Shift twice

The idea

A general repeating decimal has non-repeating digits before its repeating block; multiply x by 10ᵐ (m non-repeating digits), then by 10ⁿ (n repeating digits), subtract the two equations and solve for x.

A digit in the way

The decimal 0.1666… has a 1 that does not repeat, followed by 6s that do. Does the shift-and-subtract trick still work?

What do you get when you multiply 0.1666… by 10?

• 1.666…
• 0.1666…
• 10.666…

Two shifts

First shift so that the repeating part starts right after the point. Then shift by one more block. Subtract and solve.

10^m and 10^n

A general repeating decimal has non-repeating digits before its repeating block. Let x be the decimal, multiply by 10^m (m = number of non-repeating digits), then by 10^n (n = number of repeating digits), subtract the two equations and solve for x.

For x = 0.1666…: m = 1 and n = 1. 10x = 1.666… and 100x = 16.666…, so 100x − 10x = 90x = 15. Then x is 15/90, which is 1/6.

Notes

General repeating decimal: let x be the decimal, multiply by 10^m (m = non-repeating digits), then by 10^n (n = repeating digits), subtract, and solve for x.

Check yourself

x = 0.8333…: 10x = 8.333… and 100x = 83.333…. Subtract: 100x − 10x = 90x = ?

Answer: 75

83.333… − 8.333… = 75, so 90x = 75.

x = 2.35777…: the digits 35 do not repeat, so m = 2. What is 10^m?

Answer: 100

10² = 100. Multiplying by 100 moves the two non-repeating digits to the left of the point.

In x = 0.1666…, how many digits are non-repeating (m) and how many repeat (n)?

From 90x = 75 we get x = 75/90. In lowest terms, what is the denominator?

Answer: 6

75/90 = 5/6, so the denominator is 6.

• m = 0 and n = 2. The digit 1 comes only once, so it does not repeat.
• m = 1 and n = 1 — correct. Yes. The digit 1 does not repeat and the digit 6 repeats.
• m = 2 and n = 1. Only the digit 1 comes before the repeating 6.