We want to add the two given numbers.
0.409+0.681
Notice that both numbers are repeating decimals. To add them, we can rewrite both numbers as fractions, or we can leave them in this form. We will get the same answer either way. Let's focus on the fraction method first. Recall how to rewrite a repeating decimal as a fraction.
Write the equation x=d, where d is a repeating decimal.
Subtract the equation from the previous step from the equation 10^n x=10^n d, where n is the number of repeating digits.
Solve the equation for x.
We can use this process to rewrite our decimals as fractions, starting with the one on the left. We will substitute 0.409 for d and 2 for n, as there are two repeating digits in the decimal.
ccc x=d & ⇔ & x= 0.409 10^nx=10^n d & ⇔ & 10^2x=10^2( 0.409)
Let's simplify the second equation.
10^1x=10^2(0.409) ⇒ 100x=40.909
Now we are ready to subtract the first equation from the second one.
c r c l& 100x & = & 40.909 -( & x & = & 0.409 ) & 99x & = & 40.5
Next, we can solve for x. Let's do it!
The first number can be written as 405990. Now let's find the fraction form of the second given number.
c r c l& 100x & = & 68.181 -( & x & = & 0.681 ) & 99x & = & 67.5
Next, we will solve it for x.
