Comparing Rational Numbers - ISEE Middle Level Quantitative Reasoning

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

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

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

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

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

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

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

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

5. Which number is greater: $1\frac{1}{3}$ or $\frac{4}{3}$?

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

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

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

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

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

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

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

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

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

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

    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.

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

    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.

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

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

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

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

14. What is the rule for comparing any negative number to any positive number?

    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.

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

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

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

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

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

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

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

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

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

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

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

    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.

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

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

22. What is the rule for comparing negative numbers $-x$ and $-y$ where $x$ and $y$ are positive?

    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.

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

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

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

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

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

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

Study these flashcards