5
UNIT FRACTIONS
A UNIT FRACTION is a fraction whose numerator is 1.
| 1 2 |
, | 1 3 |
, | 1 4 |
, | 1 5 |
, etc. |
Each unit fraction is a part of 1.
| 1 2 |
is one half of 1. |
| 1 3 |
is the third part of 1. |
| 1 4 |
is the fourth part of 1. |
And so on.
| Example 1. In the fraction | 4 5 |
, what number is the unit, and how many |
of them are there?
Answer. The denominator of a fraction names the unit. The numerator tells their number -- how many.
| In the fraction | 4 5 |
, the unit is | 1 5 |
. And there are 4 of them. |
Problem 1. Which numbers are the decimal unit fractions?
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.1, .01, .001, .0001, etc.
| Problem 2. Let | 1 3 |
be the unit, and count to 2 | 1 3 |
. |
| " | 1 3 |
, | 2 3 |
, | 3 3 |
or 1. 1 | 1 3 |
, 1 | 2 3 |
, 2. 2 | 1 3 |
. | " |
We see: Each fraction is a multiple of a unit fraction.
| 3 5 |
= 3 × | 1 5 |
= | 1 5 |
+ | 1 5 |
+ | 1 5 |
Problem 3.
| a) | 4 9 |
= 4 × | 1 9 |
b) | 5 8 |
= 5 × |
1 8 |
| c) 2 | 3 5 |
= 13 × |
1 5 |
d) .35 = 35 × .01 |
Problem 4. Write each of the following as a sum of unit fractions.
| a) | 2 3 |
= | 1 3 |
+ | 1 3 |
b) | 3 8 |
= | 1 8 |
+ | 1 8 |
+ | 1 8 |
| c) 1 | 1 4 |
= | 5 4 |
= | 1 4 |
+ | 1 4 |
+ | 1 4 |
+ | 1 4 |
+ | 1 4 |
Problem 5.
| a) What number is the largest unit fraction? | 1 2 |
b) What number is the smallest? There is none.
Problem 6. 1 is equal to
| a) |
3 3 |
b) |
4 4 |
c) |
5 5 |
d) |
10 10 |
e) |
16 16 |
f) |
100 100 |
| g) 5 × | 1 5 |
h) 9 × | 1 9 |
i) 24 × | 1 24 |
We see:
1 is a multiple of every unit fraction.
We can write 1 with any denominator.
| Problem 7. What is the first multiple of | 1 32 |
that will be greater than 1? |
| 33 32 |
For more on the arithemtic of unit fractions, click here.
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