Rational Numbers

Q1: Express \(\frac{-64}{112}\) as a rational number with denominator 7.

Step 1: We are given the rational number: \(\frac{-64}{112}\)
Step 2: To convert this to a rational number with denominator 7, we first reduce the fraction to its simplest form.
We find the HCF of 64 and 112.
64 = 2 × 2 × 2 × 2 × 2 × 2
112 = 2 × 2 × 2 × 2 × 7
⇒ HCF = 2 × 2 × 2 × 2 = 16
So, \[ \frac{-64}{112} = \frac{-64 ÷ 16}{112 ÷ 16} = \frac{-4}{7} \]Step 3: Hence, the rational number with denominator 7 is:\(\frac{-4}{7}\)
Answer: \(\frac{-4}{7}\)

Q2: Express \(\frac{-48}{60}\) as a rational number with denominator 25.

Step 1: We are given the rational number: \(\frac{-48}{60}\)
Step 2: First, reduce the rational number to its simplest form.
Find the HCF of 48 and 60.
48 = 2 × 2 × 2 × 2 × 3
60 = 2 × 2 × 3 × 5
⇒ HCF = 2 × 2 × 3 = 12
So, \[ \frac{-48}{60} = \frac{-48 ÷ 12}{60 ÷ 12} = \frac{-4}{5} \]Step 3: Now, convert this rational number to one with denominator 25.
To make the denominator 25, multiply numerator and denominator by 5: \[ \frac{-4}{5} = \frac{-4 × 5}{5 × 5} = \frac{-20}{25} \]Answer: \(\frac{-20}{25}\)

Q3: Express each of the following rational numbers in standard form:

i. \(\frac{-12}{30}\)

Find HCF of 12 and 30:
12 = 2 × 2 × 3
30 = 2 × 3 × 5
HCF = 2 × 3 = 6
So, divide numerator and denominator by 6: \[ \frac{-12}{30} = \frac{-12 \div 6}{30 \div 6} = \frac{-2}{5} \] Answer: Standard form: \(\frac{-2}{5}\)

ii. \(\frac{-14}{49}\)

HCF of 14 and 49 is 7: \[ \frac{-14}{49} = \frac{-14 \div 7}{49 \div 7} = \frac{-2}{7} \] Answer: Standard form: \(\frac{-2}{7}\)

iii. \(\frac{24}{-64}\)

Negative denominator is not allowed in standard form.
Move minus sign to numerator: \[ \frac{24}{-64} = \frac{-24}{64} \] Now simplify by HCF of 8: \[ \frac{-24}{64} = \frac{-3}{8} \] Answer: Standard form: \(\frac{-3}{8}\)

iv. \(\frac{-36}{-63}\)

Both numerator and denominator are negative, which cancel out: \[ \frac{-36}{-63} = \frac{36}{63} \] Now simplify: HCF = 9 \[ \frac{36}{63} = \frac{4}{7} \] Answer: Standard form: \(\frac{4}{7}\)

Q4: Compare the following pairs of rational numbers:

i. \(\frac{15}{32}\ and\frac{17}{24}\)

To compare the rational numbers, we make the denominators same (LCM method). LCM of 32 and 24 = 96
Convert both rational numbers to like denominators:\[ \frac{15}{32} = \frac{15 \times 3}{32 \times 3} = \frac{45}{96} \\ \frac{17}{24} = \frac{17 \times 4}{24 \times 4} = \frac{68}{96} \]Now compare: \[ \frac{45}{96} \lt \frac{68}{96} \]Answer: \( \frac{15}{32} \lt \frac{17}{24} \)

ii. \(\frac{10}{11}\ and\frac{17}{18}\)

LCM of 11 and 18 = 198
Convert both rational numbers to like denominators: \[ \frac{10}{11} = \frac{10 \times 18}{11 \times 18} = \frac{180}{198} \\ \frac{17}{18} = \frac{17 \times 11}{18 \times 11} = \frac{187}{198} \]Now compare: \[ \frac{180}{198} \lt \frac{187}{198} \]Answer: \( \frac{10}{11} \lt \frac{17}{18} \)

iii. \(\frac{-5}{12}\ and\frac{-3}{4}\)

