GSEB Class 7 Maths Solutions Chapter 9 Rational Numbers Exercise 9.2

Question 1. Find the sum:
(i) \( \frac { 5 }{ 4 } + (\frac { -11 }{4}) \)
(ii) \( \frac { 5 }{ 3 } + \frac { 3 }{ 5 } \)
(iii) \( \frac { -9 }{ 10 } + \frac { 22 }{ 15 } \)
(iv) \( \frac {-3}{-11 } \) and \( \frac { 5 }{ 9 } \)
(v) \( \frac { -8 }{ 19 } + \frac { (-2) }{ 57 } \)
(vi) \( \frac {-2}{3} + 0 \)
(vii) \( - 2\frac { 1 }{ 3 } + 4\frac { 3 }{ 5 } \)
Answer:
(i) We have \( \frac { 5 }{ 4 } + (\frac { -11 }{ 4 } ) = \frac {5+(-11) }{ 4 } = \frac { -6 }{ 4 } \)
\( \implies = \frac { -3 }{ 2 } \) or \( -1\frac { 1 }{ 2 } \)
(ii) To add \( \frac {5}{ 3 } + \frac { 3 }{ 5 } \), we first find the LCM of the denominators 3 and 5, which is 15.
Now, we convert each fraction to an equivalent fraction with a denominator of 15:
\( \frac { 5 }{ 3 } = \frac { 5 \times 5 }{ 3 \times 5 } = \frac { 25 }{ 15 } \)
\( \frac { 3 }{ 5 } = \frac { 3 \times 3 }{ 5 \times 3 } = \frac { 9 }{ 15 } \)
Then, we add these equivalent fractions:
\( \frac { 25 }{ 15 } + \frac { 9 }{ 15 } = \frac { 25+9 }{ 15 } = \frac { 34 }{ 15 } \)
This can also be written as a mixed number: \( 2\frac { 4 }{ 15 } \).
Thus, \( \frac { 5 }{ 3 } + \frac { 3 }{ 5 } = \frac { 34 }{ 15 } \) or \( 2\frac { 4 }{ 15 } \).
(iii) For \( \frac { -9 }{ 10 } + \frac { 22 }{ 15 } \), we find the LCM of 10 and 15, which is 30.
We convert each fraction to have a denominator of 30:
\( \frac { -9 }{ 10 } = \frac { (-9)\times3 }{ 10\times3 } = \frac { -27 }{ 30 } \)
\( \frac { 22 }{ 15 } = \frac { 22\times2 }{ 15\times2 } = \frac { 44 }{ 30 } \)
Now, we add the fractions:
\( \frac { -9 }{ 10 } + \frac { 22 }{ 15 } = \frac { -27 }{ 30 } + \frac { 44 }{ 30 } = \frac { -27+44 }{ 30 } = \frac { 17 }{ 30 } \)
Thus, \( \frac { -9 }{ 10 } + \frac { 22 }{ 15 } = \frac { 17 }{ 30 } \).
(iv) For \( \frac {-3}{-11 } \) and \( \frac { 5 }{ 9 } \), we first simplify \( \frac {-3}{-11 } \) to \( \frac { 3 }{ 11 } \). We need to find the sum of \( \frac { 3 }{ 11 } \) and \( \frac { 5 }{ 9 } \).
The LCM of 11 and 9 is 99.
We convert each fraction to have a denominator of 99:
\( \frac { -3 }{ -11 } = \frac { 3 }{ 11 } = \frac { 3\times9 }{ 11\times9 } = \frac { 27 }{ 99 } \)
\( \frac { 5 }{ 9 } = \frac { 5\times11 }{ 9\times11 } = \frac { 55 }{ 99 } \)
