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Dividing Fractions 2b: Reciprocal Numbers
The video below explains about reciprocal numbers conceptually, and about the "rule" (how-to) for dividing fractions:
| We know that |
1
5 |
goes into 1 exactly five times. But how many times does |
2
5 |
go into 1? |
| Let's think with pictures. |
 |
goes into |
 |
two times, and then we have 1/5 left over. |
| How many times does |
2
5 |
fit into |
1
5 |
? Or, how many times does |
|
go into |
 |
? |
That is like trying to fit a
two-piece part into a hole that just fits one piece. Only 1/2 of the part
fits. So, 2/5 goes into 1/5 half a time. All in all, 2/5 fits into one exactly 2 1/2 times.
| We can write a division from this: 1 ÷ |
2
5 |
= 2 |
1
2 |
|
|
Let's look at another similar example.
| We know that |
1
4 |
goes into 1 exactly four times. But how many times does |
3
4 |
go into 1? |
| Let's look at pictures. |
 |
goes into |
 |
once, and we have 1/4 left over. |
| How many times does |
3
4 |
fit into |
1
4 |
? Or, how many times does |
|
go into |
 |
? |
We have a three-piece (3/4)
part trying to fit into one piece (1/4), so the part fits
1/3 of the way.
| Putting it all together, 3/4 fits into one 1 1/3 times. As a
division: 1 ÷ |
3
4 |
= 1 |
1
3 |
|
|
1. Solve. First think how many
times the fraction fits into 1 completely, and then think how many
times it fits into the leftover part. Write a division.
|
a.
How many times
| does |
|
go into |
 |
? |
|
1 |
÷ |
 |
= |
|
b.
[available in the book]
|
|
i.
[available in the book] |
j.
How many times
| does |
|
go into |
 |
? |
|
1 |
÷ |
|
= |
|
|
2. Write the divisions from the previous exercise |
|
|
so that the answer
is written
as a fraction, and not as a
mixed number. For example,
2 1/3 (the answer to #a)
is written as 7/3. I want you
to notice something special!
|
a.
1 |
÷ |
3
7 |
= |
7
3 |
| b.
1 |
÷ |
|
= |
|
|
|
|
g.
1 |
÷ |
|
= |
|
|
h. 1 |
÷ |
|
= |
|
|
|
|
|
|
|
3. Did you notice the special
thing? Now, let's turn it
around.
Write each division sentence
from above “backwards” as
a multiplication. The first
one is done for you.
|
a. |
4
9 |
× |
9
4 |
= 1 |
| b. |
|
× |
|
= 1 |
|
|
|
g. |
|
× |
|
= 1 |
|
h. |
|
× |
|
= 1 |
|
i. |
|
× |
|
= 1 |
|
j. |
|
× |
|
= 1 |
|
These kinds of numbers are called
reciprocal numbers!
Reciprocal numbers
Two numbers are reciprocal numbers of each other
if, when multiplied, you get 1.
|
3
4 |
is a reciprocal number of |
4
3 |
, because |
3
4 |
× |
4
3 |
= |
12
12 |
= 1. |
|
1
7 |
is a reciprocal number of |
7 |
, because |
1
7 |
× |
7 |
= |
7
7 |
= 1. |
|
| You can find a reciprocal of a fraction |
a
b |
by flipping the numerator and denominator: it is |
b
a |
. |
| This works because |
a
b
|
× |
b
a |
= 1 (a's and b's cancel each other in simplification.) |
|
|
For mixed numbers, first write them as fractions,
then find the reciprocal number.
| Since 2 |
3
4 |
= |
11
4 |
, its reciprocal number is |
4
11
|
. |
|
4. Find reciprocal numbers!
|
a. |
5
8 |
|
|
b. |
1
9 |
|
|
|
|
f. 1 |
1
23 |
|
|
[available in the book] |
|
j. |
3
1000 |
|
5. Solve the division problems.
|
a. 1 ÷ |
3
4 |
|
[available in the book] |
|
|
|
We have eight pieces trying to
fit into five... so they fit 5/8 of the way.
|
As a division, we write 1 |
÷ |
8
5 |
= |
5
8 |
. The idea of reciprocal numbers works here as well. |
|
If you check the division by multiplication, it checks:
|
5
8 |
÷ |
8
5 |
= 1. |
|
6. Solve the division problems.
|
a. 1 ÷ |
7
4 |
|
|
|
|
7. [available in the book]
8. [available in the book]
The ideas in this fraction lesson are taken from the Math Mammoth Fractions 2 book. Only a few examples of each problem type are shown; you should make more problems of each kind for the student.
Next lesson: The Shortcut for fraction division
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New! Times Tales is now on DVD!
The fast, FUN, and easy way to learn multiplication. Learn the upper times tales in two sittings using mnemonic stories.
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