Convert both to like denominators using LCM of 12 and 4 = 12 \[ \frac{-3}{4} = \frac{-3 \times 3}{4 \times 3} = \frac{-9}{12} \]Now compare: \[ \frac{-5}{12} \gt \frac{-9}{12} \]Because –5 is greater than –9 on the number line.
Answer: \( \frac{-5}{12} \gt \frac{-3}{4} \)

iv. \(\frac{-7}{24}\ and\frac{9}{-20}\)

Note: \( \frac{9}{-20} = \frac{-9}{20} \)
Now find LCM of 24 and 20 = 120 \[ \frac{-7}{24} = \frac{-7 \times 5}{24 \times 5} = \frac{-35}{120} \\ \frac{-9}{20} = \frac{-9 \times 6}{20 \times 6} = \frac{-54}{120} \]Now compare: \[ \frac{-35}{120} \gt \frac{-54}{120} \]Answer: \( \frac{-7}{24} \gt \frac{9}{-20} \)

Q5: Arrange in ascending order:

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

Step 1: Find LCM of denominators (6, 9, 12, 18)
LCM = 36
Convert all fractions to have denominator 36: \[ \frac{5}{6} = \frac{5 \times 6}{6 \times 6} = \frac{30}{36} \\ \frac{7}{9} = \frac{7 \times 4}{9 \times 4} = \frac{28}{36} \\ \frac{11}{12} = \frac{11 \times 3}{12 \times 3} = \frac{33}{36} \\ \frac{13}{18} = \frac{13 \times 2}{18 \times 2} = \frac{26}{36} \]Now arrange: \[ \frac{26}{36} \lt \frac{28}{36} \lt \frac{30}{36} \lt \frac{33}{36} \]Answer: Ascending order: \( \frac{13}{18},\ \frac{7}{9},\ \frac{5}{6},\ \frac{11}{12} \)

ii. \(\frac{5}{-7},\ \frac{-9}{14},\ \frac{-5}{6}\ and\frac{7}{-12}\)

Note: Rewrite as: \[ \frac{5}{-7} = \frac{-5}{7},\ \frac{7}{-12} = \frac{-7}{12} \]Now we compare: \[ \frac{-5}{7},\ \frac{-9}{14},\ \frac{-5}{6},\ \frac{-7}{12} \]Convert to common denominator. LCM of 7, 14, 6, 12 = 84\[ \frac{-5}{7} = \frac{-60}{84},\\ \frac{-9}{14} = \frac{-54}{84},\\ \frac{-5}{6} = \frac{-70}{84},\\ \frac{-7}{12} = \frac{-49}{84} \]Now arrange: \[ \frac{-70}{84} \lt \frac{-60}{84} \lt \frac{-54}{84} \lt \frac{-49}{84} \]Answer: Ascending order: \( \frac{-5}{6},\ \frac{-5}{7},\ \frac{-9}{14},\ \frac{-7}{12} \)

iii. \(-2,\ \frac{1}{3},\ \frac{-13}{6}\ and\frac{8}{-3}\)

Note: \( \frac{8}{-3} = \frac{-8}{3} \)
Convert all to improper fractions: \[ -2 = \frac{-12}{6},\\ \frac{1}{3} = \frac{2}{6},\\ \frac{-13}{6},\\ \frac{-8}{3} = \frac{-16}{6} \]Now list all: \[ \frac{-16}{6},\ \frac{-13}{6},\ \frac{-12}{6},\ \frac{2}{6} \]Now arrange: \[ \frac{-16}{6} \lt \frac{-13}{6} \lt \frac{-12}{6} \lt \frac{2}{6} \]Answer: Ascending order: \( \frac{8}{-3},\ \frac{-13}{6},\ -2,\ \frac{1}{3} \)

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

Step 1: Convert all to standard negative form
\[ \frac{13}{-28} = \frac{-13}{28},\quad \text{others already negative} \]Now we compare: \[ \frac{-13}{28},\quad \frac{-23}{42},\quad \frac{-4}{7},\quad \frac{-9}{14} \]Step 2: Find LCM of denominators 28, 42, 7, and 14
LCM(28, 42, 7, 14) = 84
Step 3: Convert all to like denominators (84)
\[ \frac{-13}{28} = \frac{-13 \times 3}{28 \times 3} = \frac{-39}{84} \\ \frac{-23}{42} = \frac{-23 \times 2}{42 \times 2} = \frac{-46}{84} \\ \frac{-4}{7} = \frac{-4 \times 12}{7 \times 12} = \frac{-48}{84} \\ \frac{-9}{14} = \frac{-9 \times 6}{14 \times 6} = \frac{-54}{84} \]Step 4: Arrange the converted values in ascending order:
\[ \frac{-54}{84} \lt \frac{-48}{84} \lt \frac{-46}{84} \lt \frac{-39}{84} \]Answer: Ascending order: \( \frac{-9}{14},\ \frac{-4}{7},\ \frac{-23}{42},\ \frac{13}{-28} \)

