ISEE Middle Level Quantitative Reasoning Flashcards: Comparing Rational Numbers

What this deck covers

This deck focuses on Comparing Rational Numbers, giving you a quick way to review the definitions, rules, and examples that matter most for ISEE Middle Level Quantitative Reasoning.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: What is the rule for comparing two positive fractions $\frac{a}{b}$ and $\frac{c}{d}$ using cross products?

Answer: Compare $ad$ and $bc$; if $ad>bc$ then $\frac{a}{b}>\frac{c}{d}$. Cross-multiplying compares $a/b$ and $c/d$ by checking if $a \cdot d > b \cdot c$, preserving the inequality for positive denominators.

Flashcard 2: What is the rule for comparing $\frac{a}{b}$ and $\frac{c}{d}$ when $b$ and $d$ are the same positive number?

Answer: Compare numerators: if $a>c$ then $\frac{a}{b}>\frac{c}{b}$. With identical positive denominators, the fraction with the larger numerator represents a greater value since it divides a bigger quantity by the same amount.

Flashcard 3: What is the rule for comparing $\frac{a}{b}$ and $\frac{a}{d}$ when the numerators match and denominators are positive?

Answer: Smaller denominator gives larger value: if $b<d$ then $\frac{a}{b}>\frac{a}{d}$. For positive denominators and identical numerators, a smaller denominator results in a larger fraction as it divides the numerator into fewer parts.

Flashcard 4: What is the rule for comparing any negative number to any positive number?

Answer: Any negative number is less than any positive number. Negative numbers lie to the left of positive numbers on the number line, making them smaller regardless of magnitude.

Flashcard 5: What is the rule for comparing negative numbers $-x$ and $-y$ where $x$ and $y$ are positive?

Answer: Reverse the comparison: if $x>y$ then $-x<-y$. Multiplying both sides of an inequality by -1 reverses the direction, so the negative of a larger positive is smaller.

Flashcard 6: Which symbol makes the statement true: $0.7\ \Box\ \frac{2}{3}$?

Answer: $>$. Converting to decimals shows $0.7 > 0.666\ldots$, as $2/3 \approx 0.666\ldots$.

Flashcard 7: Which symbol makes the statement true: $\frac{5}{8}\ \Box\ 0.6$?

Answer: $>$. Converting $5/8 = 0.625$ reveals it exceeds $0.6$ by $0.025$.

Flashcard 8: Which symbol makes the statement true: $-0.4\ \Box\ -\frac{3}{8}$?

Answer: $<$. On the number line, $-0.4$ is left of $-0.375$, confirming $-0.4 < -3/8$.

Flashcard 9: Which symbol makes the statement true: $\frac{7}{10}\ \Box\ \frac{2}{3}$?

Answer: $>$. Cross-multiplying gives $7 \cdot 3 = 21 > 2 \cdot 10 = 20$, so $7/10 > 2/3$.

Flashcard 10: Which symbol makes the statement true: $\frac{3}{5}\ \Box\ 0.62$?

Answer: $<$. Converting $3/5 = 0.6$ shows it is less than $0.62$ by $0.02$.

Flashcard 11: Which symbol makes the statement true: $1.25\ \Box\ \frac{5}{4}$?

Answer: $=$. Both express the same value, as $5/4 = 1.25$.

Flashcard 12: Which symbol makes the statement true: $-\frac{9}{4}\ \Box\ -2.2$?

Answer: $<$. Converting $-9/4 = -2.25$ places it left of $-2.2$ on the number line.

Flashcard 13: Which symbol makes the statement true: $\frac{11}{20}\ \Box\ 0.55$?

Answer: $=$. Dividing $11 \div 20 = 0.55$ confirms equivalence.

Flashcard 14: Which symbol makes the statement true: $0.333\ldots\ \Box\ \frac{1}{3}$?

Answer: $=$. The repeating decimal $0.333\ldots$ is the exact decimal form of $1/3$.

Flashcard 15: Which symbol makes the statement true: $0.666\ldots\ \Box\ \frac{2}{3}$?

Answer: $=$. The repeating decimal $0.666\ldots$ equals $2/3$ precisely.

Flashcard 16: Which number is greater: $\frac{4}{9}$ or $0.45$?

Answer: $0.45$. Since $4/9 \approx 0.444 < 0.45$, the decimal is larger.

Flashcard 17: Which number is greater: $\frac{13}{25}$ or $0.5$?

Answer: $\frac{13}{25}$. Converting $13/25 = 0.52$ shows it exceeds $0.5$ by $0.02$.

Flashcard 18: Which number is greater: $-\frac{5}{6}$ or $-0.8$?

Answer: $-0.8$. Comparing $-5/6 \approx -0.833 < -0.8$, the less negative value is greater.

Flashcard 19: Which number is greater: $\frac{3}{4}$ or $\frac{5}{7}$?

Answer: $\frac{3}{4}$. Cross-multiplying yields $3 \cdot 7 = 21 > 5 \cdot 4 = 20$, confirming $3/4 > 5/7$.

Flashcard 20: Which number is greater: $1\frac{1}{3}$ or $\frac{4}{3}$?

Answer: $=$. Converting $1\frac{1}{3} = 4/3$ shows both are identical.

Flashcard 21: Which number is greater: $2.05$ or $2\frac{1}{20}$?

Answer: $=$. Converting $2\frac{1}{20} = 2.05$ confirms equivalence.

Flashcard 22: Which symbol makes the statement true: $\frac{3}{8}\ \Box\ 0.375$?

Answer: $=$. Dividing $3 \div 8 = 0.375$ verifies exact equality.

Flashcard 23: Which symbol makes the statement true: $\frac{9}{16}\ \Box\ 0.56$?

Answer: $>$. Converting $9/16 = 0.5625$ exceeds $0.56$ slightly.

Flashcard 24: Which symbol makes the statement true: $-\frac{7}{5}\ \Box\ -1.35$?

Answer: $<$. Since $-7/5 = -1.4 < -1.35$, the more negative value is smaller.

Flashcard 25: Which number is least: $0.2$, $\frac{1}{6}$, or $0.18$?

Answer: $\frac{1}{6}$. Comparing decimals: $1/6 \approx 0.1667 < 0.18 < 0.2$, making $1/6$ the smallest.