Exercise: 1-D
Q1: Find a rational number between each of the following pairs of rational numbers:
i. \(\frac{7}{10}\ and\ \frac{10}{17}\)
Let’s convert both to a common denominator using LCM(10,17) = 170.\[
\frac{7}{10} = \frac{7 \times 17}{10 \times 17} = \frac{119}{170}, \quad
\frac{10}{17} = \frac{10 \times 10}{17 \times 10} = \frac{100}{170}
\]Now check:
\[
\frac{100}{170} < \frac{119}{170}
\Rightarrow \text{So } \frac{109}{170} \text{ is between them.}
\]∴ One rational number between them is \( \frac{109}{170} \)
ii. \(1\frac{3}{8}\ and\ 2\)
Convert mixed number to improper fraction:
\[
1\frac{3}{8} = \frac{11}{8}
\]Now:
\[
\frac{11}{8} < x < \frac{16}{8}
\Rightarrow \frac{12}{8}, \frac{13}{8}, \frac{14}{8}, \dots \text{ are between them.}
\]∴ One rational number between them is \( \frac{13}{8} \)
iii. \(\frac{-3}{5}\ and\ \frac{-4}{7}\)
Convert to common denominator using LCM(5,7) = 35:\[
\frac{-3}{5} = \frac{-21}{35}, \quad
\frac{-4}{7} = \frac{-20}{35}
\]Now:
\[
\frac{-21}{35} < \frac{-20}{35}
\Rightarrow \text{Try } \frac{-205}{350} \text{ which lies between them}
\Rightarrow \frac{-41}{70}
\]∴ One rational number between them is \( \frac{-41}{70} \)
iv. \(-2\ and\ \frac{-17}{21}\)
Convert \( -2 \) to same denominator as \( \frac{-17}{21} \):
\[
-2 = \frac{-42}{21}
\Rightarrow \frac{-42}{21} < x < \frac{-17}{21}
\]Choose any number between them:
\[
\frac{-30}{21} \text{ lies between } \frac{-42}{21} \text{ and } \frac{-17}{21}
\]∴ One rational number between them is \( \frac{-30}{21} \)
Q2: Find three rational numbers between:
i. \(4\ and\ 4\frac{2}{3}\)
Step 1: Convert mixed number to improper fraction:
\[
4 = \frac{4}{1},\quad 4\frac{2}{3} = \frac{14}{3}
\]Step 2: Take LCM of denominators 1 and 3 = 3
\[
\frac{12}{3},\ \frac{14}{3}
\]Step 3: To insert 3 numbers, multiply numerator and denominator of both by \(3 + 1 = 4\)\[
\frac{12 \times 4}{3 \times 4} = \frac{48}{12},\quad \frac{14 \times 4}{3 \times 4} = \frac{56}{12}
\]Now rational numbers with denominator 12 and numerators between 48 and 56 are:
\[
\frac{49}{12},\ \frac{50}{12},\ \frac{51}{12}
\]∴ Required numbers: \( \frac{49}{12},\ \frac{50}{12},\ \frac{51}{12} \)
ii. \(\frac{-1}{2}\ and\ \frac{-1}{4}\)
Step 1: Find LCM of 2 and 4 = 4\[
\frac{-1}{2} = \frac{-2}{4},\quad \frac{-1}{4} = \frac{-1}{4}
\]Step 2: Multiply numerator and denominator by \(3 + 1 = 4\)\[
\frac{-2 \times 4}{4 \times 4} = \frac{-8}{16},\quad \frac{-1 \times 4}{4 \times 4} = \frac{-4}{16}
\]Now rational numbers with denominator 16 and numerators between -8 and -4 are:
\[
\frac{-7}{16},\ \frac{-6}{16},\ \frac{-5}{16}
\]∴ Required numbers: \( \frac{-7}{16},\ \frac{-6}{16},\ \frac{-5}{16} \)
iii. \(2\ and\ 3\)
Step 1: Write as fractions with denominator 1:
\[
2 = \frac{2}{1},\quad 3 = \frac{3}{1}
\]Step 2: Multiply numerator and denominator by \(3 + 1 = 4\)\[
\frac{2 \times 4}{1 \times 4} = \frac{8}{4},\quad \frac{3 \times 4}{1 \times 4} = \frac{12}{4}
\]Now rational numbers between \( \frac{8}{4} \) and \( \frac{12}{4} \) are:
\[
\frac{9}{4},\ \frac{10}{4},\ \frac{11}{4}
\]∴ Required numbers: \( \frac{9}{4},\ \frac{10}{4},\ \frac{11}{4} \)
