Question

How do you add repeating decimals ?

Answer 1

Hint: If we have 2 or more repeating decimals we can convert the decimals to fractions and add all the fractions. We can easily convert repeating decimals into fractions. We can convert the result fraction into decimal again.

Complete step-by-step answer:
Let’s take we have 2 repeating decimals that are 2.55555…. and 3.7777…. we can convert both into fraction
So let’s convert 2.55555… into fraction
Let’s take x equal to 2.5555….
$\Rightarrow x=2.5555....$ eq 1
By multiply 10 with x we get
$\Rightarrow 10x=25.5555...$ eq 2
If we observe we can see that by subtracting eq1 from eq2 all the repeating term after the decimal sign will get cancelled out
So by subtracting we get
$\Rightarrow 9x=23$
So x is equal to $\dfrac{23}{9}$ , $\dfrac{23}{9}$ is the fraction form of 2.5555…
let’s convert 3.7777… into fraction
Let’s take x equal to 3.7777…
$\Rightarrow x=3.7777...$ eq 3
By multiply 10 with x we get
$\Rightarrow 10x=37.7777....$ eq 4
If we observe we can see that by subtracting eq3 from eq4 all the repeating term after the decimal sign will get cancelled out
So by subtracting we get
$\Rightarrow 9x=34$
So x is equal to $\dfrac{34}{9}$ , $\dfrac{34}{9}$is the fraction form of 3.7777…
Now we can add 2 fractions , $\dfrac{23}{9}$+ $\dfrac{34}{9}$ is equal to $\dfrac{57}{9}$
If we convert $\dfrac{57}{9}$ into decimal we will get 6.3333.

Note: All the fraction which are infinitely long and repeating are not irrational numbers . The non-repeating infinitely long fractions are irrational numbers for example e and $\pi $ . We can write irrational numbers as fractions .Remember that $\dfrac{22}{7}$ is not an exact value of $\pi $ , it is an approximate value of $\pi $ .