Maharashtra Board Class 7 Chapter 5 Set 23 Operations on Rational Numbers Solutions

Question 1. Write three rational numbers that lie between the two given numbers.
(i) \(\frac{2}{7}, \frac{6}{7}\)
(ii) \(\frac{4}{5}, \frac{2}{3}\)
(iii) \(-\frac{2}{3}, \frac{4}{5}\)
(iv) \(\frac{7}{9}, -\frac{5}{9}\)
(v) \(-\frac{3}{4}, \frac{+5}{4}\)
(vi) \(-\frac{7}{8}, -\frac{5}{3}\)
(vii) \(\frac{5}{7}, \frac{11}{7}\)
(viii) \(0, -\frac{3}{4}\)
Answer:
Solution:
(i) \(\frac{2}{7}, \frac{6}{7}\)
The three numbers lying between \(\frac{2}{7}\) and \(\frac{6}{7}\) are \(\frac{3}{7}, \frac{4}{7}, \frac{5}{7}\).
(ii) \(\frac{4}{5}, \frac{2}{3}\)
\(\frac{4}{5} = \frac{4 \times 6}{5 \times 6} = \frac{24}{30}\)
\(\frac{2}{3} = \frac{2 \times 10}{3 \times 10} = \frac{20}{30}\)
The three numbers between \(\frac{4}{5}\) and \(\frac{2}{3}\) are \(\frac{21}{30}, \frac{22}{30}, \frac{23}{30}\).
(iii) \(-\frac{2}{3}, \frac{4}{5}\)
\(-\frac{2}{3} = \frac{-2 \times 5}{3 \times 5} = \frac{-10}{15}\)
\(\frac{4}{5} = \frac{4 \times 3}{5 \times 3} = \frac{12}{15}\)
The three numbers between \(-\frac{2}{3}\) and \(\frac{4}{5}\) are \(-\frac{9}{15}, -\frac{7}{15}, -\frac{4}{15}\).
(iv) \(\frac{7}{9}, -\frac{5}{9}\)
The three numbers between \(\frac{7}{9}\) and \(-\frac{5}{9}\) are \(\frac{6}{9}, 0, -\frac{4}{9}\).
(v) \(-\frac{3}{4}, \frac{+5}{4}\)
The three numbers between \(-\frac{3}{4}\) and \(\frac{+5}{4}\) are \(-\frac{2}{4}, -\frac{1}{4}, \frac{3}{4}\).
(vi) \(-\frac{7}{8}, -\frac{5}{3}\)
\(-\frac{7}{8} = \frac{-7 \times 3}{8 \times 3} = \frac{-21}{24}\)
\(-\frac{5}{3} = \frac{-5 \times 8}{3 \times 8} = \frac{-40}{24}\)
The three numbers between \(-\frac{7}{8}\) and \(-\frac{5}{3}\) are \(-\frac{17}{24}, -\frac{11}{24}, -\frac{13}{24}\).
(vii) \(\frac{5}{7}, \frac{11}{7}\)
The three numbers between \(\frac{5}{7}\) and \(\frac{11}{7}\) are \(\frac{6}{7}, \frac{8}{7}, \frac{9}{7}\).
(viii) \(0, -\frac{3}{4}\)
The three numbers between \(0\) and \(-\frac{3}{4}\) are \(-\frac{1}{8}, -\frac{2}{8}, -\frac{5}{8}\).
In simple words: To find rational numbers between two given numbers, ensure they have a common denominator. Once the denominators are the same, any rational numbers whose numerators fall between the two given numerators (after converting) are valid. If not enough integers are between the numerators, multiply both the numerator and denominator by a common factor to create more space.

🎯 Exam Tip: Always make sure the denominators are common before finding numbers between them. For negative numbers, remember the order on the number line.

Maharashtra Board Class 7 Maths Chapter 5 Operations on Rational Numbers Practice Set 23 Intext Questions and Activities

Question 1. Answer the following questions: (Textbook pg. no. 36)
1. Write all the natural numbers between 2 and 9.
2. Write all the integers between -4, and 5.
3. Which rational numbers are there between \(\frac{1}{2}\) and \(\frac{3}{4}\)?
Answer:
Solution:
1. 3, 4, 5, 6, 7, 8
2. -3, -2, -1, 0, 1, 2, 3, 4
3. \(\frac{1}{2} = \frac{1 \times 2}{2 \times 2} = \frac{2}{4} = \frac{2 \times 10}{4 \times 10} = \frac{20}{40}\)
\(\frac{3}{4} = \frac{3 \times 10}{4 \times 10} = \frac{30}{40}\)
The rational numbers between \(\frac{1}{2}\) and \(\frac{3}{4}\) are \(\frac{21}{40}, \frac{22}{40}, \frac{25}{40}, \frac{27}{40}\) etc.
In simple words: Natural numbers are positive counting numbers. Integers include positive, negative, and zero. Rational numbers can be found by making their denominators common and then identifying fractions between their equivalent forms.

🎯 Exam Tip: Distinguish clearly between natural numbers, whole numbers, and integers. When finding rational numbers, always ensure common denominators for easier comparison.