Rational Numbers

Q1: Mark the following pairs of rational numbers on the separate number lines:

i. \(\frac{3}{4}\ and\ -\frac{1}{4}\)

----|----|----|----|----|----|----|----|----|----
   -1                   0                   1
                   ↑                   ↑
                 -1/4                 3/4

ii. \(\frac{2}{5}\ and\ \frac{-3}{5}\)

----|----|----|----|----|----|----|----|----|----|----|----
   -1                        0                        1
              ↑                        ↑
            -3/5                      2/5

iii. \(\frac{5}{6}\ and\ -\frac{2}{3}\)

---|---|---|---|---|---|---|---|---|---|---|---|---|---
  -1                       0                       1
           ↑                                   ↑
        -2/3                                  5/6

iv. \(\frac{2}{5}\ and\ -\frac{4}{5}\)

----|----|----|----|----|----|----|----|----|----|----|---
   -1                        0                        1
         ↑                             ↑
       -4/5                           2/5

v. \(\frac{1}{4}\ and\ -\frac{5}{4}\)

----|---|---|---|---|---|---|---|---|---|---|---|---|---
   -2              -1               0               1   
                ↑                       ↑
              -5/4                     1/4

Q2: Compare:

i. \(\frac{3}{5}\) and \(\frac{5}{7}\)
LCM of 5 and 7 = 35 \[ \frac{3}{5} = \frac{21}{35}, \quad \frac{5}{7} = \frac{25}{35} \] \(\frac{3}{5} < \frac{5}{7}\)

ii. \(\frac{-7}{2}\) and \(\frac{5}{2}\)
Same denominator 2: \[ -7 < 5 \] \(\frac{-7}{2} < \frac{5}{2}\)

iii. \(-3\) and \(2\frac{3}{4}\)
\[ -3 < 2\frac{3}{4} \] \(-3 < 2\frac{3}{4}\)

iv. \(-1\frac{1}{2}\) and \(0\)
\[ -1.5 < 0 \] \(-1\frac{1}{2} < 0\)

v. \(0\) and \(\frac{3}{4}\)
\[ 0 < \frac{3}{4} \] \(0 < \frac{3}{4}\)

vi. \(3\) and \(-1\)
\[ 3 > -1 \] \(3 > -1\)

Q3: Compare:

i. \(-\frac{1}{4}\) and \(0\) \[ -\frac{1}{4} < 0 \] \(-\frac{1}{4} < 0\)

ii. \(\frac{1}{4}\) and \(0\) \[ \frac{1}{4} > 0 \] \(\frac{1}{4} > 0\)

iii. \(-\frac{3}{8}\) and \(\frac{2}{5}\)
LCM of 8 and 5 = 40 \[ -\frac{3}{8} = -\frac{15}{40}, \quad \frac{2}{5} = \frac{16}{40} \] \[ -15 < 16 \] \(-\frac{3}{8} < \frac{2}{5}\)

iv. \(\frac{-5}{8}\) and \(\frac{7}{-12}\)
LCM of 8 and 12 = 24 \[ \frac{-5}{8} = \frac{-15}{24}, \quad \frac{7}{-12} = \frac{-14}{24} \] \[ -15 < -14 \] \(\frac{-5}{8} < \frac{7}{-12}\)

v. \(\frac{5}{-9}\) and \(\frac{-5}{-9}\)
Simplify: \[ \frac{5}{-9} = -\frac{5}{9}, \quad \frac{-5}{-9} = \frac{5}{9} \] \[ -\frac{5}{9} < \frac{5}{9} \] \(\frac{5}{-9} < \frac{-5}{-9}\)

vi. \(\frac{-7}{8}\) and \(\frac{5}{-6}\)
LCM of 8 and 6 = 24 \[ \frac{-7}{8} = \frac{-21}{24}, \quad \frac{5}{-6} = \frac{-20}{24} \] \[ -21 < -20 \] \(\frac{-7}{8} < \frac{5}{-6}\)

vii. \(\frac{2}{7}\) and \(\frac{-3}{-8}\)
Simplify: \[ \frac{-3}{-8} = \frac{3}{8} \] LCM of 7 and 8 = 56 \[ \frac{2}{7} = \frac{16}{56}, \quad \frac{3}{8} = \frac{21}{56} \] \[ 16 < 21 \] \(\frac{2}{7} < \frac{-3}{-8}\)

Q4: Arrange the given rational numbers in ascending order:

i. \(\frac{7}{10}, \frac{-11}{-30}, \frac{5}{-15}\)
First, simplify: \[ \frac{-11}{-30} = \frac{11}{30}, \quad \frac{5}{-15} = -\frac{1}{3} \]Convert to decimals: \[ \frac{7}{10} = 0.7, \quad \frac{11}{30} \approx 0.366, \quad -\frac{1}{3} \approx -0.333 \]Ascending order: \[ \frac{5}{-15}, \frac{-11}{30}, \frac{7}{10} \]Answer: \(\frac{5}{-15}, \frac{-11}{30}, \frac{7}{10}\)

ii. \(\frac{4}{-9}, \frac{-5}{12}, \frac{2}{-3}\)
Simplify: \[ \frac{4}{-9} = -\frac{4}{9}, \quad \frac{2}{-3} = -\frac{2}{3} \]Convert to decimals: \[ -\frac{4}{9} \approx -0.444, \quad -\frac{5}{12} \approx -0.416, \quad -\frac{2}{3} \approx -0.666 \]Ascending order: \[ \frac{2}{-3}, \frac{4}{-9}, \frac{-5}{12} \]Answer: \(\frac{2}{-3}, \frac{4}{-9}, \frac{-5}{12}\)