Q6: Represent each of the following numbers on the number line:

i. \(\frac{5}{6}\)

This is a positive proper fraction between 0 and 1.
We divide the segment from 0 to 1 into 6 equal parts and mark the 5th point.
Answer: \( \frac{5}{6} \) lies between 0 and 1, at the 5th mark when the interval is divided into 6 parts.

ii. \(\frac{14}{3}\)

Convert to mixed number: \[ \frac{14}{3} = 4\frac{2}{3} \]It lies between 4 and 5.
Divide the segment from 4 to 5 into 3 equal parts and mark the 2nd part.
Answer: \( \frac{14}{3} = 4\frac{2}{3} \) lies between 4 and 5, at the second division.

iii. \(\frac{-3}{7}\)

Negative proper fraction, lies between –1 and 0.
Divide segment from –1 to 0 into 7 parts and mark the 3rd division from 0 moving left.
Answer: \( \frac{-3}{7} \) lies between –1 and 0, at the 3rd mark from 0.

iv. \(\frac{-17}{5}\)

Convert to mixed number: \[ \frac{-17}{5} = -3\frac{2}{5} \]It lies between –4 and –3.
Divide the segment from –4 to –3 into 5 parts and mark the 2nd part from –4.
Answer: \( \frac{-17}{5} = -3\frac{2}{5} \) lies between –4 and –3, at the 2nd division.

v. \(-2\frac{2}{7}\)

Already in mixed form.
It lies between –3 and –2.
Divide the segment from –3 to –2 into 7 equal parts and mark the 2nd from –3.
Answer: \( -2\frac{2}{7} \) lies between –3 and –2, at the 2nd mark from –3.

Q7: State whether each of the following statements is true or false:

i. Every whole number is a rational number.

Whole numbers = {0, 1, 2, 3, …}
Rational numbers = Numbers of the form \( \frac{p}{q} \), where \( q \ne 0 \)
Since every whole number can be written as \( \frac{a}{1} \), it is a rational number.
Answer: TRUE

ii. Every integer is a rational number.

Integers = {…, –3, –2, –1, 0, 1, 2, 3, …}
All can be written as \( \frac{p}{1} \), so they are rational.
Answer: TRUE

iii. \(\frac{2}{3}=\frac{2+4}{3+4}\)

Left-hand side: \[ \frac{2}{3} \]Right-hand side: \[ \frac{2+4}{3+4} = \frac{6}{7} \]Clearly, \[ \frac{2}{3} \ne \frac{6}{7} \]Answer: FALSE

iv. \( \frac{-5}{6} < \frac{-6}{5} \)

Let’s compare both values:\[ \frac{-5}{6} \approx -0.833,\quad \frac{-6}{5} = -1.2 \]Since –0.833 is greater than –1.2, the statement is false.
Answer: FALSE

v. \(\frac{-3}{-4}\) is a negative rational number.

Double negatives cancel out: \[ \frac{-3}{-4} = \frac{3}{4} \]It is a positive rational number.
Answer: FALSE

vi. 0 is a whole number but it is not a rational number.

0 is a whole number. Also, \[ 0 = \frac{0}{1} \] which is a valid rational number.
Answer: FALSE

Q8: Represent \(\frac{13}{5}\) and \(\frac{-13}{5}\) on the number line.

Step-by-step Explanation:
We are given the rational numbers:     \(\frac{13}{5}\)
and     \(-\frac{13}{5}\)
First, convert them into mixed fractions to understand their positions:
⇒ \(\frac{13}{5} = 2 \frac{3}{5}\)
⇒ \(-\frac{13}{5} = -2 \frac{3}{5}\)
So on the number line:
• \(\frac{13}{5}\)
lies between 2 and 3 • \(-\frac{13}{5}\)

lies between –2 and –3

       -13/5                                                                         13/5
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  -3     ↑       -2             -1              0              1              2        ↑     3