Rational Numbers

Q1: Which of the two rational numbers is greater in each of the following pairs?

i. \(\frac{3}{-7}\) or \(\frac{1}{7}\)

Step 1: \(\frac{3}{-7} = -\frac{3}{7}\) (negative), \(\frac{1}{7}\) (positive).
Step 2: Positive numbers are always greater than negative numbers.
Answer: \(\frac{1}{7}\) is greater.

ii. \(\frac{11}{-18}\) or \(\frac{-5}{18}\)

Step 1: \(\frac{11}{-18}\) is negative, \(\frac{-5}{18}\) is negative.
Step 2: -5 is greater than -11.
Answer: \(\frac{-5}{18}\) is greater.

iii. \(\frac{7}{10}\) or \(\frac{-9}{10}\)

Step 1: \(\frac{7}{10}\) is positive, \(\frac{-9}{10}\) is negative.
Step 2: Positive number is greater.
Answer: \(\frac{7}{10}\) is greater.

iv. \(0\) or \(\frac{-3}{4}\)

Step 1: \(0\) is greater than any negative number.
Answer: 0 is greater.

v. \(\frac{1}{12}\) or \(0\)

Step 1: \(\frac{1}{12}\) is positive, \(0\) is neutral.
Answer: \(\frac{1}{12}\) is greater.

vi. \(\frac{18}{-19}\) or \(0\)

Step 1: \(\frac{18}{-19} = -\frac{18}{19}\) is negative.
Answer: 0 is greater.

vii. \(\frac{7}{8}\) or \(\frac{11}{16}\)

Step 1: Find common denominator: 16.
\(\frac{7}{8} = \frac{7 \times 2}{8 \times 2} = \frac{14}{16}\)
Step 2: Compare numerators: \(14 > 11\).
Answer: \(\frac{7}{8}\) is greater.

viii. \(\frac{11}{-12}\) or \(\frac{-10}{11}\)

Step 1: \(\frac{11}{-12} = -\frac{11}{12}\), \(\frac{-10}{11}\) is also negative.
Step 2: Convert to decimals:
\(-\frac{11}{12} \approx -0.9167\), \(-\frac{10}{11} \approx -0.9091\)
Step 3: Greater is the one closer to zero:
\(-0.9091 > -0.9167\)
Answer: \(\frac{-10}{11}\) is greater.

ix. \(\frac{-13}{5}\) or \(-4\)

Step 1: Convert to decimals:
\(\frac{-13}{5} = -2.6\), \(-4\) is \(-4.0\)
Step 2: \(-2.6 > -4\)
Answer: \(\frac{-13}{5}\) is greater.

x. \(\frac{17}{-6}\) or \(\frac{-13}{4}\)

Step 1: \(\frac{17}{-6} = -\frac{17}{6} \approx -2.8333\)
\(\frac{-13}{4} = -3.25\)
Step 2: Compare decimals:
\(-2.8333 > -3.25\)
Answer: \(\frac{17}{-6}\) is greater.

xi. \(\frac{7}{-9}\) or \(\frac{-5}{8}\)

Step 1: \(\frac{7}{-9} = -\frac{7}{9} \approx -0.7778\)
\(\frac{-5}{8} = -0.625\)
Step 2: Compare decimals:
\(-0.625 > -0.7778\)
Answer: \(\frac{-5}{8}\) is greater.

xii. \(\frac{-3}{-8}\) or \(\frac{5}{9}\)

Step 1: \(\frac{-3}{-8} = \frac{3}{8} = 0.375\)
\(\frac{5}{9} \approx 0.5556\)
Step 2: Compare decimals:
\(0.5556 > 0.375\)
Answer: \(\frac{5}{9}\) is greater.

