Rational Numbers

Q1: Multiple Choice Type

i. In the following number line, points A and B represent:

                           A                                           B
←──|─|─|─|─|─●─|─|─|─|─|─●─|─|─|─|─●─|─|─|─|─●─|─|─|─|─●─|─|─|─|─●─|─|─|─|─|─|─|─|─|──→
            -3          -2        -1         0         1         2         3

🔹 Point A: Point A is 1 step left of -2: \[ A = -2 – \frac{1}{5} = -\frac{10}{5} – \frac{1}{5} = -\frac{11}{5} = -1\frac{4}{5} \]🔹 Point B: Point B is 3 steps right of 2: \[ B = 2 + \frac{3}{5} = \frac{10}{5} + \frac{3}{5} = \frac{13}{5} = 2\frac{3}{5} \]Answer: c. \(-1\frac{4}{5}\ and\ 2\frac{3}{5}\)

ii. Using the number line, given above; the length of line segment AB is:

Line segment AB = Distance between A and B \[ AB = \left| \frac{13}{5} – \left( -\frac{11}{5} \right) \right| = \left| \frac{13}{5} + \frac{11}{5} \right| = \left| \frac{24}{5} \right| = 4\frac{4}{5} \]Answer: c. \(4\frac{4}{5}\)

iii. The rational number between \(\frac{a}{b}\ and\ \frac{c}{d}\) is:

To find a rational number between two given rational numbers, we use: \[ \text{Midpoint} = \frac{a+c}{b+d} \]Answer: c. \(\frac{a+c}{b+d}\)

iv. Two rational numbers between \(\frac{1}{3}\ and\ \frac{1}{2}\) are:

Let’s convert to like denominators: \[ \frac{1}{3} = \frac{8}{24}, \quad \frac{1}{2} = \frac{12}{24} \] So values between them could be: \(\frac{9}{24} = \frac{3}{8}, \quad \frac{10}{24} = \frac{5}{12}\)
Thus a valid pair is: \(\frac{3}{8}\) and \(\frac{2}{5}\)
Answer: a. \(\frac{3}{7}\) and \(\frac{3}{8}\)

v. The rational numbers \(-\frac{7}{4}\ and\frac{3}{4}\) are represented by:

           A   B     C                 D       E             F         G
←──|─|─|─|─●─|─●─|─|─●─|─|─|─|─|─|─|─|─●─|─|─|─●─|─|─|─|─|─|─●─|─|─|─|─●─|─|─|──→
        -3        -2        -1         0         1         2         3

🔹 Let’s find \(-\frac{7}{4}\): We know: \[ -\frac{7}{4} = -1\frac{3}{4} \]From 0, go 1 full unit left to -1, then go 3 more divisions (¼ each) left → lands on Point C.
✅ So, \(-\frac{7}{4} = \textbf{Point C}\)🔹 Now find \(\frac{3}{4}\): \[ \frac{3}{4} = 0 + \frac{3}{4} \]From 0, go 3 divisions (¼ each) right → lands on Point E.
✅ So, \(\frac{3}{4} = \textbf{Point E}\)Answer: c. C and E respectively

Q2: Draw a number line and mark \(\frac{3}{4},\frac{7}{4},\frac{-3}{4}\ and\ \frac{-7}{4}\) on it.

Step 1: Convert all the given numbers to decimal form for better placement on the number line. \[ \frac{3}{4} = 0.75,\quad \frac{7}{4} = 1.75,\quad \frac{-3}{4} = -0.75,\quad \frac{-7}{4} = -1.75 \]Step 2: Draw a number line ranging from -2 to 2 and mark the above positions approximately.
Marked Points:
✓ -1.75 → -7/4
✓ -0.75 → -3/4
✓ 0.75 → 3/4
✓ 1.75 → 7/4

←────●──|──|──|──●──|──|──|──●──|──|──|──●──|──|──|──●──|──|──|──●──|──|──|──→
    -3          -2 -1¾      -1 -¾        0        ¾  1       1¾  2 
                    ↑           ↑                 ↑           ↑
            -7/4 marked     -3/4 marked       3/4 marked     7/4 marked

Q3: On a number line mark the points \(\frac{2}{3},\frac{-8}{3},\frac{7}{3},\frac{-2}{3}\ and\ -2\).

