Fractions & Decimals

What you need to know

• Which decimals end. Put $\frac{p}{q}$ in lowest terms first. Its decimal ends (terminates) exactly when $q$ has no prime factors except $2$ and $5$. If $q = 2^a 5^b$, the number of decimal places is the larger of $a$ and $b$. Example: $\frac{3}{40} = 0.075$ has $3$ places, since $40 = 2^3\cdot5$.
• Repeating decimals. A bar means the digits repeat forever. If a block of $k$ digits repeats right after the decimal point, the fraction is the block over $k$ nines, that is, over $10^k-1$. Example: $0.\overline{36} = \frac{36}{99} = \frac{4}{11}$.
• Digits before the repeat. For $0.3\overline{6} = 0.3666\ldots$, the numerator is the digits through the first block ($36$) minus the digits before the repeat ($3$). The denominator has a $9$ per repeating digit, then a $0$ per digit before the repeat. Example: $0.3\overline{6} = \frac{36-3}{90} = \frac{11}{30}$.
• Length of the repeat. For $\frac1q$ with $q$ sharing no factor with $10$, the repeating block's length (the period) is the smallest $k$ for which $10^k$ leaves remainder $1$ when divided by $q$, written $10^k \equiv 1 \pmod q$. When $q$ is prime, the period divides $q-1$. Example: $\frac17 = 0.\overline{142857}$ has period $6$, which divides $7-1$.

Periods worth knowing:

• Unit fractions. $\frac1n-\frac1{n+1} = \frac{1}{n(n+1)}$, so $\frac1n = \frac1{n+1}+\frac{1}{n(n+1)}$. Example: $\frac13-\frac14 = \frac1{12}$.
• Moving the decimal point $k$ places right multiplies by $10^k$. Example: $0.0375$ becomes $3.75$, $100$ times as much.
• Fractions and decimals match. Every rational number (a fraction of two integers) has a decimal that ends or, from some point on, repeats; and every such decimal is a fraction. Example: $\frac16 = 0.1\overline{6}$, while $\sqrt2 = 1.41421\ldots$ never repeats, so it is not a fraction.

How AMC 10 tests it

• "Which digit is in the $2026$th decimal place of $\frac{3}{7}$?" Find the period, then the remainder of the position divided by the period (problems 5–12).
• Convert a repeating decimal to a fraction in lowest terms, then give the sum of the numerator and denominator.
• "How many fractions $\frac{n}{k}$ with $1 \le n \le N$ terminate?" Cancel the primes other than $2$ and $5$ 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 about a number and its decimal point moved, such as $10^4 x = \frac{4}{x}$.

Standard approaches

1. Change decimals to fractions at once; $\frac{4}{11}$ is safer to work with than $0.\overline{36}$.
2. Reduce to lowest terms before judging whether the decimal ends or reading off a period.
3. For the $n$th digit, find the block, then divide $n$ by the period. The remainder is the position in the block, and remainder $0$ means the last digit. Example: $\frac37 = 0.\overline{428571}$ and $2026 = 6\cdot337+4$, so the $2026$th digit is the $4$th one, $5$.
4. For telescoping, split each term into two unit fractions so the middle terms cancel: $\frac{1}{n(n+1)} = \frac1n-\frac1{n+1}$ and $\frac{1}{(2n-1)(2n+1)} = \frac12\left(\frac{1}{2n-1}-\frac{1}{2n+1}\right)$.
5. To count terminating $\frac{n}{k}$, divide all the $2$s and $5$s out of $k$; the decimal ends exactly when what is left divides $n$. Example: $120 = 2^3\cdot3\cdot5$ leaves $3$, so $\frac{n}{120}$ ends exactly when $3$ divides $n$.

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) $\frac{11}{10}$

Step 1: find $x$. $100x = 36.\overline{36}$, which is $36+x$. So $99x = 36$ and $x = \frac{36}{99} = \frac{4}{11}$.

Step 2: find $y$. $10y = 3.666\ldots$ and $100y = 36.666\ldots$ have the same digits after the point. Subtracting, $90y = 33$, so $y = \frac{33}{90} = \frac{11}{30}$.

Step 3: divide. Dividing by $\frac{11}{30}$ means multiplying by $\frac{30}{11}$:
$$\frac{x}{y} = \frac{4}{11} \cdot \frac{30}{11} = \frac{120}{121}.$$

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

Pitfalls

• Using $10^k$ instead of $10^k-1$: $0.\overline{36}$ is $\frac{36}{99}$, not $\frac{36}{100}$. Also watch digits before the repeat: $0.3\overline{6}$ is $\frac{33}{90}$, not $\frac{36}{99}$.
• Judging whether the decimal ends before reducing. Example: $\frac{6}{30} = \frac15 = 0.2$ ends, though $30$ has a factor of $3$.
• Reading the first digit of the block when the remainder is $0$; it should be the last. Example: the $6$th digit of $\frac37 = 0.\overline{428571}$ is $1$, not $4$.
• Rounding decimals partway through. 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)