Q3: Find five rational numbers between:
i. \(\frac{3}{5}\ and\ \frac{2}{3}\)
Step 1: Make denominators same (LCM of 5 and 3 = 15)
\[
\frac{3}{5} = \frac{9}{15}, \quad \frac{2}{3} = \frac{10}{15}
\]
Step 2: Choose equivalent fractions with larger denominator (multiply numerator and denominator by 10)
\[
\frac{9}{15} = \frac{90}{150}, \quad \frac{10}{15} = \frac{100}{150}
\]
Step 3: Pick five numbers between 90 and 100 (denominator 150)
\[
\frac{91}{150}, \frac{92}{150}, \frac{93}{150}, \frac{94}{150}, \frac{95}{150}
\]
Answer: \( \frac{91}{150}, \frac{92}{150}, \frac{93}{150}, \frac{94}{150}, \frac{95}{150} \)
ii. \(-2\ and\ -1\frac{1}{2}\)
Step 1: Convert to improper fractions
\[
-2 = \frac{-4}{2}, \quad -1\frac{1}{2} = \frac{-3}{2}
\]
Step 2: Multiply numerator and denominator by 10
\[
\frac{-4}{2} = \frac{-40}{20}, \quad \frac{-3}{2} = \frac{-30}{20}
\]
Step 3: Pick five values between -40 and -30 with denominator 20
\[
\frac{-39}{20}, \frac{-38}{20}, \frac{-37}{20}, \frac{-36}{20}, \frac{-35}{20}
\]
Answer: \( \frac{-39}{20}, \frac{-38}{20}, \frac{-37}{20}, \frac{-36}{20}, \frac{-35}{20} \)
iii. \(-3\ and\ -2\)
Step 1: Write with same denominator
\[
-3 = \frac{-30}{10}, \quad -2 = \frac{-20}{10}
\]
Step 2: Pick values between -30 and -20 with denominator 10
\[
\frac{-29}{10}, \frac{-28}{10}, \frac{-27}{10}, \frac{-26}{10}, \frac{-25}{10}
\]
Answer: \( \frac{-29}{10}, \frac{-28}{10}, \frac{-27}{10}, \frac{-26}{10}, \frac{-25}{10} \)
Q4: Find 10 rational numbers between \(\frac{-3}{4}\ and\ \frac{5}{6}\).
Step 1: Find LCM of denominators 4 and 6
\[
\text{LCM}(4, 6) = 12
\]Step 2: Convert both fractions to have denominator 12
\[
\frac{-3}{4} = \frac{-9}{12}, \quad \frac{5}{6} = \frac{10}{12}
\]Step 3: Multiply numerator and denominator by 10 for better spacing
\[
\frac{-9}{12} = \frac{-90}{120}, \quad \frac{10}{12} = \frac{100}{120}
\]Step 4: Choose 10 rational numbers between -90 and 100 (with denominator 120)
Pick 10 values between -90 and 100:
\[
\frac{-80}{120}, \frac{-60}{120}, \frac{-40}{120}, \frac{-20}{120}, \frac{0}{120}, \frac{20}{120}, \frac{40}{120}, \frac{60}{120}, \frac{80}{120}, \frac{90}{120}
\]Answer: \( \frac{-80}{120}, \frac{-60}{120}, \frac{-40}{120}, \frac{-20}{120}, \frac{0}{120}, \frac{20}{120}, \frac{40}{120}, \frac{60}{120}, \frac{80}{120}, \frac{90}{120} \)
Q5: Find 12 rational numbers between \(-1\ and\ 2\).
Step 1: Write -1 and 2 as rational numbers with same denominator
Let’s choose 13 as denominator to find 12 values in between (more spacing).
\[
-1 = \frac{-13}{13}, \quad 2 = \frac{26}{13}
\]Step 2: List 12 rational numbers between \( \frac{-13}{13} \) and \( \frac{26}{13} \)
We now choose 12 values between -13 and 26:\[
\frac{-12}{13}, \frac{-8}{13}, \frac{-4}{13}, \frac{0}{13}, \frac{2}{13}, \frac{5}{13}, \frac{9}{13}, \frac{12}{13}, \frac{15}{13}, \frac{18}{13}, \frac{21}{13}, \frac{24}{13}
\]Answer: \( \frac{-12}{13}, \frac{-8}{13}, \frac{-4}{13}, \frac{0}{13}, \frac{2}{13}, \frac{5}{13}, \frac{9}{13}, \frac{12}{13}, \frac{15}{13}, \frac{18}{13}, \frac{21}{13}, \frac{24}{13} \)
