Question

15. Express the repeating decimal 4.61 as an exact fraction using a geometric series with 0.01 being the repeating decimal.​

Answer 1

• 4 11/18

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We have a repeating decimal 4.6(1).

Let's express it as a GP:

• 4.6(1) = 4.6 + 0.01 + 0.001 + 0.0001 + ...

Fund the sum of infinite GP, with the first term of a = 0.01 and common ratio of r = 0.1:

• S = a/(1 - r)
• S = 0.01/(1 - 0.1) = 0.01/0.9 = 1/90

Add 4.6 to the sum:

• 4.6 + 1/90 =
• 4 + 0.6 + 1/90 =
• 4 + 6/10 + 1/90 =
• 4 + 54/90 + 1/90 =
• 4 + 55/90 =
• 4 + 11/18
• 4 11/18

Hence the fraction is 4 11/18.