QUANTITATIVE REASONING · GMAT-Q-FR-01

Fractions, Decimals and Rational Operations

GMAT lesson development preview · DRAFT · Original learning and assessment material

Concepts and problem-solving strategy

\frac{a}{b} = a \div b, b \neq = 0

\frac{a}{b} + \frac{c}{d} = \frac{a d + b c}{b d}

\frac{a}{b} > \frac{c}{d} ⟺ a d > b c (b, d > 0)

\frac{8}{15} \cdot \frac{35}{72} = \frac{7}{27}

\frac{\frac{3}{4}}{\frac{9}{10}} = \frac{3}{4} \cdot \frac{10}{9} = \frac{5}{6}

\frac{1}{8} = 0.125 = 12.5%, \frac{3}{8} = 0.375 = 37.5%

x = 0. 27 ⟹ 99 x = 27 ⟹ x = \frac{3}{11}

\frac{49}{101} < \frac{1}{2} < \frac{51}{101}

\frac{a + b}{c} = \frac{a}{c} + \frac{b}{c}, \frac{a}{b + c} \neq = \frac{a}{b} + \frac{a}{c}

Before calculating, inspect denominators, convert units consistently, and look for a faster comparison or simplification.

Compare fractions by cross-multiplication only when denominators are positive.

Cancel common factors before multiplying fractions.

Convert repeating decimals into rational numbers using an equation.

When fractions have an awkward denominator, compare them with one-half or another familiar benchmark.

For algebraic-looking relative quantities, consider substituting smart values that preserve every constraint.

Treat a fraction bar as parentheses around the entire numerator and the entire denominator.

Evaluate $\frac{3}{4} + \frac{5}{6}$ .

\frac{3}{4} + \frac{5}{6} = \frac{9}{12} + \frac{10}{12} = \frac{19}{12}

Use the least common denominator 12.

Compare $\frac{7}{12}$ and $\frac{5}{9}$ .

7 \cdot 9 = 63 > 60 = 5 \cdot 12

Cross-products show 7/12 is larger.

Calculate $(\frac{8}{15}) \times (\frac{35}{72})$ without large multiplication.

\frac{8}{15} \cdot \frac{35}{72} = \frac{1 \cdot 7}{3 \cdot 9} = \frac{7}{27}

Cancel across numerator and denominator before multiplication: 8 with 72, and 35 with 15.

Convert 0.272727... into a fraction.

x = 0. 27, 100 x - x = 27, x = \frac{3}{11}

Multiplying by 100 aligns the repeating block; subtract to eliminate the repeating tail.

Estimate whether $\frac{49}{101}$ is above or below one-half.

2 (49) = 98 < 101 ⟹ \frac{49}{101} < \frac{1}{2}

Compare to the benchmark 1/2 without long division.

Why is a/(b+c) not generally equal to a/b+a/c?

\frac{6}{2 + 4} = 1, \frac{6}{2} + \frac{6}{4} = 4.5

A denominator sum is one entire grouping; never distribute division over a sum.

Do not add denominators when adding fractions.
A negative denominator changes the direction of a comparison after multiplication.
Finite decimals correspond to rational numbers whose reduced denominators have only factors 2 and 5.
Cross-cancel factors in multiplication only, never across addition terms.
A fraction a/(b+c) cannot be split into a/b+a/c.
When choosing smart numbers, avoid values that accidentally make two distinct answer expressions equal.