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Question Video: Expressing a Recurring Decimal as a Fraction Using the Sum of an Infinite Geometric Series Mathematics • 10th Grade

By finding the sum of an infinite geometric sequence, express 0.375 recurring as a common fraction.

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Video Transcript

By finding the sum of an infinite geometric sequence, express 0.375 recurring as a common fraction.

In this question, we are given a decimal with a line above the three, seven, and five. This means that these three digits repeat or recur. 0.375 recurring is equal to 0.375375375 and so on. It is also worth noting that this can be written with a dot above the first and last digits that recur. There are several ways of rewriting a recurring decimal as a fraction. In this question, we are asked to do so using our knowledge of geometric sequences. We begin by splitting 0.375 recurring as follows. It is equal to 0.375 plus 0.000375 plus 0.000000375 and so on. The right-hand side of our equation is an example of a geometric series, where the first term is 𝑎, the second term 𝑎𝑟, third term 𝑎𝑟 squared, and so on. The terms of the series could also be written as a geometric sequence with first term 𝑎 equal to 0.375.

We can calculate the value of the common ratio 𝑟 by dividing any term by the preceding one. 𝑟 is therefore equal to 0.000375 divided by 0.375. This is equal to one thousandth or 0.001. Since the absolute value of the common ratio is less than one, we can find the sum of this infinite geometric sequence. We can calculate the sum by dividing the first term 𝑎 by one minus 𝑟. Substituting in our values of 𝑎 and 𝑟, we have 0.375 divided by one minus 0.001. This is equal to 0.375 over 0.999, and this is equivalent to 375 over 999. Both the numerator and denominator of this fraction are divisible by three. So our fraction simplifies to 125 over 333. We can therefore conclude that 0.375 recurring written as a common fraction in its simplest form is 125 over 333.

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