Comparing Fractions - ISEE Lower Level: Quantitative Reasoning

$\frac{12}{25}$. Cross-multiplying shows $12 \times 11 = 132 > 5 \times 25 = 125$, confirming $\frac{12}{25}$ is greater.

$\frac{15}{16}$. Cross-multiplying results in $8 \times 16 = 128 < 15 \times 9 = 135$, confirming $\frac{15}{16}$ is greater.

$\frac{7}{20}$. Converting to twentieths yields $\frac{3}{10} = \frac{6}{20} < \frac{7}{20}$, or cross-multiplying $3 \times 20 = 60 < 7 \times 10 = 70$.

$\frac{1}{4}$. Cross-multiplying gives $2 \times 4 = 8 < 1 \times 9 = 9$, indicating $\frac{1}{4}$ is greater.

$\frac{7}{15}$. Cross-multiplying results in $7 \times 20 = 140 > 9 \times 15 = 135$, indicating $\frac{7}{15}$ is greater.

$\frac{1}{2}$. Cross-multiplying yields $10 \times 2 = 20 < 1 \times 21 = 21$, so $\frac{1}{2}$ is greater.

$\frac{5}{7}$. Cross-multiplying gives $9 \times 7 = 63 < 5 \times 14 = 70$, confirming $\frac{5}{7}$ is greater.

$\frac{3}{7}$. Converting to fourteenths gives $\frac{3}{7} = \frac{6}{14} > \frac{5}{14}$, or cross-multiplying $5 \times 7 = 35 < 3 \times 14 = 42$.

$\frac{1}{2}$. Cross-multiplying yields $1 \times 9 = 9 > 4 \times 2 = 8$, so $\frac{1}{2}$ is greater.

$\frac{6}{11}$. Cross-multiplying shows $6 \times 13 = 78 > 7 \times 11 = 77$, indicating $\frac{6}{11}$ is greater.

$\frac{4}{9}$. With the same numerator, $\frac{4}{9}$ is greater because its smaller denominator divides the whole into fewer parts, making each part larger.

$\frac{5}{7}$. With the same denominator, $\frac{5}{7}$ is greater because its numerator is larger, representing more sevenths.

$\frac{5}{8}$. Cross-multiplying yields $7 \times 8 = 56 < 5 \times 12 = 60$, confirming $\frac{5}{8}$ is greater.

$\frac{2}{3}$. Cross-multiplying gives $2 \times 5 = 10 > 3 \times 3 = 9$, so $\frac{2}{3}$ is greater.

Cross-multiply: compare $ad$ and $bc$. This method determines which fraction is larger by comparing the cross products $ad$ and $bc$, where if $ad > bc$ then $\frac{a}{b} > \frac{c}{d}$ for positive denominators.

Greater numerator gives the greater fraction. When denominators are identical and positive, the fraction with the larger numerator represents a greater value since it takes a bigger portion of the same whole.

Smaller denominator gives the greater fraction. When numerators are identical and denominators positive, the fraction with the smaller denominator is larger because the whole is divided into fewer parts.

$\frac{6}{7}$. Cross-multiplying shows $5 \times 7 = 35 < 6 \times 6 = 36$, so $\frac{6}{7}$ is greater.

$\frac{9}{10}$. Converting to twentieths gives $\frac{9}{10} = \frac{18}{20} > \frac{17}{20}$, or cross-multiplying $9 \times 20 = 180 > 17 \times 10 = 170$.

$\frac{3}{4}$. Cross-multiplying results in $11 \times 4 = 44 < 3 \times 15 = 45$, indicating $\frac{3}{4}$ is greater.

$\frac{7}{9}$. Cross-multiplying gives $13 \times 9 = 117 < 7 \times 18 = 126$, so $\frac{7}{9}$ is greater.

$\frac{13}{15}$. Cross-multiplying shows $4 \times 15 = 60 < 13 \times 5 = 65$, so $\frac{13}{15}$ is greater.

$\frac{5}{12}$. Cross-multiplying yields $5 \times 8 = 40 > 3 \times 12 = 36$, confirming $\frac{5}{12}$ is greater.