Now, we add the fractions:
\( \frac { 27 }{ 99 } + \frac { 55 }{ 99 } = \frac { 27+55 }{ 99 } = \frac { 82 }{ 99 } \)
Thus, \( \frac {-3}{-11 } + \frac { 5 }{ 9 } = \frac { 82 }{ 99 } \).
(v) For \( \frac { -8 }{ 19 } + \frac { (-2) }{ 57 } \), we find the LCM of 19 and 57, which is 57.
We convert the first fraction to have a denominator of 57:
\( \frac { -8 }{ 19 } = \frac { (-8)\times3 }{ 19\times3 } = \frac { -24 }{ 57 } \)
Now, we add the fractions:
\( \frac { -8 }{ 19 } + \frac { (-2) }{ 57 } = \frac { -24 }{ 57 } + \frac { (-2) }{ 57 } = \frac { (-24)+(-2) }{ 57 } = \frac { -26 }{ 57 } \)
Thus, \( \frac { -8 }{ 19 } + \frac { (-2) }{ 57 } = \frac { -26 }{ 57 } \).
(vi) For \( \frac {-2}{3} + 0 \), when any number is added to zero, the result is the number itself.
So, \( \frac { -2 }{ 3 } + 0 = \frac { -2 }{ 3 } \).
This can also be shown as: \( \frac { -2 }{ 3 } + \frac { 0 }{ 3 } = \frac { -2+0 }{ 3 } = \frac { -2 }{ 3 } \).
Thus, \( \frac {-2}{3} + 0 = \frac { -2 }{ 3 } \).
(vii) For \( - 2\frac { 1 }{ 3 } + 4\frac { 3 }{ 5 } \), we first convert the mixed fractions to improper fractions.
\( -2\frac { 1 }{ 3 } = - \frac { (2\times3)+1 }{ 3 } = - \frac { 7 }{ 3 } \)
\( 4\frac { 3 }{ 5 } = \frac { (4\times5)+3 }{ 5 } = \frac { 23 }{ 5 } \)
Now we need to add \( - \frac { 7 }{ 3 } + \frac { 23 }{ 5 } \). The LCM of 3 and 5 is 15.
Convert each fraction to have a denominator of 15:
\( - \frac { 7 }{ 3 } = - \frac { 7\times5 }{ 3\times5 } = - \frac { 35 }{ 15 } \)
\( \frac { 23 }{ 5 } = \frac { 23\times3 }{ 5\times3 } = \frac { 69 }{ 15 } \)
Now, add the converted fractions:
\( - \frac { 35 }{ 15 } + \frac { 69 }{ 15 } = \frac { -35+69 }{ 15 } = \frac { 34 }{ 15 } \)
This can also be written as a mixed number: \( 2\frac { 4 }{ 15 } \).
Thus, \( -2\frac { 1 }{ 3 } + 4\frac { 3 }{ 5 } = 2\frac { 4 }{ 15 } \).
In simple words: To find the total sum, you either add the fractions directly if they have the same bottom number (denominator). If the bottom numbers are different, you first find a common bottom number (LCM), change both fractions to use it, and then add them up. Sometimes, mixed numbers need to be turned into improper fractions first. Adding zero to any number simply gives you the same number back.

