CONVERTING RECURRING DECIMALS TO FRACTIONS
About "Converting Recurring Decimals to Fractions"
Converting Recurring Decimals to Fractions :
In this section, we are going to learn, how to convert recurring decimals into fractions.
How to convert recurring decimals into fractions ?
Step 1 :
Let x = Given decimal number
For example,
If the given decimal number is 2.0343434.........
then, let x = 2.0343434...........
Step 2 :
Identify the repeated pattern
For example,
In 2.0343434..........., the repeated pattern is 34
(Because 34 is being repeated)
Step 3 :
Identify the first repeated pattern and second repeated pattern as as explained in the example given below.
Step 4 :
Count the number of digits between the decimal point and first repeated pattern as given in the picture below.
Step 5 :
Since there is 1 digit between the decimal point and the first repeated pattern, we have to multiply the given decimal by 10 as given in the picture below.
(If there are two digits -----------> multiply by 100,
three digits -----------> multiply by 1000 and so on )
Note :
In (1), we have only repeated patterns after the decimal.
Step 6 :
Count the number of digits between the decimal point and second repeated pattern as given in the picture below.
Step 7 :
Since there are 3 digits between the decimal point and the second repeated pattern, we have to multiply the given decimal by 1000 as given in the picture below.
Note :
In (2), we have only repeated patterns after the decimal.
Step 8 :
Now, we have to subtract the result of step 5 from step 7 as given in the picture below.
Now we got the fraction which is equal to the given decimal.
Converting recurring decimals to fractions - Practice Problems
Problem 1 :
Covert the given repeating decimal into fraction
32.03256256256..........
Solution :
Let x = 32.03256256256.............
Here, the repeated pattern is 256
No. of digits between the 1st repeated pattern and decimal = 2
So, multiply the given decimal by 100. Then, we have
100x = 3203.256256256...............----------(1)
No. of digits between the 2nd repeated pattern and decimal = 5
So, multiply the given decimal by 100000. Then, we have
100000x = 3203256.256256256...............---------(2)
Subtracting (1) from (2), we get
(2) - (1) --------> 99900x = 3200053
x = 3200053/99900
Hence, 32.03256256256.......... = 3200053/99900.
Problem 2 :
Covert the given repeating decimal into fraction
0.01232222........
Solution :
Let x = 0.01232222.............
Here, the repeated pattern is 2
No. of digits between the 1st repeated pattern and decimal = 4
(Here, the first repeated pattern starts after four digits of the decimal)
So, multiply the given decimal by 10000. Then, we have
10000X = 123.2222...............----------(1)
No. of digits between the 2nd repeated pattern and decimal = 5
So, multiply the given decimal by 100000. Then, we have
100000x = 1232.2222...............----------(2)
Subtracting (1) from (2), we get
(2) - (1) --------> 90000x = 1109
x = 1109/90000
Hence, 0.01232222........... = 1109/90000.
Problem 3 :
Covert the given repeating decimal into fraction
2.03323232..........
Solution :
Let x = 2.03323232.............
Here, the repeated pattern is 32
No. of digits between the 1st repeated pattern and decimal = 2
(Here, the first repeated pattern starts after two digits of the decimal)
So, multiply the given decimal by 100. Then, we have
100x = 203.323232...............----------(1)
No. of digits between the 2nd repeated pattern and decimal = 4
So, multiply the given decimal by 10000. Then, we have
10000x = 20332.323232...............----------(2)
Subtracting (1) from (2), we get
(2) - (1) --------> 9900x = 20129
x = 9900/20129
Hence, 2.03323232.......... = 9900/20129.
Problem 4 :
Covert the given repeating decimal into fraction
0.252525..........
Solution :
Let x = 0.252525.............
Here, the repeated pattern is 25
No. of digits between the 1st repeated pattern and decimal = 0
So, multiply the given decimal by 1. Then, we have
x = 0.252525...............----------(1)
No. of digits between the 2nd repeated pattern and decimal = 2
So, multiply the given decimal by 100. Then, we have
100x = 25.252525...............----------(2)
Subtracting (1) from (2), we get
(2) - (1) --------> 99x = 25
x = 25/99
Hence, 0.252525.......... = 25/99.
Problem 5 :
Covert the given repeating decimal into fraction
3.3333..........
Solution :
Let x = 3.3333.............
Here, the repeated pattern is 3
No. of digits between the 1st repeated pattern and decimal = 0
(Here, the first repeated pattern is "3" which comes right after the decimal point)
So, multiply the given decimal by 1. Then, we have
x = 3.3333...............----------(1)
No. of digits between the 2nd repeated pattern and decimal = 1
(Here, the second repeated pattern is "3" which comes one digit after the decimal point)
So, multiply the given decimal by 10. Then, we have
10x = 33.3333...............----------(2)
Subtracting (1) from (2), we get
(2) - (1) --------> 9x = 30
x = 30/9 = 10/3
Hence, 3.3333.............. = 10/3.
Problem 6 :
Covert the given repeating decimal into fraction
1.023562562562..........
Solution :
Let x = 1.023562562562.............
Here, the repeated pattern is 562
No. of digits between the 1st repeated pattern and decimal = 3
So, multiply the given decimal by 1000. Then, we have
1000x = 1023.562562562...............----------(1)
No. of digits between the 2nd repeated pattern and decimal = 6
So, multiply the given decimal by 1000000. Then, we have
1000000x = 1023562.562562562...............----------(2)
Subtracting (1) from (2), we get
(2) - (1) --------> 999000x = 1022538
x = 1022539/999000
Hence, 1.023562562562.......... = 1022539/999000
After having gone through the stuff and examples, we hope that the students would have understood, "Converting recurring decimals to fractions"
Apart from the stuff and examples, if you want to know more about, "Converting recurring decimals to fractions", please click here.
Apart from the stuff "Converting recurring decimals to fractions", if you need any other stuff in math, please use our google custom search here.
WORD PROBLEMS
HCF and LCM word problems
Word problems on simple equations
Word problems on linear equations
Word problems on quadratic equations
Area and perimeter word problems
Word problems on direct variation and inverse variation
Word problems on comparing rates
Converting customary units word problems
Converting metric units word problems
Word problems on simple interest
Word problems on compound interest
Word problems on types of angles
Complementary and supplementary angles word problems
Markup and markdown word problems
Word problems on mixed fractrions
One step equation word problems
Linear inequalities word problems
Ratio and proportion word problems
Word problems on sets and venn diagrams
Pythagorean theorem word problems
Percent of a number word problems
Word problems on constant speed
Word problems on average speed
Word problems on sum of the angles of a triangle is 180 degree
OTHER TOPICS
Time, speed and distance shortcuts
Ratio and proportion shortcuts
Domain and range of rational functions
Domain and range of rational functions with holes
Graphing rational functions with holes
Converting repeating decimals in to fractions
Decimal representation of rational numbers
Finding square root using long division
L.C.M method to solve time and work problems
Translating the word problems in to algebraic expressions
Remainder when 2 power 256 is divided by 17
Remainder when 17 power 23 is divided by 16
Sum of all three digit numbers divisible by 6
Sum of all three digit numbers divisible by 7
Sum of all three digit numbers divisible by 8
Sum of all three digit numbers formed using 1, 3, 4
Sum of all three four digit numbers formed with non zero digits
Sum of all three four digit numbers formed using 0, 1, 2, 3
Sum of all three four digit numbers formed using 1, 2, 5, 6
