EXPRESSING DECIMALS AS RATIONAL NUMBERS
About "Expressing decimals as rational numbers"
Expressing decimals as rational numbers :
A rational number is any number that can be written as a ratio in the form a/b, where a and b are integers and b is not 0.
Examples of rational numbers are 2/3 and 1/5.
We all know that 6 is an integer. But 6 also can be considered as rational number.
Because, 6 can be written as 6/1.
We can express terminating and repeating decimals as rational numbers.
Expressing decimals as rational numbers - Examples
Example 1 :
Write the decimal 0.825 as a fraction in simplest form.
Solution :
The decimal 0.825 means “825 thousandths.” Write this as a fraction.
To write “825 thousandths”, put 825 over 1000.
825 / 1000
Then simplify the fraction.
Both 825 and 1000 are the multiples of 25. So, divide both the numerator and the denominator by 25.
(825 ÷ 25) / (1000 ÷ 25) = 33 / 40
0.825 = 33 / 40
Hence, the fraction equal to 0.825 is 33/40.
Example 2 :
Write the decimal 0.12 as a fraction in simplest form.
Solution :
The decimal 0.12 means “12 hundredths.” Write this as a fraction.
To write “12 hundredths”, put 12 over 100.
12/100
Then simplify the fraction.
Both 12 and 100 are the multiples of 4. So, divide both the numerator and the denominator by 4.
(12 ÷ 4) / (100 ÷ 4) = 3 / 25
0.12 = 3 / 25
Hence, the fraction equal to 0.12 is 3/25.
Example 3 :
Convert the following repeating decimal as fraction
0.474747...........
Solution :
Let x = 0.474747............... ------(1)
Here repeating number of digits = 2. So we have to multiply 100 on both sides.
100 x = 47.4747.............. ------(2)
(2) - (1)
aaaaaaaaaaaaaaaaa100 x = 47.4747.........aaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaax = 0.4747...............aaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaa(-)aaaaaa(-)aaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaa------------------------aaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaa99 x = 47.0000aaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaa-----------------------aaaaaaaaaaaaaaaaa
x = 47/99
Hence, the fraction equal to 0.474747........... is 47/99.
Example 4 :
Convert the following repeating decimal as fraction
0.57777..........
Solution :
Let x = 0.5777777............... ------(1)
Here repeating number of digits = 1. So we have to multiply 10 on both sides.
10 x = 0.57777777.............. ------(2)
(2) - (1)
aaaaaaaaaaaaaaaa10 x = 5.77777777.........aaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaax = 0.57777777...............aaaaaaaaaaaa
aaaaaaaaaaaaaaaaaa(-)aaaaaa(-)aaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaa------------------------aaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaa9 x = 5.20000aaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaa-----------------------aaaaaaaaaaaaaaaaa
x = 5.2/9
Multiply the numerator and denominator by 10
x = 52/90
x = 26/45
Hence, the fraction equal to 0.57777........... is 26/45.
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