Write the following repeating decimals as a fraction in simplest form. 2.34141414141...

Question

only the 4141 are repeating.

Answer 1

2.3414141...

let x=2.3414141...

multiply both sides by 100 because 2 digits repeat (100=10^2)

100x=234.14141414...

x= 2.34141414...

subtract

99x=231.8

x=231.8/99

multiply both terms by 10

x=2318/990

simplify

x=1159/495

another solution you might not have seen...

2.3414141...

first only work with the decimal part

0.341414141...

multiply by 10 to get to the repetend and beyond the lag of 3, then divide by 10 so as not to change the value of the number (multiply by the power of 10 that gets you beyond the lag-don't forget to divide by this power of 10 also)

(3.41414141...)/10

you should have realized that there is a pattern in repeating decimals:

assuming there is no lag

if one digit repeats put the digit over 9

if two digits repeat put them over 99

if three digits repeat put them over 999

and so on

0.7777...=7/9

0.63636363...=63/99=7/11

0.123123123123...=123/999=41/333

back to the problem

(3.414141...)/10=

(3 41/99)/10=

change the mixed number into an improper fraction

(338/99)/10=

338/990=

169/495

now add the 2 back into the problem

2 169/495

change the mixed number into an improper fraction

1159/495

it is not as hard as it looks, especially if there is no whole number to worry about

Answer 2

Start by finding multiples of the number so that everything after the decimal point is identical.

x=2.3414141...

10x = 23.414141...

1000x = 2341.414141...

1000x - 10x = 2341.414141.... - 23.414141....

=> 990x = 2341-23

=> 990x = 2318

=> x = 2318/990 = 1159/495