Fractions & Decimals

What you need to know

• A fraction $\frac{p}{q}$ in lowest terms has a terminating decimal iff $q$ has no prime factors other than $2$ and $5$. The number of decimal places is $\max(a, b)$ where $q = 2^a 5^b$.
• Repeating decimals: $0.\overline{d_1 \ldots d_k} = \frac{d_1 \ldots d_k}{10^k - 1}$, e.g. $0.\overline{36} = \frac{36}{99} = \frac{4}{11}$. A mixed one such as $0.3\overline{6}$ equals $\frac{36 - 3}{90}$.
• The period of $\frac{1}{q}$ (with $\gcd(q, 10) = 1$) is the smallest $k$ with $10^k \equiv 1 \pmod q$; it divides $q - 1$ when $q$ is prime. Useful periods: $\frac{1}{3}, \frac{1}{9}$ period $1$; $\frac{1}{11}$ period $2$; $\frac{1}{7}, \frac{1}{13}$ period $6$; $\frac{1}{37}$ period $3$.
• Unit fraction identities: $\frac{1}{n} - \frac{1}{n+1} = \frac{1}{n(n+1)}$ and $\frac{1}{n} = \frac{1}{n+1} + \frac{1}{n(n+1)}$.
• Moving the decimal point $k$ places right multiplies by $10^k$.
• Every rational number is a terminating or eventually repeating decimal, and conversely.

How AMC 10 tests it

• "Which digit is in the $2026$th decimal place of $\frac{3}{7}$?" Find the period, reduce the position mod the period (problems 5–12).
• Convert a repeating decimal to a fraction, then ask for the sum of numerator and denominator in lowest terms.
• "How many fractions $\frac{n}{k}$ with $1 \le n \le N$ terminate?" Cancel primes other than $2, 5$ from $k$ (problems 10–16).
• Telescoping sums of $\frac{1}{n(n+1)}$ or $\frac{1}{(2n-1)(2n+1)}$ hidden in a word problem or a series.
• Equations involving a number and its decimal shift, such as $10^4 x = \frac{4}{x}$.

Standard approaches

1. Convert decimals to fractions immediately; arithmetic on $\frac{4}{11}$ is safer than on $0.\overline{36}$.
2. Reduce to lowest terms before deciding on termination or reading off a period.
3. For the $n$th digit, compute the repeating block, then use $n \bmod (\text{period})$, remembering remainder $0$ means the last digit of the block.
4. For telescoping, write each term as a difference of unit fractions and cancel.
5. To count terminating fractions $\frac{n}{k}$, require the "bad" part of $k$ to divide $n$.

Worked example

Let $x = 0.\overline{36}$ and $y = 0.3\overline{6}$. What is $\frac{x}{y}$?

(A) $\frac{110}{121}$ (B) $\frac{120}{121}$ (C) $1$ (D) $\frac{121}{120}$ (E) $\frac{11}{10}$

For $x$: $100x = 36.\overline{36} = 36 + x$, so $x = \frac{36}{99} = \frac{4}{11}$. For $y$: $10y = 3.\overline{6}$ and $100y = 36.\overline{6}$, so $90y = 33$ and $y = \frac{33}{90} = \frac{11}{30}$. Then

$$\frac{x}{y} = \frac{4}{11} \cdot \frac{30}{11} = \frac{120}{121}.$$

The answer is $\boxed{\textbf{(B)}\ \tfrac{120}{121}}$.

Pitfalls

• Using $10^k$ instead of $10^k - 1$ in the denominator, or mishandling a non-repeating prefix like the $3$ in $0.3\overline{6}$.
• Judging termination before reducing: $\frac{6}{30} = \frac{1}{5}$ terminates even though $30$ has the factor $3$.
• Taking the $n$th digit as position $n \bmod p$ and reading digit $0$ of the block when the remainder is $0$.
• Rounding intermediate decimals; keep exact fractions until the last step.

Traps that recur

• Counting only the nine solutions with a = b = c, or allowing b or c to be 0 (all three digits must be nonzero). (2022 AMC 10A #17)
• Forgetting that a fraction may be added to itself (such as 1/2 + 1/2 = 1), or listing a sum twice (8 arises as 4 + 4 and as 3/2 + 13/2). (2021 AMC Fall 10B #7)
• Counting the zeros in 1/10^20 only (19 or 20) and ignoring the extra factor 1/2^20, or miscounting the shift by one. (2020 AMC 10B #12)
• Writing the difference as ab/99 (forgetting the two-place shift), which gives ab = 0.75 and no valid digits. (2021 AMC 10A #8)