My homework is converting repeating decimals into fractions. I can't figure out the steps for this problem: 0.083 and the 3 is the repeating number. Can someone please explain the steps to me?
one digit is repeating
... so multiply the number by 10^1
10 n = .8333...
n = .08333...
subtracting ... 9 n = .75 (the 3's cancel)
n = .75 / 9 = (3/4) / 9 = 3 / 36 = 1 / 12
general method:
e.g. .34567567567... (the 567 repeats)
for the numerator:
1. Without the decimal, write down the digits to the end of the first repeat:
-----> 34567
2. From that , subtract all the digits that don't repeat:
---> 34567-34 = 34533
for the denominator:
1. write down a 9 for each of the repeating digits
there are 3 digits repeating, so ---> 999
2. tag on a 0 for each of the non-repeating digits
there are two of those, so ---> 99900
fraction is 34533/99900
or reduced to 3837/11100 = 1279/3700
check with a calculator
another e.g.
.666666....
numerator = 6 - 0 = 6
denominator = 9 , there is a single digit repeating, so one 9, and there are no non-repeating, so no zero has to be tagged on
= 6/9
= 2/3
For your case:
0.0833333....
= (83 - 08)/900
= 75/900
= 1/12
check: 1/12 = .08333...
Sure! Converting repeating decimals into fractions involves a few steps. Let's break it down for the decimal 0.083, where the 3 is the repeating number.
Step 1: Identify the repeating part
Notice that the number 3 repeats, so we can represent it as 0.0833... or 0.083(repeating).
Step 2: Set up equations
Let's assume x is the fraction we want to find. To set up the equation, we'll multiply both sides by 1000 (since there are three digits after the decimal point):
1000x = 83.333...
Step 3: Subtract the repeating part
Now, let's subtract x from both sides of the equation to eliminate the repeating part on the right side:
1000x - x = 83.333... - 0.083...
This simplifies to:
999x = 83.25
Step 4: Solve for x
To find the value of x, we divide both sides of the equation by 999:
x = 83.25 / 999
Now, to simplify this fraction further, we can find the greatest common divisor (GCD) of 83.25 and 999. In this case, the GCD is 249. Dividing both the numerator and denominator by 249 gives us:
x = 1 / 12
Therefore, the fraction equivalent of the repeating decimal 0.083 is 1/12.