Step 1: Convert the rational numbers to decimals. \[ \frac{2}{3} \approx 0.666,\quad \frac{-8}{3} \approx -2.666,\quad \frac{7}{3} \approx 2.333,\quad \frac{-2}{3} \approx -0.666,\quad -2 = -2.0 \]Step 2: Draw a number line ranging from -3 to 3.
Marked Points:
✓ -2.666 → -8/3
✓ -2.0 → -2
✓ -0.666 → -2/3
✓ 0.666 → 2/3
✓ 2.333 → 7/3

←──●───|───|───●───|───|───●───|───|───●───|───|───●───|───|───●───|───|───●───|───|───●───|──→
  -4          -3          -2          -1           0           1           2           3   
       ↑                   ↑               ↑               ↑                   ↑
   -8/3 marked        -2 marked       -2/3 marked     2/3 marked           7/3 marked

Q4: Insert one rational number between

i. \(\frac{3}{5}\ and\frac{5}{8}\)

Step 1: Add numerator with numerator and denominator with denominator: \[ \frac{3+5}{5+8} = \frac{8}{13} \]Step 2: A number between \(\frac{3}{5}\ and\frac{5}{8}\) is: \(\frac{8}{13}\)
Answer: One rational number = \(\frac{8}{13}\)

ii. \(\frac{1}{2}\ and\ 2\)

Convert to decimal: \(\frac{1}{2} = 0.5, 2 = 2.0\)
Any number between 0.5 and 2 is valid.→ (1) is between 1/2 and 2
Answer: 1

Q5: Insert two rational numbers between: \(\frac{5}{7}\ and\frac{3}{8}\).

Step 1: LCM of 7 and 8 = 56 \[ \frac{5}{7} = \frac{40}{56},\quad \frac{3}{8} = \frac{21}{56} \]Since \(40 > 21\), we reverse order: \[ \frac{21}{56},\ \frac{40}{56} \]Step 2: Multiply numerator and denominator by \(2 + 1 = 3\) \[ \frac{21 \times 3}{56 \times 3} = \frac{63}{168},\quad \frac{40 \times 3}{56 \times 3} = \frac{120}{168} \]Now numbers between 63 and 120 with denominator 168 are: \[ \frac{64}{168},\ \frac{65}{168} \]∴ Required numbers: \( \frac{64}{168},\ \frac{65}{168} \)

Q6: Insert three rational numbers between: \(\frac{8}{11}\ and\ \frac{4}{9}\).

LCM of 11 and 9 = 99 \[ \frac{8}{11} = \frac{72}{99},\quad \frac{4}{9} = \frac{44}{99} \]Order: \( \frac{44}{99},\ \frac{72}{99} \)Multiply both by \(3 + 1 = 4\): \[ \frac{44 \times 4}{99 \times 4} = \frac{176}{396},\quad \frac{72 \times 4}{99 \times 4} = \frac{288}{396} \]Now pick 3 values between 176 and 288: \[ \frac{200}{396},\ \frac{220}{396},\ \frac{250}{396} \]∴ Required numbers: \( \frac{200}{396},\ \frac{220}{396},\ \frac{250}{396} \)

Q7: Insert five rational numbers between \(\frac{3}{5}\ and\frac{2}{3}\).

LCM of 5 and 3 = 15 \[ \frac{3}{5} = \frac{9}{15},\quad \frac{2}{3} = \frac{10}{15} \]Multiply by \(5 + 1 = 6\): \[ \frac{9 \times 6}{15 \times 6} = \frac{54}{90},\quad \frac{10 \times 6}{15 \times 6} = \frac{60}{90} \]Pick 5 values between 54 and 60: \[ \frac{55}{90},\ \frac{56}{90},\ \frac{57}{90},\ \frac{58}{90},\ \frac{59}{90} \]∴ Required numbers: \( \frac{55}{90},\ \frac{56}{90},\ \frac{57}{90},\ \frac{58}{90},\ \frac{59}{90} \)

Q8: Insert six rational numbers between \(\frac{5}{6}\ and\frac{8}{9}\).

LCM of 6 and 9 = 18 \[ \frac{5}{6} = \frac{15}{18},\quad \frac{8}{9} = \frac{16}{18} \]Multiply by \(6 + 1 = 7\): \[ \frac{15 \times 7}{18 \times 7} = \frac{105}{126},\quad \frac{16 \times 7}{18 \times 7} = \frac{112}{126} \]Pick 6 values between 105 and 112: \[ \frac{106}{126},\ \frac{107}{126},\ \frac{108}{126},\ \frac{109}{126},\ \frac{110}{126},\ \frac{111}{126} \]∴ Required numbers: \( \frac{106}{126},\ \frac{107}{126},\ \frac{108}{126},\ \frac{109}{126},\ \frac{110}{126},\ \frac{111}{126} \)

Q9: Insert seven rational numbers between 2 and 3.

Write: \(2 = \frac{2}{1},\quad 3 = \frac{3}{1}\)
Multiply by \(7 + 1 = 8\): \[ \frac{2 \times 8}{1 \times 8} = \frac{16}{8},\quad \frac{3 \times 8}{1 \times 8} = \frac{24}{8} \]Now pick 7 values between 16 and 24: \[ \frac{17}{8},\ \frac{18}{8},\ \frac{19}{8},\ \frac{20}{8},\ \frac{21}{8},\ \frac{22}{8},\ \frac{23}{8} \]∴ Required numbers: \( \frac{17}{8},\ \frac{18}{8},\ \frac{19}{8},\ \frac{20}{8},\ \frac{21}{8},\ \frac{22}{8},\ \frac{23}{8} \)