Maharashtra Board Class 7 Chapter 5 Set 22 Operations on Rational Numbers Solutions

Question 1. Carry out the following additions of rational numbers:
(i) \( \frac{5}{36} + \frac{6}{42} \)
(ii) \( 1\frac{2}{3} + 2\frac{4}{5} \)
(iii) \( \frac{11}{17} + \frac{13}{19} \)
(iv) \( 2\frac{3}{11} + 1\frac{3}{77} \)
Answer:
(i) \( \frac{5}{36} + \frac{6}{42} \)
\( = \frac{5 \times 7}{36 \times 7} + \frac{6 \times 6}{42 \times 6} \)
\( = \frac{35}{252} + \frac{36}{252} \)
\( = \frac{35+36}{252} \)
\( = \frac{71}{252} \)
(ii) \( 1\frac{2}{3} + 2\frac{4}{5} \)
\( = \frac{5}{3} + \frac{14}{5} \)
\( = \frac{5 \times 5}{3 \times 5} + \frac{14 \times 3}{5 \times 3} \)
\( = \frac{25}{15} + \frac{42}{15} \)
\( = \frac{25+42}{15} \)
\( = \frac{67}{15} \)
(iii) \( \frac{11}{17} + \frac{13}{19} \)
\( = \frac{11 \times 19}{17 \times 19} + \frac{13 \times 17}{19 \times 17} \)
\( = \frac{209}{323} + \frac{221}{323} \)
\( = \frac{209+221}{323} \)
\( = \frac{430}{323} \)
(iv) \( 2\frac{3}{11} + 1\frac{3}{77} \)
\( = \frac{25}{11} + \frac{80}{77} \)
\( = \frac{25 \times 7}{11 \times 7} + \frac{80}{77} \)
\( = \frac{175}{77} + \frac{80}{77} \)
\( = \frac{175+80}{77} \)
\( = \frac{255}{77} \)
In simple words: To add rational numbers, first convert mixed fractions to improper fractions. Then, find a common denominator (LCM) for all fractions, convert them to equivalent fractions with the common denominator, and finally add the numerators while keeping the denominator the same.

🎯 Exam Tip: Always simplify fractions to their lowest terms at the end of calculations to secure full marks.

Question 2. Carry out the following subtractions involving rational numbers.
(i) \( \frac{7}{11} - \frac{3}{7} \)
(ii) \( \frac{13}{36} - \frac{2}{40} \)
(iii) \( 1\frac{2}{3} - 3\frac{5}{6} \)
(iv) \( 4\frac{1}{2} - 3\frac{1}{3} \)
Answer:
(i) \( \frac{7}{11} - \frac{3}{7} \)
\( = \frac{7 \times 7}{11 \times 7} - \frac{3 \times 11}{7 \times 11} \)
\( = \frac{49}{77} - \frac{33}{77} \)
\( = \frac{49-33}{77} \)
\( = \frac{16}{77} \)
(ii) \( \frac{13}{36} - \frac{2}{40} \)
\( = \frac{13 \times 10}{36 \times 10} - \frac{2 \times 9}{40 \times 9} \)
\( = \frac{130}{360} - \frac{18}{360} \)
\( = \frac{130-18}{360} \)
\( = \frac{112}{360} \)
\( = \frac{112 \div 8}{360 \div 8} \)
\( = \frac{14}{45} \)
(iii) \( 1\frac{2}{3} - 3\frac{5}{6} \)
\( = \frac{5}{3} - \frac{23}{6} \)
\( = \frac{5 \times 2}{3 \times 2} - \frac{23}{6} \)
\( = \frac{10}{6} - \frac{23}{6} \)
\( = \frac{10-23}{6} \)
\( = \frac{-13}{6} \)
(iv) \( 4\frac{1}{2} - 3\frac{1}{3} \)
\( = \frac{9}{2} - \frac{10}{3} \)
\( = \frac{9 \times 3}{2 \times 3} - \frac{10 \times 2}{3 \times 2} \)
\( = \frac{27}{6} - \frac{20}{6} \)
\( = \frac{27-20}{6} \)
\( = \frac{7}{6} \)
In simple words: To subtract rational numbers, convert mixed fractions to improper fractions and then find a common denominator. Convert fractions to equivalent forms with the common denominator and subtract the numerators. Simplify the final fraction if possible.

🎯 Exam Tip: Pay close attention to negative signs, especially when converting mixed numbers or subtracting fractions, as sign errors are common.

Question 3. Multiply the following rational numbers.
(i) \( \frac{3}{11} \times \frac{2}{5} \)
(ii) \( \frac{12}{5} \times \frac{4}{15} \)
(iii) \( \frac{(-8)}{9} \times \frac{3}{4} \)
(iv) \( \frac{0}{6} \times \frac{3}{4} \)
Answer:
(i) \( \frac{3}{11} \times \frac{2}{5} \)
\( = \frac{3 \times 2}{11 \times 5} \)
\( = \frac{6}{55} \)
(ii) \( \frac{12}{5} \times \frac{4}{15} \)
\( = \frac{4 \times 3}{5} \times \frac{4}{5 \times 3} \)
\( = \frac{4 \times 4}{5 \times 5} \)
\( = \frac{16}{25} \)
(iii) \( \frac{(-8)}{9} \times \frac{3}{4} \)
\( = \frac{(-2) \times 4}{3 \times 3} \times \frac{3}{4} \)
\( = \frac{-2}{3} \)
(iv) \( \frac{0}{6} \times \frac{3}{4} \)
\( = 0 \times \frac{3}{4} \)
\( = 0 \)
In simple words: To multiply rational numbers, multiply the numerators together and multiply the denominators together. Simplify the resulting fraction by canceling out common factors before or after multiplication. Any number multiplied by zero equals zero.

