Maharashtra Board Class 7 Chapter 5 Set 24 Operations on Rational Numbers Solutions

Question 1. Write the following rational numbers in decimal form.
(i) \( \frac{13}{4} \)
(ii) \( \frac{-7}{8} \)
(iii) \( 7\frac{3}{5} \)
(iv) \( \frac{5}{12} \)
(v) \( \frac{22}{7} \)
(vi) \( \frac{4}{3} \)
(vii) \( \frac{7}{9} \)
Answer:
Solution:
(i) \( \frac{13}{4} \)
\[ \begin{array}{r} 3.25 \\ 4)\overline{13.00} \\ -12\phantom{00} \\ \hline 1\,0\phantom{0} \\ -\,8\phantom{0} \\ \hline \,20 \\ -20 \\ \hline \,0 \end{array} \]
\( \implies \frac{13}{4} = 3.25 \)
(ii) \( \frac{-7}{8} \)
\[ \begin{array}{r} 0.875 \\ 8)\overline{7.000} \\ -\,0\phantom{000} \\ \hline 70\phantom{00} \\ -64\phantom{00} \\ \hline \,60\phantom{0} \\ -56\phantom{0} \\ \hline \,40 \\ -40 \\ \hline \,0 \end{array} \]
\( \implies \frac{-7}{8} = (-1) \times 0.875 \)
\( \implies = -0.875 \)
(iii) \( 7\frac{3}{5} \)
\( \implies 7\frac{3}{5} = \frac{7 \times 5 + 3}{5} = \frac{35+3}{5} = \frac{38}{5} \)
\[ \begin{array}{r} 7.6 \\ 5)\overline{38.0} \\ -35\phantom{0} \\ \hline \,30 \\ -30 \\ \hline \,0 \end{array} \]
\( \implies 7\frac{3}{5} = 7.6 \)
(iv) \( \frac{5}{12} \)
\[ \begin{array}{r} 0.416\overline{6} \\ 12)\overline{5.0000} \\ -\,0\phantom{0000} \\ \hline 50\phantom{000} \\ -48\phantom{000} \\ \hline \,20\phantom{00} \\ -12\phantom{00} \\ \hline \,80\phantom{0} \\ -72\phantom{0} \\ \hline \,80 \\ -72 \\ \hline \,8 \end{array} \]
\( \implies \frac{5}{12} = 0.41\overline{6} \)
(v) \( \frac{22}{7} \)
\[ \begin{array}{r} 3.142857... \\ 7)\overline{22.000000} \\ -21\phantom{000000} \\ \hline \,10\phantom{00000} \\ -\,7\phantom{00000} \\ \hline \,30\phantom{0000} \\ -28\phantom{0000} \\ \hline \,20\phantom{000} \\ -14\phantom{000} \\ \hline \,60\phantom{00} \\ -56\phantom{00} \\ \hline \,40\phantom{0} \\ -35\phantom{0} \\ \hline \,50 \\ -49 \\ \hline \,1 \end{array} \]
\( \implies \frac{22}{7} = 3.142857... \)
(vi) \( \frac{4}{3} \)
\[ \begin{array}{r} 1.3\overline{3} \\ 3)\overline{4.00} \\ -3\phantom{00} \\ \hline 10\phantom{0} \\ -\,9\phantom{0} \\ \hline \,10 \\ -\,9 \\ \hline \,1 \end{array} \]
\( \implies \frac{4}{3} = 1.\overline{3} \)
(vii) \( \frac{7}{9} \)
\[ \begin{array}{r} 0.7\overline{7} \\ 9)\overline{7.00} \\ -\,0\phantom{00} \\ \hline 70\phantom{0} \\ -63\phantom{0} \\ \hline \,70 \\ -63 \\ \hline \,7 \end{array} \]
\( \implies \frac{7}{9} = 0.\overline{7} \)
In simple words: To convert a rational number to decimal form, perform division of the numerator by the denominator. Continue the division until the remainder is zero (terminating decimal) or a repeating pattern of digits emerges (non-terminating repeating decimal). For mixed fractions, first convert them to improper fractions before dividing.

🎯 Exam Tip: Ensure precise long division calculations, especially for recurring decimals, to avoid errors. Correct placement of the decimal point and recurring bar is crucial for full marks.

Maharashtra Board Class 7 Maths Chapter 5 Operations On Rational Numbers Practice Set 24 Intext Questions And Activities

Question 1. Without using division, can we tell from the denominator of a fraction, whether the decimal form of the fraction will be a terminating decimal? Find out. (Textbook pg. no. 40)
Answer: Solution:
If the prime factorization of the denominator of a fraction has only factors as 2 or 5 or a combination of 2 and 5 then the decimal form of that fractional will be a terminating decimal form.
Consider the fractions \( \frac{17}{20} \) and \( \frac{19}{6} \)
Now, \( 20 = 2 \times 2 \times 5 \), and \( 6 = 2 \times 3 \)
\( \therefore \frac{17}{20} \) is terminating decimal form while \( \frac{19}{6} \) is recurring decimal form.
In simple words: Yes, a fraction's decimal form is terminating if and only if the prime factors of its denominator (in simplest form) are only 2s and/or 5s. If any other prime factor exists, it will be a non-terminating, repeating decimal.

🎯 Exam Tip: Remember to simplify the fraction to its lowest terms before prime factorizing the denominator. This method helps quickly identify terminating or non-terminating decimals without performing actual division, which is a key concept in number theory.