Express 0.75 recurring as a rational number in its simplest form.
We recall that this bar notation tells us that the digits seven and five recur or repeat. So 0.75 recurring is 0.757575 and so on. The question is telling us to express this as a rational number. Well, that’s in a fraction form. We say that a rational number is in the form of 𝑎 over 𝑏 where 𝑎 and 𝑏 are integers. They’re whole numbers. So how do we perform this process? Well, we begin by defining an unknown? We’ll call it 𝑥, and it’s equal to our decimal. So 𝑥 here is 0.75 recurring. Now, we can either leave it in this notation or express it as 0.757575 and so on. Leaving it in this form can make the next steps a little easier.
Our job is to find two decimals where the digits after the decimal point are identical. Now, the seven and the five are both recurring. And so if we multiply 0.75 recurring by 100, we move the digits to the left two places. And we get another number where the digits after the decimal point are 75 recurring. In fact, we get 75.7575 and so on. Now, when we multiply the left-hand side of this equation by 100, we get 100𝑥.
Our next job is to eliminate that recurring bit after the decimal. Well, what happens here is if we subtract 0.75 recurring from 75.75 recurring, the bit after the decimal point becomes zero. And so 75.75 recurring minus 0.75 recurring is just 75. 100𝑥 take away 𝑥 is 99𝑥. So we have an equation 99𝑥 equals 75.
Our last job is to solve this equation for 𝑥. And so we divide through by 99. 99𝑥 divided by 99 is 𝑥. And 75 divided by 99 we can write as 75 over 99. But be careful, the question wants us to write our answer in its simplest form. So we notice that both 75 and 99 are multiples of three. And we divide both our numerator and denominator by three. 75 divided by three is 25 and 99 divided by three is 33. So we found 𝑥 is equal to 25 over 33. But remember, we defined 𝑥 to be equal to 0.75 recurring. So in fact, we’ve shown that 0.75 recurring is equal to 25 over 33.