For GMAT, we must know how to convert non-terminating repeating decimals into rational
numbers. We know how to do vice versa i.e. given a rational number, we can divide the numerator
by the denominator to find its decimal equivalent. For example,
1/3 = 0.333333333… (infinite number of 3s)
We write this as 0. .
Similarly, 1/6 = .16666666666… (infinite number of 6s)
We write this as 0.1 .
1/7 = 0.142857142857…
We write this as 0.
.
The problem arises when we are given a decimal which we need to convert to a fraction. We know
that every non-terminating repeating decimal can be written cleanly and then used in calculations
by converting it to fraction. But given, say 1. , how do we know which fraction it represents? We
can approximate and say that 1.88888… is almost 1.9 but approximation may not be suitable in
every question or you might be asked for the actual value of the fraction.
So how do you convert 1. to a fraction? A terminating decimal is easy to handle. Say, a decimal
such as 1.8. We get rid of the decimal sign by dividing by 10 i.e. 1.8 = 18/10.
When we have a decimal such as 1. , the problem is that we have infinite 8s so we will need infinite
0s in the denominator.
1.888 = 1888/1000
1.888888 = 1888888/1000000
But what do we do when we have infinite 8s? It is very hard for us to fathom infinite numbers and
harder still to work with them. We need to get rid of the infinite sequence in some way. The good
thing about the infinite sequence is that even if we pull away one 8 out of it, the sequence still
remains infinite.
X = 1.
10X = 18. (When you multiply by 10, the decimal moves one place to the right but you still have
infinite 8s leftover)
Subtract the first equation from the second to get,
9X = 18. - 1. = 17 (Infinite 8s subtract out the infinite 8s and you are left with 18 – 1 = 17)
So X = 17/9
So the exact value of 1. is 17/9.
To sum it, the manipulation required is this – we need the infinite sequence (and only the infinite
sequence) to the right of decimal in both equations. We subtract out the equations to get rid of the
infinite sequence. We are left with a clean fraction.
We use the exact same logic to convert any number with a non terminating repeating part into a
fraction. Let’s take another example. Say,
X=
Note that multiplying by 10 or 100 here doesn’t help us. We need the part after the decimal to be
the same in both the equations so that we can cancel it off. Note how we work around this problem:
100X =
10000X =
Now we have two equations where we have only the infinite sequence to the right of the decimal
point. When we subtract these two equations, we get rid of the repetitive part and get,
9900X = 21133
X = 21133/9900
The point is to separate out the repetitive part and then cancel it. You can now convert any
repeating non-terminating decimal into a fraction.
For practice, try out these numbers.
1. 0.08
2. 34.02
3. 1.