Exam Tip: When adding rational numbers, always find the Least Common Multiple (LCM) of the denominators. Convert fractions to equivalent forms with the LCM as the denominator before performing addition or subtraction, and simplify your final answer if possible.

Question 2. Find:
(i) \( \frac { 7 }{ 24 } – \frac { 17 }{ 36 } \)
(ii) \( \frac {5}{ 63 } - (\frac { -6 }{ 21 }) \)
(iii) \( \frac {-6}{ 13 } – \frac { -7 }{ 15 } \)
(iv) \( \frac {-3}{ 8 } – \frac { 7 }{11} \)
(v) \( -2\frac {1}{9}-6 \)
Answer:
(i) To find \( \frac { 7 }{ 24 } – \frac { 17 }{ 36 } \), we first find the LCM of 24 and 36, which is 72.
Now, we convert each fraction to an equivalent fraction with a denominator of 72:
\( \frac { 7 }{ 24 } = \frac { 7\times3 }{ 24\times3 } = \frac { 21 }{ 72 } \)
\( \frac { 17 }{ 36 } = \frac { 17\times2 }{ 36\times2 } = \frac { 34 }{ 72 } \)
Then, we subtract these equivalent fractions:
\( \frac { 7 }{ 24 } - \frac { 17 }{ 36 } = \frac { 21 }{ 72 } - \frac { 34 }{ 72 } = \frac { 21-34 }{ 72 } = \frac { -13 }{ 72 } \)
Thus, \( \frac { 7 }{ 24 } – \frac { 17 }{ 36 } = \frac { -13 }{ 72 } \).
(ii) To find \( \frac {5}{ 63 } - (\frac { -6 }{ 21 }) \), subtracting a negative number is the same as adding a positive number, so this becomes \( \frac {5}{ 63 } + \frac { 6 }{ 21 } \).
We find the LCM of 63 and 21, which is 63.
Convert the second fraction to have a denominator of 63:
\( \frac { 6 }{ 21 } = \frac { 6\times3 }{ 21\times3 } = \frac { 18 }{ 63 } \)
Now, we add the fractions:
\( \frac { 5 }{ 63 } + \frac { 18 }{ 63 } = \frac { 5+18 }{ 63 } = \frac { 23 }{ 63 } \)
Thus, \( \frac {5}{ 63 } - (\frac { -6 }{ 21 }) = \frac { 23 }{ 63 } \).
(iii) To find \( \frac {-6}{ 13 } – \frac { -7 }{ 15 } \), we change the subtraction of a negative to addition of a positive: \( \frac {-6}{ 13 } + \frac { 7 }{ 15 } \).
The LCM of 13 and 15 is 195.
Convert each fraction to have a denominator of 195:
\( \frac { -6 }{ 13 } = \frac { (-6)\times15 }{ 13\times15 } = \frac { -90 }{ 195 } \)
\( \frac { 7 }{ 15 } = \frac { 7\times13 }{ 15\times13 } = \frac { 91 }{ 195 } \)
Now, add the fractions:
\( \frac { -90 }{ 195 } + \frac { 91 }{ 195 } = \frac { -90+91 }{ 195 } = \frac { 1 }{ 195 } \)
Thus, \( \frac {-6}{ 13 } – \frac { -7 }{ 15 } = \frac { 1 }{ 195 } \).
(iv) To find \( \frac {-3}{ 8 } – \frac { 7 }{11} \), we find the LCM of 8 and 11, which is 88.
Convert each fraction to have a denominator of 88:
\( \frac { -3 }{ 8 } = \frac { (-3)\times11 }{ 8\times11 } = \frac { -33 }{ 88 } \)
\( \frac { 7 }{ 11 } = \frac { 7\times8 }{ 11\times8 } = \frac { 56 }{ 88 } \)
Now, subtract the fractions:
\( \frac { -33 }{ 88 } - \frac { 56 }{ 88 } = \frac { -33-56 }{ 88 } = \frac { -89 }{ 88 } \)
Thus, \( \frac {-3}{ 8 } – \frac { 7 }{11} = \frac { -89 }{ 88 } \) or \( -1\frac { 1 }{ 88 } \).
(v) To find \( -2\frac {1}{9}-6 \), we first convert the mixed fraction to an improper fraction and express 6 as a fraction.
\( -2\frac { 1 }{ 9 } = - \frac { (2\times9)+1 }{ 9 } = - \frac { 19 }{ 9 } \)
\( 6 = \frac { 6 }{ 1 } \)
Now we need to subtract \( - \frac { 19 }{ 9 } - \frac { 6 }{ 1 } \). The LCM of 9 and 1 is 9.
Convert the second fraction to have a denominator of 9:
\( \frac { 6 }{ 1 } = \frac { 6\times9 }{ 1\times9 } = \frac { 54 }{ 9 } \)
Now, perform the subtraction:
\( - \frac { 19 }{ 9 } - \frac { 54 }{ 9 } = \frac { -19-54 }{ 9 } = \frac { -73 }{ 9 } \)
This can also be written as a mixed number: \( -8\frac { 1 }{ 9 } \).
Thus, \( -2\frac {1}{9}-6 = -8\frac { 1 }{ 9 } \).
In simple words: To find the difference between rational numbers, you first make sure they have the same bottom number (denominator) by finding the LCM. Then, you subtract the top numbers (numerators). If you subtract a negative number, it's the same as adding a positive one. Mixed numbers should be converted to improper fractions before doing calculations.

Exam Tip: Remember that subtracting a negative number is equivalent to adding its positive counterpart. Always simplify mixed numbers to improper fractions or whole numbers to make calculations easier before finding the LCM and proceeding with operations.

