Which of the following fractions has a recurring decimal expansion? Is it (A) one-half, (B) three-quarters, (C) three-fifths, (D) three-eighths, or (E) eight thirteenths?
We know that the line in a fraction means divide. So, to convert any fraction of the type 𝑎 over 𝑏, we need to divide 𝑎, the numerator, by 𝑏, the denominator. In this question, we could divide one by two, three by four, and so on. We could do this using the short division bus stop method. We are trying to find a recurring decimal, and a recurring decimal exists when a decimal number repeats forever.
Our first two fractions, one-half and three-quarters, are common fractions that we should know the conversion of. One-half is equal to 0.5 as one divided by two is 0.5. As previously mentioned, we could prove this using the bus stop method. Two does not divide into one. So, we carry a one to the tenths column. 10 divided by two is five, giving us an answer of 0.5. One-quarter is a half of a half. It is equal to 0.25. This means that three-quarters is equal to 0.75. Both of these decimals terminate. This means that they do not repeat forever or recur. We can, therefore, eliminate option (A) and option (B).
Three-fifths is equivalent to the fraction six-tenths as we can multiply the numerator and denominator by two. Six divided by 10 is equal to 0.6. The fraction three-fifths is, therefore, equal to 0.6 and once again is not a recurring decimal. We can split a shape into eighths by splitting quarters in half. This means that three-eighths is equal to one-quarter plus one-eighth. We know that a quarter is equal to 0.25 and one-eighth is half of this. So, it is equal to 0.125. Adding these gives us 0.375. This is the decimal that is equal to three-eighths.
As this is also not a recurring decimal, it suggests that our correct answer is (E) eight thirteenths. Eight thirteenths is more difficult to turn or convert into a decimal. We will do this by dividing eight by 13 using the short division bus stop method. We begin by carrying the eight to the tenth column. 80 divided by 13 is equal to six remainder two. Dividing 20 by 13 gives us one remainder seven. 70 divided by 13 is equal to five remainder five. Dividing 50 by 13 gives us three remainder 11. 110 divided by 13 is equal to eight remainder six. 60 divided by 13 is equal to four remainder eight.
At this stage, that doesn’t appear to be an obvious pattern. However, our next step is to divide 80 by 13. We already know that this is equal to six remainder two. The pattern, therefore, continues, as shown. The digits six, one, five, three, eight, four repeat. We can, therefore, conclude that the fraction eight thirteenths is equal to 0.615384 recurring. This is denoted by a bar or dot above the digits. It is important that the bar extends from the first repeating number to the last one.
The fraction that has a recurring decimal expansion is eight thirteenths, option (E).