Prove algebraically that the recurring decimal 0.3409, where zero and nine are
recurring digits, can be written as 15 over 44 or fifteen forty-fourths.
As the digits zero and nine are recurring, the decimal can be rewritten as 0.34090909
and so on. In order to write this recurring decimal as a fraction, we’ll first let 𝑟 equal
0.34090909. Multiplying both sides of this equation by 100 gives us 100𝑟 is equal to 34.090909
and so on.
We now have two equations which we’re going to subtract. When we subtract equation two from equation one, everything to the right-hand side of
the dotted line will cancel. 100𝑟 minus 𝑟 is equal to 99𝑟 and 34.09 minus 0.34 is equal to 33.75. Dividing both sides of this equation by 99 give us 𝑟 is equal to 33.75 divided by
99. This means that the recurring decimal 0.3409 with zero nine recurring can be
rewritten as 33.75 over 99.
There are two issues with this answer. Firstly, we cannot have any decimals in a fraction. And secondly, we we’re asked to prove that the fraction was equal to fifteen
forty-fourths. One way to simplify this fraction would be to multiply the top and the bottom by
100. Remember whatever we did to the top, we must do to the bottom. This gives us 3375 over 9900 and eliminates the decimal point.
We can cancel or simplify this fraction by looking for common factors. For example, five and 25 both divide into the top and the bottom. However, the highest common factor of 3375 and 9900 is 225. This means that we can divide the numerator and the denominator by 225. Dividing the numerator by 225 gives us 15 and dividing the denominator by 225 gives
us 44. This means that the recurring decimal can be written as 15 over 44 or fifteen
We’ll now look at an alternative method to simplify the fraction 33.75 over 99. Multiplying the top and bottom by four gives us 135 over four multiplied by 99. We can split the numerator 135 into three multiplied by 45. And going one step further, we can split 45 into three multiplied by 15. On the denominator, 99 can be written as three multiplied by 33 and 33 can be written
as three multiplied by 11. We can now cancel the threes on the top and the bottom. This leaves us with 15 over four multiplied by 11. Four multiplied by 11 is equal to 44. Therefore, the fraction is 15 over 44.
There are lots of different ways to simplify a fraction. And it’s important that you find a way that works best for you.