In fact, the decimal will be exact only if we can make the denominator a power of 10. See Section 2.
Example 5. Write |
9 25 |
as a decimal. |
Solution. 25 is not a power of 10, but we can easily make it a power of 10 -- we can make it 100 -- by multiplying it by 4. We must also, then, multiply the numerator by 4.
Example 6. Write |
4 5 |
as a decimal. |
We can make 5 into 10 by multiplying it -- and 4 -- by 2.
Example 7. Write as a decimal: |
7 200 |
Solution. |
7 200 |
= |
35 1000 |
= .035 |
We can make 200 into 1000 by multiplying it -- and 7 -- by 5.
Alternatively,
7 200 |
= |
3.5 100 |
, on dividing both terms by 2, |
|
|
= |
.035 |
, on dividing 3.5 by 100. |
Example 8. Write as a decimal: |
8 200 |
Solution. |
8 200 |
= |
4 100 |
= .04 |
Here, we can change 200 into a power of 10 by dividing it by 2. We can do this because 8 also is divisible by 2.
Or, again,
8 200 |
= |
_ 8 _ 2 × 100 |
= |
4 100 |
= .04 |
Example 9. Write as a decimal: |
12 400 |
Solution. |
12 400 |
= |
3 100 |
= .03 |
We can change 400 to 100 by dividing it -- and 12 -- by 4.
To summarize: We go from a larger denominator to a smaller by dividing (Examples 8 and 9); from a smaller denominator to a larger by multiplying (Example 5).
Example 10. Express 11 ÷ 20 as a decimal.
Solution. Upon using the division bar:
We made 20 into 100 by multiply it by 5. We therefore had to multiply 11 by 5 also.
Example 11.
a) We know that 5% is 5 out of 100 (Lesson 4). .5%, then, is 5 out of how many?
Answer. We can change .5% into the decimal
.005 (Lesson 4), which in
turn is equal to the fraction |
5 1000 |
. |
Therefore, .5% is 5 out of 1000.
b) .05% is 5 out of how many?
Solution. .05% = .0005 = |
5 10,000 |
. Therefore, .05% is 5 out of 10,000. |
Compare Lesson 18, Example 7.
Frequent decimals
In the actual practice of arithmetic, changing a fraction to a decimal is an extremely rare event. (We change a fraction to a percent directly: Lesson 27, Question 3.) The following are the only fractions whose decimal equivalents come up with any frequency. The student should know them.
Next, |
1 4 |
. But |
1 4 |
is half of |
1 2 |
. |

Therefore, its decimal will be half of .50 --
As for |
3 4 |
, since |
3 4 |
= 3 × |
1 4 |
, then |
Next, |
1 8 |
. But |
1 8 |
is half of |
1 4 |
. |

(Compare Lesson 15.)
Therefore its decimal will be half of .25 or .250 (Lesson 3) --
The decimals for the rest of the eighths will be multiples of .125.
Since 3 × 125 = 375,
5 8 |
will be 5 × |
1 8 |
= 5 × .125. |
5 × 125 = 5 × 100 + 5 × 25 = 500 + 125 = 625.
(Lesson 9) Therefore,
7 × 125 = 7 × 100 + 7 × 25 = 700 + 175 = 875.
These decimals come up frequently. The student should know how to generate them quickly.
Example 12. Write as a decimal: |
5 16 |
Solution. |
5 16 |
will be 5 times |
1 16 |
. |
|
Now, |
1 16 |
is half of |
1 8 |
(Lesson 15), which we saw is .125, or, .1250. |
5 × 625 |
= |
5 × 600 + 5 × 25 |
|
|
= |
3000 + 125 |
|
|
= |
3125. |
Therefore,
Fifths
The student should also know the decimals for the fifths:
The rest will be the multiples of .2 --
2 5 |
= |
2 × |
1 5 |
= 2 × .2 = .4 |
|
3 5 |
= 3 × .2 = .6 |
|
4 5 |
= 4 × .2 = .8 |
Example 13. Write as a decimal: 8 |
3 4 |
The whole number does not change. Simply replace the common
fraction |
3 4 |
with the decimal .75. |
Example 14. Write as a decimal: |
7 2 |
Solution. First change an improper fraction to a mixed number:
"2 goes into 7 three (3) times (6) with 1 left over."
Example 15. How many times is .25 contained in 3?
Answer. .25 = |
1 4 |
. And |
1 4 |
is contained in 1 four times. (Lesson 21.)
|
Therefore, |
1 4 |
, or .25, will be contained in 3 three times as many times. It will |
be contained 3 × 4 = 12 times.

Example 16. How many times is .125 contained in 5?
Answer. .125 = |
1 8 |
. And |
1 8 |
is contained in 1 eight times. Therefore, |
1 8 |
, |
or .125, will be contained in 5 five times as many times. It will be contained 5 × 8 = 40 times.
As for |
1 3 |
and |
2 3 |
, neither one can be expressed exactly as a decimal. |
However,
1 3 |
 |
.333 |
and
2 3 |
 |
.667 |
See Section 2, Question 3.
Frequent percents
From the decimal equivalent of a fraction, we can easily derive the percent: Move the decimal point two digits right. Again, the student should know these. They come up frequently.
1 2 |
= |
.50 |
= |
50% |
|
1 4 |
= |
.25 |
= |
25% |
|
3 4 |
= |
.75 |
= |
75% |
|
1 8 |
= |
.125 |
= |
12.5% |
(Half of |
1 4 |
. |
|
See Lesson 27, Example 7.) |
|
3 8 |
= |
.375 |
= |
37.5%. |
See above. |
|
5 8 |
= |
.625 |
= |
62.5% |
|
7 8 |
= |
.875 |
= |
87.5% |
|
1 5 |
= |
.2 |
= |
20% |
|
2 5 |
= |
.4 |
= |
40% |
|
3 5 |
= |
.6 |
= |
60% |
|
4 5 |
= |
.8 |
= |
80% |
In addition, the student should know
1 3 |
= 33 |
1 3 |
% |
|
2 3 |
= 66 |
2 3 |
% |
(Lesson 16)
At this point, please "turn" the page and do some Problems.
or
Continue on to the next Section.
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