GSEB Class 7 Maths Solutions Chapter 9 સંમેય સંખ્યાઓ Exercise 9.2

Gujarat Board Textbook Solutions Class 7 Maths Chapter 9 સંમેય સંખ્યાઓ Ex 9.2

Question 1. સરવાળો શોધોઃ
(i) \( \frac{5}{4}+\left(\frac{-11}{4}\right) \)
(ii) \( \frac{5}{3}+\frac{3}{5} \)
(iii) \( \frac{-9}{10}+\frac{22}{15} \)
(iv) \( \frac{-3}{-11}+\frac{5}{9} \)
(v) \( \frac{-8}{19}+\frac{(-2)}{57} \)
(vi) \( \frac{-2}{3} + 0 \)
(vii) \( -2\frac{1}{3} + 4\frac{3}{5} \)
Answer:
(i) \( \frac{5}{4}+\left(\frac{-11}{4}\right) \)
\( = \frac{5+(-11)}{4} \)
\( = \frac{-6}{4} \)
\( = \frac{-3}{2} \)
\( = -1\frac{1}{2} \) The sum is negative one and a half.
(ii) \( \frac{5}{3}+\frac{3}{5} \)
For 3 and 5, the Least Common Multiple (LCM) is 15.
\( \therefore \frac{5}{3} = \frac{5 \times 5}{3 \times 5} = \frac{25}{15} \) and \( \frac{3}{5} = \frac{3 \times 3}{5 \times 3} = \frac{9}{15} \)
\( \therefore \frac{5}{3}+\frac{3}{5} = \frac{25}{15}+\frac{9}{15} \)
\( = \frac{25+9}{15} \)
\( = \frac{34}{15} \)
\( = 2\frac{4}{15} \) The total of these fractions is two and four-fifteenths.
(iii) \( \frac{-9}{10}+\frac{22}{15} \)
For 10 and 15, the Least Common Multiple (LCM) is 30.
\( \therefore \frac{-9}{10} = \frac{(-9) \times 3}{10 \times 3} = \frac{-27}{30} \) and \( \frac{22}{15} = \frac{22 \times 2}{15 \times 2} = \frac{44}{30} \)
\( \therefore \frac{-9}{10}+\frac{22}{15} = \frac{-27}{30}+\frac{44}{30} \)
\( = \frac{-27+44}{30} \)
\( = \frac{17}{30} \) The result of adding these numbers is seventeen-thirtieths.
(iv) \( \frac{-3}{-11}+\frac{5}{9} \)
For 11 and 9, the Least Common Multiple (LCM) is 99.
\( \therefore \frac{-3}{-11} = \frac{(-3) \times 9}{(-11) \times 9} = \frac{-27}{-99} = \frac{27}{99} \) and \( \frac{5}{9} = \frac{5 \times 11}{9 \times 11} = \frac{55}{99} \)
\( \therefore \frac{-3}{-11}+\frac{5}{9} = \frac{27}{99}+\frac{55}{99} \)
\( = \frac{27+55}{99} \)
\( = \frac{82}{99} \) The sum of the fractions is eighty-two ninety-ninths.
(v) \( \frac{-8}{19}+\frac{(-2)}{57} \)
For 19 and 57, the Least Common Multiple (LCM) is 57.
\( \therefore \frac{-8}{19} = \frac{(-8) \times 3}{19 \times 3} = \frac{-24}{57} \)
\( \therefore \frac{-8}{19}+\frac{(-2)}{57} = \frac{-24}{57}+\frac{(-2)}{57} \)
\( = \frac{-24-2}{57} \)
\( = \frac{-26}{57} \) The result of this addition is negative twenty-six fifty-sevenths.
(vi) \( \frac{-2}{3} + 0 \)
\( = \frac{-2+0}{3} \)
\( = \frac{-2}{3} \) Adding zero to a number gives the same number.
(vii) \( -2\frac{1}{3} + 4\frac{3}{5} \)
\( -2\frac{1}{3} = \frac{-(2 \times 3 + 1)}{3} = \frac{-7}{3} \) and \( 4\frac{3}{5} = \frac{4 \times 5 + 3}{5} = \frac{23}{5} \)
For 3 and 5, the Least Common Multiple (LCM) is 15.
\( \therefore \frac{-7}{3} = \frac{-7 \times 5}{3 \times 5} = \frac{-35}{15} \) and \( \frac{23}{5} = \frac{23 \times 3}{5 \times 3} = \frac{69}{15} \)
\( \therefore \frac{-7}{3}+\frac{23}{5} = \frac{-35}{15}+\frac{69}{15} \)
\( = \frac{-35+69}{15} \)
\( = \frac{34}{15} \)
\( = 2\frac{4}{15} \) The sum of the mixed fractions is two and four-fifteenths.
In simple words: To add fractions, first make sure the bottom numbers (denominators) are the same. If they're not, find the smallest number that both denominators can divide into (this is the LCM). Change both fractions to have this new bottom number, then add the top numbers (numerators). Simplify your answer if you can.

