The recurring decimal 1.68
recurring is equal to the fraction 76 over 45. Use algebra to prove this.
There are two ways of proving that
recurring decimal is equal to a specific fraction. The first way is to perform long
division on the fraction. Here, that would be dividing 76 by
45 and then showing that the answer recurs. The question has told us to use
algebra though. So we’ll use an alternative
For this method, we’ll give our
recurring decimal a name. Let’s call it 𝑥. 𝑥 is equal to 1.68 recurring. It can be useful to add the extra
digits instead of the recurring symbol as it helps us spot what’s happening as we go
through this process. With this method, our aim is to
create two decimals with the exact same set of recurring digits after the decimal
To do this, we multiply our number
at least once by some power of 10. Let’s begin by multiplying it by
10. When we multiply by 10, the digits
move to the left one place. 10𝑥 is, therefore, equal to 16.8
Notice how we haven’t yet created
any numbers with identical digits after the decimal point. What we need to do then is multiply
this new equation by 10. That’s the same as multiplying the
original equation by 100. 10𝑥 multiplied by 10 is 100𝑥. So our new equation is 100𝑥 is
equal to 168.8 recurring.
Once we have two numbers with the
same digits after the decimal point, we can subtract the smaller from the
larger. It’s helpful to write the larger
above the smaller before completing this step. 100𝑥 minus 10𝑥 is 90𝑥. Let’s use the column method to see
what happens when we subtract 16.8 recurring from 168.8 recurring.
Every single eight cancels since
eight take away eight is zero. Eight take away six is two. Six take away one is five. And one take away zero is just
one. Our equation then is 90𝑥 is equal
to 152. We can solve this equation by
dividing both sides by 90, giving us 𝑥 is equal to 152 over 90.
Finally, we need to simplify this
equation by recognizing that both 152 and 90 share a common factor of two. 152 divided by two is 76 and 90
divided by two is 45. Remember we said at the start that
𝑥 was equal to 1.68 recurring. And now, we’ve also shown that it’s
equal to 76 over 45. That means then that 1.68 recurring
must be equal to 76 over 45, which is what we were required to show.