Question 3. Find the product:
(i) \( \frac { 9 }{ 2 } \times ( \frac { -7 }{ 4 }) \)
(ii) \( \frac { 3 }{ 10 } \times (-9) \)
(iii) \( \frac { -6 }{ 5 } \times \frac { 9 }{ 11 } \)
(iv) \( \frac { 3 }{ 7 } \times \frac { -2 }{ 5 } \)
(v) \( \frac { 3 }{ 11 } \times \frac { 2 }{ 5 } \)
(vi) \( \frac { 3 }{ -5 } \times \frac { -5 }{ 3 } \)
Answer:
(i) To find the product \( \frac { 9 }{ 2 } \times ( \frac { -7 }{ 4 }) \), we multiply the numerators together and the denominators together.
\( \frac { 9 }{ 2 } \times \frac { -7 }{ 4 } = \frac { 9\times(-7) }{ 2\times4 } = \frac { -63 }{ 8 } \)
This can also be expressed as a mixed number: \( -7\frac { 7 }{ 8 } \).
(ii) To find the product \( \frac { 3 }{ 10 } \times (-9) \), we can write -9 as \( \frac { -9 }{ 1 } \).
\( \frac { 3 }{ 10 } \times \frac { -9 }{ 1 } = \frac { 3\times(-9) }{ 10\times1 } = \frac { -27 }{ 10 } \)
This can also be expressed as a mixed number: \( -2\frac { 7 }{ 10 } \).
(iii) To find the product \( \frac { -6 }{ 5 } \times \frac { 9 }{ 11 } \), we multiply the numerators and the denominators.
\( \frac { -6 }{ 5 } \times \frac { 9 }{ 11 } = \frac { (-6)\times9 }{ 5\times11 } = \frac { -54 }{ 55 } \)
(iv) To find the product \( \frac { 3 }{ 7 } \times \frac { -2 }{ 5 } \), we multiply the numerators and the denominators.
\( \frac { 3 }{ 7 } \times \frac { -2 }{ 5 } = \frac { 3\times(-2) }{ 7\times5 } = \frac { -6 }{ 35 } \)
(v) To find the product \( \frac { 3 }{ 11 } \times \frac { 2 }{ 5 } \), we multiply the numerators and the denominators.
\( \frac { 3 }{ 11 } \times \frac { 2 }{ 5 } = \frac { 3\times2 }{ 11\times5 } = \frac { 6 }{ 55 } \)
(vi) To find the product \( \frac { 3 }{ -5 } \times \frac { -5 }{ 3 } \), we can simplify by cancelling common factors or multiply directly.
\( \frac { 3 }{ -5 } \times \frac { -5 }{ 3 } = \frac { 3\times(-5) }{ (-5)\times3 } = \frac { -15 }{ -15 } = 1 \)
In simple words: To find the product of fractions, just multiply the top numbers together and the bottom numbers together. If there's a negative sign, make sure to keep track of it. When a number is multiplied by its reciprocal, the result is always 1.

Exam Tip: When multiplying rational numbers, always simplify by canceling common factors in the numerator and denominator before multiplying to reduce the complexity of calculations.

Question 4. Draw the number line and represent the following rational numbers on it:
(i) \( (-4) \div \frac { 2 }{ 3 } \)
(ii) \( \frac {-3}{5} \div 2 \)
(iii) \( \frac { -4 }{ 5 } \div (-3) \)
(iv) \( \frac { -1 }{ 8 } \div \frac { 3 }{ 4 } \)
(v) \( \frac { -2 }{ 13 } \div \frac { 1 }{7} \)
(vi) \( \frac { -7 }{12} \div (\frac { -2 }{ 13 }) \)
(vii) \( \frac { 3 }{13} \div (\frac { -4 }{65}) \)
Answer:
(i) To find \( (-4) \div \frac { 2 }{ 3 } \), we multiply -4 by the reciprocal of \( \frac { 2 }{ 3 } \). The reciprocal of \( \frac { 2 }{ 3 } \) is \( \frac { 3 }{ 2 } \).
\( (-4) \div \frac { 2 }{ 3 } = (-4) \times \frac { 3 }{ 2 } = \frac { -4\times3 }{ 1\times2 } = \frac { -12 }{ 2 } = -6 \)
To represent -6 on a number line, locate the point 6 units to the left of 0.

(ii) To find \( \frac {-3}{5} \div 2 \), we multiply \( \frac {-3}{5} \) by the reciprocal of 2. The reciprocal of 2 is \( \frac { 1 }{ 2 } \).
\( \frac {-3}{5} \div 2 = \frac { -3 }{ 5 } \times \frac { 1 }{ 2 } = \frac { (-3)\times1 }{ 5\times2 } = \frac { -3 }{ 10 } \)
To represent \( \frac {-3}{10} \) on a number line, divide the segment between 0 and -1 into 10 equal parts and mark the third part from 0 towards -1.

(iii) To find \( \frac { -4 }{ 5 } \div (-3) \), we multiply \( \frac { -4 }{ 5 } \) by the reciprocal of -3. The reciprocal of -3 is \( \frac { 1 }{ -3 } \).
\( \frac { -4 }{ 5 } \div (-3) = \frac { -4 }{ 5 } \times \frac { 1 }{ -3 } = \frac { (-4)\times1 }{ 5\times(-3) } = \frac { -4 }{ -15 } = \frac { 4 }{ 15 } \)
To represent \( \frac { 4 }{ 15 } \) on a number line, divide the segment between 0 and 1 into 15 equal parts and mark the fourth part from 0 towards 1.