Q2: Fill in the blanks with the correct symbol out of >, = or <:

i. \(\frac{-17}{4}\) ___ \(\frac{-15}{4}\)

Step 1: Both fractions have the same denominator \(4\). Compare numerators:
\(-17 < -15\)
Answer: \(\frac{-17}{4} < \frac{-15}{4}\)

ii. \(0\) ___ \(\frac{-1}{-2}\)

Step 1: \(\frac{-1}{-2} = \frac{1}{2} = 0.5\), which is positive.
Answer: \(0 < \frac{-1}{-2}\)

iii. \(\frac{4}{-3}\) ___ \(\frac{-8}{7}\)

Step 1: Convert both to decimals:
\(\frac{4}{-3} = -\frac{4}{3} \approx -1.333\)
\(\frac{-8}{7} \approx -1.1429\)
Step 2: Compare decimals:
\(-1.333 < -1.1429\)
Answer: \(\frac{4}{-3} < \frac{-8}{7}\)

iv. \(\frac{-5}{12}\) ___ \(\frac{7}{-16}\)

Step 1: Convert both to decimals:
\(\frac{-5}{12} \approx -0.4167\)
\(\frac{7}{-16} = -\frac{7}{16} \approx -0.4375\)
Step 2: Compare decimals:
\(-0.4167 > -0.4375\)
Answer: \(\frac{-5}{12} > \frac{7}{-16}\)

v. \(\frac{-7}{8}\) ___ \(\frac{-8}{9}\)

Step 1: Convert to decimals:
\(\frac{-7}{8} = -0.875\)
\(\frac{-8}{9} \approx -0.8889\)
Step 2: Compare decimals:
\(-0.875 > -0.8889\)
Answer: \(\frac{-7}{8} > \frac{-8}{9}\)

vi. \(\frac{1}{-10}\) ___ \(\frac{-4}{-5}\)

Step 1: Convert to decimals:
\(\frac{1}{-10} = -0.1\)
\(\frac{-4}{-5} = \frac{4}{5} = 0.8\)
Step 2: Compare decimals:
\(-0.1 < 0.8\)
Answer: \(\frac{1}{-10} < \frac{-4}{-5}\)

Q3: Arrange the following rational numbers in ascending order:

i. \(\frac{3}{4},\ \frac{5}{8},\ \frac{11}{16},\ \frac{21}{32}\)

Step 1: Find LCM of denominators \(4, 8, 16, 32\).
LCM(4, 8, 16, 32) = 32.
Step 2: Express each fraction with denominator 32: \[ \frac{3}{4} = \frac{24}{32}, \\ \frac{5}{8} = \frac{20}{32}, \\ \frac{11}{16} = \frac{22}{32}, \\ \frac{21}{32} = \frac{21}{32} \] Step 3: Compare numerators:
\(20 < 21 < 22 < 24\)
Answer: \(\frac{5}{8} < \frac{21}{32} < \frac{11}{16} < \frac{3}{4}\)

ii. \(\frac{-2}{5},\ \frac{7}{-10},\ \frac{-8}{15},\ \frac{17}{-30}\)

Step 1: Write all fractions with negative signs in numerator: \[ \frac{-2}{5}, \quad \frac{-7}{10}, \quad \frac{-8}{15}, \quad \frac{-17}{30} \] Step 2: Find LCM of denominators \(5, 10, 15, 30\).
LCM = 30.
Step 3: Express each fraction with denominator 30: \[ \frac{-12}{30}, \quad \frac{-21}{30}, \quad \frac{-16}{30}, \quad \frac{-17}{30} \] Step 4: Arrange numerators ascending:
\(-21 < -17 < -16 < -12\)
Answer: \(\frac{-7}{10} < \frac{-17}{30} < \frac{-8}{15} < \frac{-2}{5}\)

iii. \(\frac{5}{-12},\ \frac{-2}{3},\ \frac{-7}{9},\ \frac{11}{-18}\)

Step 1: Write fractions with negative sign in numerator: \[ \frac{-5}{12}, \quad \frac{-2}{3}, \quad \frac{-7}{9}, \quad \frac{-11}{18} \] Step 2: Find LCM of denominators \(12, 3, 9, 18\).
LCM = 36.
Step 3: Express fractions with denominator 36: \[ \frac{-15}{36}, \quad \frac{-24}{36}, \quad \frac{-28}{36}, \quad \frac{-22}{36} \] Step 4: Arrange numerators ascending:
\(-28 < -24 < -22 < -15\)
Answer: \(\frac{-7}{9} < \frac{-2}{3} < \frac{-11}{18} < \frac{-5}{12}\)

iv. \(\frac{-4}{7},\ \frac{13}{-28},\ \frac{9}{14},\ \frac{23}{42}\)