Exam Tip: When adding fractions with different denominators, always find the Least Common Multiple (LCM) to ensure the simplest common denominator for calculations.

Question 2. શોધોઃ
(i) \( \frac{7}{24}-\frac{17}{36} \)
(ii) \( \frac{5}{63}-\left(\frac{-6}{21}\right) \)
(iii) \( \frac{-6}{13}-\left(\frac{-7}{15}\right) \)
(iv) \( \frac{-3}{8}-\frac{7}{11} \)
(v) \( -2\frac{1}{9} - 6 \)
Answer:
(i) \( \frac{7}{24}-\frac{17}{36} \)
For 24 and 36, the Least Common Multiple (LCM) is 72.
\( \therefore \frac{7}{24} = \frac{7 \times 3}{24 \times 3} = \frac{21}{72} \) and \( \frac{17}{36} = \frac{17 \times 2}{36 \times 2} = \frac{34}{72} \)
Now, \( \frac{7}{24}-\frac{17}{36} = \frac{21}{72}-\frac{34}{72} \)
\( = \frac{21-34}{72} \)
\( = \frac{-13}{72} \) The result of this subtraction is negative thirteen seventy-seconds.
(ii) \( \frac{5}{63}-\left(\frac{-6}{21}\right) \)
For 63 and 21, the Least Common Multiple (LCM) is 63.
\( \therefore \frac{5}{63} = \frac{5 \times 1}{63 \times 1} = \frac{5}{63} \) and \( \frac{6}{21} = \frac{6 \times 3}{21 \times 3} = \frac{18}{63} \)
Now, \( \frac{5}{63}-\left(\frac{-6}{21}\right) = \frac{5}{63}+\frac{6}{21} \) [Because subtracting a negative is like adding a positive]
\( = \frac{5}{63}+\frac{18}{63} \)
\( = \frac{5+18}{63} \)
\( = \frac{23}{63} \) The difference between these numbers is twenty-three sixty-thirds.
(iii) \( \frac{-6}{13}-\left(\frac{-7}{15}\right) \)
For 13 and 15, the Least Common Multiple (LCM) is 195.
\( \therefore \frac{-6}{13} = \frac{-6 \times 15}{13 \times 15} = \frac{-90}{195} \) and \( \frac{-7}{15} = \frac{-7 \times 13}{15 \times 13} = \frac{-91}{195} \)
Now, \( \frac{-6}{13}-\left(\frac{-7}{15}\right) = \frac{-90}{195}-\left(\frac{-91}{195}\right) \)
\( = \frac{-90}{195}+\frac{91}{195} \) [Because subtracting a negative is like adding a positive]
\( = \frac{-90+91}{195} \)
\( = \frac{1}{195} \) The result of this calculation is one one-hundred-ninety-fifth.
(iv) \( \frac{-3}{8}-\frac{7}{11} \)
For 8 and 11, the Least Common Multiple (LCM) is 88.
\( \therefore \frac{-3}{8} = \frac{(-3) \times 11}{8 \times 11} = \frac{-33}{88} \) and \( \frac{7}{11} = \frac{7 \times 8}{11 \times 8} = \frac{56}{88} \)
Now, \( \frac{-3}{8}-\frac{7}{11} = \frac{-33}{88}-\frac{56}{88} \)
\( = \frac{-33-56}{88} \)
\( = \frac{-89}{88} \)
\( = -1\frac{1}{88} \) The final answer is negative one and one eighty-eighth.
(v) \( -2\frac{1}{9} - 6 \)
\( -2\frac{1}{9} = \frac{-(2 \times 9 + 1)}{9} = \frac{-19}{9} \) and \( 6 = \frac{6}{1} \)
For 9 and 1, the Least Common Multiple (LCM) is 9.
\( \therefore \frac{-19}{9} = \frac{-19 \times 1}{9 \times 1} = \frac{-19}{9} \) and \( \frac{6}{1} = \frac{6 \times 9}{1 \times 9} = \frac{54}{9} \)
Now, \( -2\frac{1}{9} - 6 = \frac{-19}{9}-\frac{6}{1} \)
\( = \frac{-19}{9}-\frac{54}{9} \)
\( = \frac{-19-54}{9} \)
\( = \frac{-73}{9} \)
\( = -8\frac{1}{9} \) The difference is negative eight and one-ninth.
In simple words: When subtracting fractions, make sure they have the same bottom number. If a number is negative, remember that subtracting a negative number is the same as adding a positive one. Convert mixed numbers to improper fractions first.

Exam Tip: Be careful with negative signs when subtracting fractions. Remember that \( a - (-b) \) is equivalent to \( a + b \).