Q5: Arrange the given rational numbers in descending order:

i. \(\frac{5}{8}, \frac{13}{-16}, \frac{-7}{12}\)
Simplify: \[ \frac{13}{-16} = -\frac{13}{16} \]Convert to decimals: \[ \frac{5}{8} = 0.625, \quad -\frac{13}{16} \approx -0.812, \quad -\frac{7}{12} \approx -0.583 \]Descending order: \[ 0.625, -0.583, -0.812 \]Answer: \(\frac{5}{8}, \frac{-7}{12}, \frac{13}{-16}\)

ii. \(\frac{3}{-10}, \frac{-13}{30}, \frac{8}{-20}\)
Simplify: \[ \frac{3}{-10} = -\frac{3}{10}, \quad \frac{8}{-20} = -\frac{2}{5} \]Convert to decimals: \[ -\frac{3}{10} = -0.3, \quad -\frac{13}{30} \approx -0.433, \quad -\frac{2}{5} = -0.4 \]Descending order: \[ -0.3, -0.4, -0.433 \]Answer: \(\frac{3}{-10}, \frac{8}{-20}, \frac{-13}{30}\)

Q6: Fill in the blanks:

i. \(\frac{5}{8}\) and \(\frac{3}{10}\) are on the same side of zero.

ii. \(-\frac{5}{8}\) and \(\frac{3}{10}\) are on opposite sides of zero.

iii. \(-\frac{5}{8}\) and \(-\frac{3}{10}\) are on the same side of zero.

iv. \(\frac{5}{8}\) and \(-\frac{3}{10}\) are on opposite sides of zero.

Q7: Insert three rational numbers between:

(a) Between \(-\frac{2}{3}\) and \(\frac{3}{4}\)
Step 1: Find LCM of denominators 3 and 4.
LCM of 3 and 4 = 12
Step 2: Convert both fractions to have denominator 12: \[ -\frac{2}{3} = \frac{-8}{12},\quad \frac{3}{4} = \frac{9}{12} \]Step 3: Now between \(-8\) and \(9\) (numerators), we can pick numbers like \(-5, -2, 2\) etc.
Thus, three rational numbers between \(-\frac{2}{3}\) and \(\frac{3}{4}\) are: \[ \frac{-5}{12},\quad \frac{-2}{12},\quad \frac{2}{12} \] or simplified: \[ \frac{-5}{12},\quad \frac{-1}{6},\quad \frac{1}{6} \]

(b) Between \(\frac{5}{7}\) and \(\frac{7}{9}\)
Step 1: Find LCM of 7 and 9.
LCM of 7 and 9 = 63
Step 2: Convert both fractions: \[ \frac{5}{7} = \frac{45}{63},\quad \frac{7}{9} = \frac{49}{63} \]Step 3: Between 45 and 49, we have 46, 47, 48.
Thus, three rational numbers are: \[ \frac{46}{63},\quad \frac{47}{63},\quad \frac{48}{63} \]

(c) Between \(-\frac{5}{8}\) and \(-\frac{1}{6}\)
Step 1: Find LCM of 8 and 6.
LCM of 8 and 6 = 24
Step 2: Convert both fractions: \[ -\frac{5}{8} = \frac{-15}{24},\quad -\frac{1}{6} = \frac{-4}{24} \]Step 3: Between -15 and -4, we have -14, -12, -10, etc.
Thus, three rational numbers are: \[ \frac{-14}{24} = \frac{-7}{12},\quad \frac{-12}{24} = \frac{-1}{2},\quad \frac{-10}{24} = \frac{-5}{12} \]

Q8: Insert four rational numbers between:

(a) Between \(-\frac{3}{8}\) and \(-\frac{5}{6}\)
Step 1: Find LCM of 8 and 6.
LCM = 24
Step 2: Convert fractions: \[ -\frac{3}{8} = \frac{-9}{24},\quad -\frac{5}{6} = \frac{-20}{24} \]Step 3: Numbers between -20 and -9 are -19, -18, -17, -16, etc.
Thus, four rational numbers are: \[ \frac{-19}{24},\quad \frac{-18}{24}=\frac{-3}{4},\quad \frac{-17}{24},\quad \frac{-16}{24}=\frac{-2}{3} \]

(b) Between \(-\frac{4}{5}\) and \(\frac{2}{3}\)
Step 1: LCM of 5 and 3 = 15
Step 2: Convert fractions: \[ -\frac{4}{5} = \frac{-12}{15},\quad \frac{2}{3} = \frac{10}{15} \]Step 3: Numbers between -12 and 10 are -10, -8, -6, -4, etc.
Thus, four rational numbers are: \[ \frac{-10}{15}=\frac{-2}{3},\quad \frac{-8}{15},\quad \frac{-6}{15}=\frac{-2}{5},\quad \frac{-4}{15} \]

(c) Between \(-3\) and \(6\)
Step 1: Simple integers and fractions between -3 and 6:
Examples of four rational numbers are: \[ -2,\quad 0,\quad 2,\quad 5 \]

(d) Between \(0\) and \(6\)
Simple rational numbers between 0 and 6:
Examples of four rational numbers are: \[ 1,\quad 2,\quad 3,\quad 4 \]