🎯 Exam Tip: Always look for opportunities to cross-cancel common factors in the numerator and denominator before multiplying to simplify calculations and avoid large numbers.

Question 4. Write the multiplicative inverse of.
(i) \( \frac{2}{5} \)
(ii) \( \frac{-3}{8} \)
(iii) \( \frac{-17}{39} \)
(iv) \( 7 \)
(v) \( -7\frac{1}{3} \)
Answer:
(i) \( \frac{5}{2} \)
(ii) \( \frac{8}{-3} \)
(iii) \( \frac{39}{-17} \)
(iv) \( \frac{1}{7} \)
(v) \( -7\frac{1}{3} = -\frac{22}{3} \implies \) Multiplicative inverse is \( \frac{-3}{22} \)
In simple words: The multiplicative inverse (or reciprocal) of a number is what you multiply the number by to get 1. For a fraction \( \frac{a}{b} \), its inverse is \( \frac{b}{a} \). For a whole number \( x \), its inverse is \( \frac{1}{x} \). For a negative number, its inverse will also be negative.

🎯 Exam Tip: Remember to convert mixed numbers to improper fractions first before finding their multiplicative inverse. The sign of the number does not change when finding the inverse.

Question 5. Carry out the divisions of rational numbers:
(i) \( \frac{40}{12} \div \frac{10}{4} \)
(ii) \( \frac{-10}{11} \div \frac{-11}{10} \)
(iii) \( \frac{-7}{8} \div \frac{-3}{6} \)
(iv) \( \frac{2}{3} \div (-4) \)
(v) \( 2\frac{1}{5} \div \frac{5}{11} \)
(vi) \( \frac{-5}{13} \div \frac{7}{26} \)
(vii) \( \frac{9}{11} \div (-8) \)
(viii) \( 5 \div \frac{2}{5} \)
Answer:
(i) \( \frac{40}{12} \div \frac{10}{4} \)
\( = \frac{40}{12} \times \frac{4}{10} \)
\( = \frac{4}{3} \)
(ii) \( \frac{-10}{11} \div \frac{-11}{10} \)
\( = \frac{-10}{11} \times \frac{10}{-11} \)
\( = \frac{100}{121} \)
(iii) \( \frac{-7}{8} \div \frac{-3}{6} \)
\( = \frac{-7}{8} \times \frac{6}{-3} \)
\( = \frac{-7}{8} \times (-2) \)
\( = \frac{7}{4} \)
(iv) \( \frac{2}{3} \div (-4) \)
\( = \frac{2}{3} \times \frac{1}{-4} \)
\( = \frac{1}{3} \times \frac{1}{-2} \)
\( = \frac{-1}{6} \)
(v) \( 2\frac{1}{5} \div \frac{5}{11} \)
\( = \frac{11}{5} \div \frac{5}{11} \)
\( = \frac{11}{5} \times \frac{11}{5} \)
\( = \frac{121}{25} \)
(vi) \( \frac{-5}{13} \div \frac{7}{26} \)
\( = \frac{-5}{13} \times \frac{26}{7} \)
\( = \frac{-5 \times 2}{7} \)
\( = \frac{-10}{7} \)
(vii) \( \frac{9}{11} \div (-8) \)
\( = \frac{9}{11} \times \frac{1}{-8} \)
\( = \frac{-9}{88} \)
(viii) \( 5 \div \frac{2}{5} \)
\( = \frac{5}{1} \times \frac{5}{2} \)
\( = \frac{25}{2} \)
In simple words: To divide rational numbers, you multiply the first fraction by the multiplicative inverse (reciprocal) of the second fraction. Remember to convert whole numbers or mixed fractions into improper fractions first.

🎯 Exam Tip: Reciprocating the divisor (the second fraction) is the most crucial step in division of rational numbers; ensure this is done correctly before multiplying.

Maharashtra Board Class 7 Maths Chapter 5 Operations On Rational Numbers Practice Set 22 Intext Questions And Activities

Question 1. Complete the table given below. (Textbook pg. no. 34)
Answer:

In simple words: This table classifies different numbers into Natural Numbers, Integers, and Rational Numbers. Natural numbers are positive whole numbers (1, 2, 3...). Integers include positive and negative whole numbers and zero (...-2, -1, 0, 1, 2...). Rational numbers are any number that can be expressed as a fraction \( \frac{p}{q} \) where p and q are integers and q is not zero.

🎯 Exam Tip: Understand the definitions of natural numbers, integers, and rational numbers thoroughly, as classifying numbers is a fundamental concept in mathematics.

Question 2. Discuss the characteristics of various groups of numbers in class and complete the table below. In front of each group, write the inference you make after carrying out the operations of addition, subtraction, multiplication and division, using a (√) or a (x). Remember that you cannot divide by zero. (Textbook pg. no. 35)
Answer:

In simple words: This table demonstrates the closure property of different number sets under various arithmetic operations. A set is 'closed' under an operation if performing that operation on any two numbers in the set always results in a number that is also in the set. Rational numbers are closed under all four basic operations (excluding division by zero).

🎯 Exam Tip: Understanding the closure property helps in comprehending why certain operations yield results within a number system while others do not, which is a key concept in number theory.