(iv) To find \( \frac { -1 }{ 8 } \div \frac { 3 }{ 4 } \), we multiply \( \frac { -1 }{ 8 } \) by the reciprocal of \( \frac { 3 }{ 4 } \). The reciprocal of \( \frac { 3 }{ 4 } \) is \( \frac { 4 }{ 3 } \).
\( \frac { -1 }{ 8 } \div \frac { 3 }{ 4 } = \frac { -1 }{ 8 } \times \frac { 4 }{ 3 } = \frac { -1\times4 }{ 8\times3 } = \frac { -4 }{ 24 } = \frac { -1 }{ 6 } \)
To represent \( \frac {-1}{6} \) on a number line, divide the segment between 0 and -1 into 6 equal parts and mark the first part from 0 towards -1.

(v) To find \( \frac { -2 }{ 13 } \div \frac { 1 }{7} \), we multiply \( \frac { -2 }{ 13 } \) by the reciprocal of \( \frac { 1 }{ 7 } \). The reciprocal of \( \frac { 1 }{ 7 } \) is \( \frac { 7 }{ 1 } \).
\( \frac { -2 }{ 13 } \div \frac { 1 }{ 7 } = \frac { -2 }{ 13 } \times \frac { 7 }{ 1 } = \frac { (-2)\times7 }{ 13\times1 } = \frac { -14 }{ 13 } \)
This can also be expressed as a mixed number: \( -1\frac { 1 }{ 13 } \).
To represent \( -1\frac { 1 }{ 13 } \) on a number line, locate -1 and then move \( \frac { 1 }{ 13 } \) more unit to the left, dividing the segment between -1 and -2 into 13 equal parts and marking the first part.

(vi) To find \( \frac { -7 }{12} \div (\frac { -2 }{ 13 }) \), we multiply \( \frac { -7 }{12} \) by the reciprocal of \( \frac { -2 }{ 13 } \). The reciprocal of \( \frac { -2 }{ 13 } \) is \( \frac { 13 }{ -2 } \).
\( \frac { -7 }{ 12 } \div (\frac { -2 }{ 13 } ) = \frac { -7 }{ 12 } \times (\frac { 13 }{ -2 } ) = \frac { (-7)\times13 }{ 12\times(-2) } = \frac { -91 }{ -24 } = \frac { 91 }{ 24 } \)
This can also be expressed as a mixed number: \( 3\frac { 19 }{ 24 } \).
To represent \( 3\frac { 19 }{ 24 } \) on a number line, locate 3 and then move \( \frac { 19 }{ 24 } \) more unit to the right, dividing the segment between 3 and 4 into 24 equal parts and marking the 19th part.

(vii) To find \( \frac { 3 }{13} \div (\frac { -4 }{65}) \), we multiply \( \frac { 3 }{13} \) by the reciprocal of \( \frac { -4 }{ 65 } \). The reciprocal of \( \frac { -4 }{ 65 } \) is \( \frac { 65 }{ -4 } \).
\( \frac { 3 }{ 13 } \div (\frac { -4 }{ 65 }) = \frac { 3 }{ 13 } \times (\frac { 65 }{ -4 } ) = \frac { 3\times65 }{ 13\times(-4) } = \frac { 195 }{ -52 } = - \frac { 195 }{ 52 } \)
This can be simplified by dividing both numerator and denominator by 13: \( \frac { -15 }{ 4 } \).
This can also be expressed as a mixed number: \( -3\frac { 3 }{ 4 } \).
To represent \( -3\frac { 3 }{ 4 } \) on a number line, locate -3 and then move \( \frac { 3 }{ 4 } \) more unit to the left, dividing the segment between -3 and -4 into 4 equal parts and marking the third part.

In simple words: To divide by a fraction, you flip the second fraction (find its reciprocal) and then multiply it by the first fraction. Always simplify the answer if you can. Then, you place the resulting fraction or whole number accurately on the number line, remembering that negative numbers are to the left of zero.

Exam Tip: Remember that dividing by a fraction is the same as multiplying by its reciprocal. When representing rational numbers on a number line, ensure accurate subdivision of units to place the points correctly, especially for mixed numbers.