Fractions and decimals are just different ways of writing the
same numbers |
·
Every fraction can be written as a
decimal. ·
Every decimal can be written as a
fraction. |
FRACTION |
DECIMAL |
_{} |
0.5 |
_{} |
0.3333_{} |
_{} |
0.25 |
_{} |
0.2 |
_{} |
0.1666_{} |
_{} |
0._{}4285_{} |
_{} |
0.125 |
_{} |
0.111_{} |
_{} |
0.1 |
Figure 1. The decimal and
fraction form for some numbers. |
For
example _{}=0.75 and _{} = 0._{}8571_{}
(the string 285714 keeps repeating itself all the time).
Normally
a fraction produces a recurring decimal
in which a string of numbers keeps repeating itself
and never ends. For example
_{} = 0._{}_{}_{}
(a bar on top of the digit indicates where the recurring
string starts and ends).
·
If 2 and 5 are the only prime factors of the denominator è non-recurring
decimal ·
If the denominator has prime factors other than 2 and 5 è recurring decimal |
Example 1. Which of the fractions 4/15,
2/25, 8/35 will produce recurring decimals are which will not?
Answer. 3 is a prime factor of 15 è 4/15 recurring decimal.
5
is the only prime factor of 25 è 2/25 ending decimal.
7
is a prime factor of 35 è 8/35 recurring decimal. §
Decimals from fractions
Fraction means dividing. To find the decimal for any fraction, just divide |
You
need to get enough digits to see what the recurring string is going to be.
Fractions from decimals
Non-recurring decimals
To go from decimal to fraction if the decimal is
non-recurring just multiply and divide by 10, 100, 1000… and simplify
Example 2. Find the fraction form of
2.75.
Answer. If we multiply by 100 we get 275, which is a counting number and good for
the numerator. Therefore,
2.75
= _{} = _{}§
Example 3. Find the fraction form of
67.987.
Answer. If we multiply by 1000 we get 67987, which we use for the numerator.
Therefore,
67.987
= _{}. This fraction cannot be simplified§
Pure recurring decimals
For pure recurring decimals, i.e. decimals numbers
with no terminating part the rule is:
0. _{}ecurrin_{} |
= |
recurring |
As many 9s as
digits are there in ‘recurring’ |
Example 4. Write 0._{} as a fraction.
Answer. 0._{} = _{} (Check it!) §
Example 4. Write 0.0_{} as a fraction.
Answer. 0.0_{} = _{} (Check it!) §
Example 5. Write 0._{} as
a fraction.
Answer. 0._{} = _{} (Check it!) §
Example 6. Write 0.0_{} as a fraction.
Answer. 0.0_{} = _{} (Check it!) §
Example 7. Write 0._{}_{} as a fraction.
Answer. 0._{}_{} = _{} (Check it!) §
Example 8. Write 0._{}6_{} as a fraction.
Answer. 0._{}6_{} = _{} (Check it!) §
Recurring decimals with a non-recurring part
If the decimal is recurring but it has a
non-recurring part to find the fraction do first the recurring part,
then the non-recurring part and add them
together..
Example 9. Find the fraction form of
1.5_{}.
Answer: 1.5_{} = 1.5 + 0.0_{}= _{} + _{} = _{} + _{} = _{}. §
Example 10. Find the fraction form of
8.9_{}4_{}.
Answer: 8.9_{}4_{} = 8.9 + 0.0_{}4_{} = _{} + _{} = _{}. §