
2 ÷ |
3
4 |
= ? |
|
Or, how many times
does |
|
go into |
|
? |
|
First, let’s solve how many
times |
|
goes into |
|
. |
|
Since 1 ÷ |
3
4 |
= |
|
, it goes into one |
|
times. |
|
|
If 3/4 fits into
_____
times, then it fits
into
double that many times, or _____ times. |
|
We get the same answer by using the shortcut:
|
2 |
÷ |
3
4 |
|
| |
↓ |
↓ |
|
|
2 |
× |
|
= |
|
10. Fill in.
|
|
÷ |
|
= ? |
|
Or, how many times
does |
|
go into |
 |
? |
|
First, let’s solve how many
times |
|
goes into |
|
. |
|
Since 1 ÷ |
|
= |
|
, it goes into one |
|
times. |
|
|
If 2/7 fits into
_____
times, then it fits
into
exactly 5/6 as many times as it fits into 1,
|
|
We get the same answer by using the shortcut:
|
This lesson is taken from Maria Miller's book Math Mammoth Fractions 2, and posted at www.HomeschoolMath.net with permission from the author. Copyright © Maria Miller.
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.
=> Learn more and see the free samples!
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