Step 1: Write negative fractions with negative numerator: \[ \frac{-4}{7}, \quad \frac{-13}{28}, \quad \frac{9}{14}, \quad \frac{23}{42} \] Step 2: Find LCM of denominators \(7, 28, 14, 42\).
LCM = 84.
Step 3: Express all fractions with denominator 84: \[ \frac{-48}{84}, \quad \frac{-39}{84}, \quad \frac{54}{84}, \quad \frac{46}{84} \] Step 4: Arrange numerators ascending:
\(-48 < -39 < 46 < 54\)
Answer: \(\frac{-4}{7} < \frac{-13}{28} < \frac{23}{42} < \frac{9}{14}\)

Q4: Arrange the following rational numbers in descending order:

i. \(\frac{11}{12},\ \frac{13}{18},\ \frac{5}{6},\ \frac{7}{9}\)

Step 1: Find LCM of denominators \(12, 18, 6, 9\).
LCM(12, 18, 6, 9) = 36.
Step 2: Express each fraction with denominator 36: \[ \frac{11}{12} = \frac{33}{36}, \\ \frac{13}{18} = \frac{26}{36}, \\ \frac{5}{6} = \frac{30}{36}, \\ \frac{7}{9} = \frac{28}{36} \] Step 3: Compare numerators descending:
\(33 > 30 > 28 > 26\)
Answer: \(\frac{11}{12} > \frac{5}{6} > \frac{7}{9} > \frac{13}{18}\)

ii. \(\frac{-11}{20},\ \frac{3}{-10},\ \frac{17}{-30},\ \frac{-7}{15}\)

Step 1: Write all fractions with negative numerators: \[ \frac{-11}{20}, \quad \frac{-3}{10}, \quad \frac{-17}{30}, \quad \frac{-7}{15} \] Step 2: Find LCM of denominators \(20, 10, 30, 15\).
LCM = 60.
Step 3: Express each fraction with denominator 60: \[ \frac{-33}{60}, \quad \frac{-18}{60}, \quad \frac{-34}{60}, \quad \frac{-28}{60} \] Step 4: Compare numerators descending:
\(-18 > -28 > -33 > -34\)
Answer: \(\frac{3}{-10} > \frac{-7}{15} > \frac{-11}{20} > \frac{17}{-30}\)

iii. \(\frac{9}{-24},\ -1,\ \frac{2}{-3},\ \frac{-7}{-6}\)

Step 1: Write fractions with negative numerator: \[ \frac{-9}{24}, \quad -1, \quad \frac{-2}{3}, \quad \frac{7}{6} \] Step 2: Write all in fraction form with denominator 24 (LCM of 24, 1, 3, 6 is 24): \[ \frac{-9}{24}, \quad \frac{-24}{24}, \quad \frac{-16}{24}, \quad \frac{28}{24} \] Step 3: Compare numerators descending:
\(28 > -9 > -16 > -24\)
Answer: \(\frac{-7}{-6} > \frac{9}{-24} > \frac{2}{-3} > -1\)

iv. \(\frac{7}{-10},\ \frac{11}{15},\ \frac{-17}{-30},\ \frac{-2}{5}\)

Step 1: Write fractions with negative numerators: \[ \frac{-7}{10}, \quad \frac{11}{15}, \quad \frac{17}{30}, \quad \frac{-2}{5} \] Step 2: Find LCM of denominators \(10, 15, 30, 5\).
LCM = 30.
Step 3: Express all fractions with denominator 30: \[ \frac{-21}{30}, \quad \frac{22}{30}, \quad \frac{17}{30}, \quad \frac{-12}{30} \] Step 4: Compare numerators descending:
\(22 > 17 > -12 > -21\)
Answer: \(\frac{11}{15} > \frac{-17}{-30} > \frac{-2}{5} > \frac{7}{-10}\)