3.2 Decimals, Place Value & Terminating Decimals

Quick Answer: A reduced fraction $\frac{p}{q}$ terminates if and only if $q = 2^a \cdot 5^b$. Any other prime in $q$ produces a repeating decimal. Convert a pure repeating block of $k$ digits by placing that block over $k$ nines. Place-value arithmetic — shifting by powers of 10 — replaces the calculator the GMAT does not allow.

Place Value and Base-10 Expansion

Every real number has a positional base-10 expansion. Digits to the left of the decimal point are nonnegative powers of 10; digits to the right are negative powers.

So $3.142 = 3 + \frac{1}{10} + \frac{4}{100} + \frac{2}{1{,}000}$. Comparing decimals is comparing these place values from the left: $0.409 > 0.41$ is false, because $0.410 > 0.409$. Trailing zeros after the decimal point do not change value ($0.41 = 0.410$), but they do change how many decimal digits you count when multiplying.

Decimal Arithmetic Without a Calculator

Multiplying decimals

1. Count the total number of digits to the right of the decimal point across all factors.
2. Multiply the numbers as integers, ignoring the points.
3. Place the decimal point that many positions from the right of the product.

Worked example. $(0.004) \times (0.07) = 4 \times 10^{-3} \times 7 \times 10^{-2} = 28 \times 10^{-5} = 0.00028$. Four thousandths times seven hundredths is twenty-eight hundred-thousandths — three zeros after the decimal point, then 28, which displays as $0.00028$.

A product of a number with $m$ decimal digits and a number with $n$ decimal digits has $m + n$ decimal digits before any trailing zeros are dropped. $1.5 \times 0.2 = 0.30 = 0.3$, but the unreduced placement still used $1 + 1 = 2$ digits.

Dividing decimals

Shift the decimal point in both the divisor and the dividend by the same number of places until the divisor is an integer. That is multiplication of numerator and denominator by the same power of 10, so the quotient is unchanged.

$$\frac{0.0054}{0.018} = \frac{5.4}{18} = \frac{54}{180} = \frac{3}{10} = 0.3.$$

Multiplying a decimal by $10^n$ moves the point $n$ places right; dividing by $10^n$ moves it $n$ places left. That shift is the entire content of scientific-notation items that stay inside arithmetic.

Terminating vs Repeating Decimals

Terminating Decimal Theorem. Let $\frac{p}{q}$ be in lowest terms, with $q > 0$. The decimal expansion of $\frac{p}{q}$ terminates if and only if the prime factorization of $q$ contains no prime other than $2$ and $5$:

$$q = 2^a \cdot 5^b \quad (a, b \ge 0).$$

If reduced $q$ still contains $3$, $7$, $11$, $13$, or any other prime, the expansion is non-terminating and repeating.

Number of decimal digits

When $\frac{p}{q}$ terminates with $q = 2^a \cdot 5^b$, the number of digits after the decimal point is $\max(a, b)$. Pair extra $2$s or $5$s into the numerator until the denominator is a power of $10$:

$$\frac{7}{2^4 \times 5^2} = \frac{7 \times 5^2}{2^4 \times 5^4} = \frac{175}{10^4} = 0.0175,$$

which has $\max(4, 2) = 4$ decimal digits.

The reduction trap

Always cancel before inspecting $q$. For $\frac{21}{150}$, the unreduced denominator $150 = 2 \times 3 \times 5^2$ appears to contain a $3$, but $\frac{21}{150} = \frac{7}{50}$ and $50 = 2 \times 5^2$, so the decimal terminates ($0.14$). Conversely, $\frac{2}{15}$ does not cancel the $3$, so $0.1\overline{3}$ repeats.

Converting Repeating Decimals to Fractions

Repeating decimals are rationals. Bar notation marks the repeating block: $0.\overline{3} = 0.333\ldots$ and $0.\overline{27} = 0.272727\ldots$.

Pure repeating decimals, including $0.\overline{abc}$

A pure repeating decimal with a $k$-digit block equals that block over $k$ nines:

$$0.\overline{d_1 d_2 \ldots d_k} = \frac{\text{the } k\text{-digit integer}}{10^k - 1}.$$

• One digit: $0.\overline{7} = \frac{7}{9}$.
• Two digits: $0.\overline{45} = \frac{45}{99} = \frac{5}{11}$.
• Three digits ($0.\overline{abc}$): $0.\overline{142} = \frac{142}{999}$.

Derivation for a three-digit repeat. Let $x = 0.\overline{abc} = 0.abcabcabc\ldots$. Then $1000x = abc.\overline{abc}$. Subtract:

$$1000x - x = abc \implies 999x = abc \implies x = \frac{abc}{999}.$$

Reduce if possible. For $0.\overline{123}$, $\frac{123}{999} = \frac{41}{333}$. The same subtraction with $10^k$ produces $k$ nines in the denominator for any block length.

Mixed repeating decimals

If non-repeating digits sit before the repeating block, align two multiples of $x$ so the repeating tails cancel.

Worked example. Convert $x = 0.1\overline{6} = 0.1666\ldots$.

• $10x = 1.666\ldots$
• $100x = 16.666\ldots$
• $100x - 10x = 16.666\ldots - 1.666\ldots = 15$, so $90x = 15$ and $x = \frac{15}{90} = \frac{1}{6}$.

The first shift ($10x$) moves the point just past the non-repeating digit; the second shift ($100x$) moves it past one full repeating block. Subtraction kills the infinite tail.

Common Fraction-to-Decimal Reference

Memorizing the short list below removes a surprising amount of arithmetic on Problem Solving.

Scientific Notation

The official Reference Sheet groups scientific notation with arithmetic and decimals, and it is pure place-value bookkeeping. A number in scientific notation is written $A \times 10^n$ with exactly one nonzero digit to the left of $A$'s decimal point.

• To convert $A \times 10^n$ back to ordinary decimal form, move $A$'s decimal point $n$ places right when $n$ is positive and $|n|$ places left when $n$ is negative.
• To convert a decimal into scientific notation, move the point until exactly one nonzero digit remains on its left. If you moved left by $n$ places, multiply by $10^n$; if you moved right by $n$ places, multiply by $10^{-n}$.

Worked examples. $3.6 \times 10^4 = 36{,}000$. $5.02 \times 10^{-3} = 0.00502$. Going the other way, $0.00047 = 4.7 \times 10^{-4}$ because the point moved four places right, and $284{,}000 = 2.84 \times 10^5$ because it moved five places left.

On a no-calculator section the payoff is multiplication and division of very large or very small numbers: separate the leading factors from the powers of ten and use the exponent laws from Chapter 4. For instance, $(6 \times 10^{8}) \div (2 \times 10^{5}) = 3 \times 10^{3} = 3{,}000$, and $(4 \times 10^{-3})(2.5 \times 10^{6}) = 10 \times 10^{3} = 1 \times 10^{4}$ — note the renormalization step, since $10 \times 10^3$ is not yet in scientific form.

Rounding on the GMAT follows the usual half-up rule at the stated place, but exact values are preferred whenever an option is a fraction. Convert repeating decimals to fractions before adding or multiplying; adding $0.333\ldots + 0.666\ldots$ as truncated decimals is how $1$ becomes $0.999$ on scratch paper.