Question 3. ગુણાકાર શોધોઃ
(i) \( \frac{9}{2}\times\left(\frac{-7}{4}\right) \)
(ii) \( \frac{3}{10}\times(-9) \)
(iii) \( \frac{-6}{11}\times\left(\frac{5}{3}\right) \)
(iv) \( \frac{3}{7}\times\left(\frac{-2}{5}\right) \)
(v) \( \frac{3}{11}\times\frac{2}{5} \)
(vi) \( \frac{3}{-5}\times\left(\frac{-5}{3}\right) \)
Answer:
(i) \( \frac{9}{2}\times\left(\frac{-7}{4}\right) \)
\( = \frac{9 \times (-7)}{2 \times 4} \)
\( = \frac{-63}{8} \)
\( = -7\frac{7}{8} \) The product is negative seven and seven-eighths.
(ii) \( \frac{3}{10}\times(-9) \)
\( = \frac{3}{10}\times\frac{-9}{1} \)
\( = \frac{3 \times (-9)}{10 \times 1} \)
\( = \frac{-27}{10} \)
\( = -2\frac{7}{10} \) The multiplication gives negative two and seven-tenths.
(iii) \( \frac{-6}{11}\times\left(\frac{5}{3}\right) \)
\( = \frac{(-6) \times 5}{11 \times 3} \)
\( = \frac{-30}{33} \)
\( = \frac{-10}{11} \) After simplifying, the answer is negative ten-elevenths.
(iv) \( \frac{3}{7}\times\left(\frac{-2}{5}\right) \)
\( = \frac{3 \times (-2)}{7 \times 5} \)
\( = \frac{-6}{35} \) The result of multiplying these fractions is negative six thirty-fifths.
(v) \( \frac{3}{11}\times\frac{2}{5} \)
\( = \frac{3 \times 2}{11 \times 5} \)
\( = \frac{6}{55} \) The product is six fifty-fifths.
(vi) \( \frac{3}{-5}\times\left(\frac{-5}{3}\right) \)
\( = \frac{3 \times (-5)}{(-5) \times 3} \)
\( = \frac{-15}{-15} \)
\( = 1 \) When you multiply these, the answer is one.
In simple words: To multiply fractions, you just multiply the top numbers together and then multiply the bottom numbers together. If there are negative signs, count them; an even number of negatives makes the answer positive, and an odd number makes it negative. Always simplify your answer if possible.

Exam Tip: Before multiplying, look for opportunities to cross-cancel common factors between numerators and denominators to simplify the calculation.

Question 4. કિંમત શોધોઃ
(i) \( (-4)\div\frac{2}{3} \)
(ii) \( \frac{-3}{5}\div2 \)
(iii) \( \frac{-4}{5}\div(-3) \)
(iv) \( \frac{-1}{8}\div\frac{3}{4} \)
(v) \( \frac{-2}{13}\div\frac{1}{7} \)
(vi) \( \frac{-7}{12}\div\left(\frac{-2}{13}\right) \)
(vii) \( \frac{3}{13}\div\left(\frac{-4}{65}\right) \)
Answer:
(i) \( (-4)\div\frac{2}{3} \)
\( = (-4)\times\frac{3}{2} \)
\( = \frac{-4}{1}\times\frac{3}{2} \)
\( = \frac{(-4) \times 3}{1 \times 2} \)
\( = \frac{-12}{2} \)
\( = -6 \) Dividing by a fraction is the same as multiplying by its inverse, giving negative six.
(ii) \( \frac{-3}{5}\div2 \)
\( = \frac{-3}{5}\div\frac{2}{1} \)
\( = \frac{-3}{5}\times\frac{1}{2} \)
\( = \frac{(-3) \times 1}{5 \times 2} \)
\( = \frac{-3}{10} \) The result of this division is negative three-tenths.
(iii) \( \frac{-4}{5}\div(-3) \)
\( = \frac{-4}{5}\div\frac{-3}{1} \)
\( = \frac{-4}{5}\times\frac{-1}{3} \)
\( = \frac{(-4) \times (-1)}{5 \times 3} \)
\( = \frac{4}{15} \) Dividing by a negative number results in a positive answer, which is four-fifteenths.
(iv) \( \frac{-1}{8}\div\frac{3}{4} \)
\( = \frac{-1}{8}\times\frac{4}{3} \)
\( = \frac{(-1) \times 4}{8 \times 3} \)
\( = \frac{-4}{24} \)
\( = \frac{-1}{6} \) After simplification, the quotient is negative one-sixth.
(v) \( \frac{-2}{13}\div\frac{1}{7} \)
\( = \frac{-2}{13}\times\frac{7}{1} \)
\( = \frac{(-2) \times 7}{13 \times 1} \)
\( = \frac{-14}{13} \)
\( = -1\frac{1}{13} \) The result of dividing these fractions is negative one and one-thirteenth.
(vi) \( \frac{-7}{12}\div\left(\frac{-2}{13}\right) \)
\( = \frac{-7}{12}\times\frac{13}{-2} \)
\( = \frac{(-7) \times 13}{12 \times (-2)} \)
\( = \frac{-91}{-24} \)
\( = \frac{91}{24} \)
\( = 3\frac{19}{24} \) The division yields three and nineteen twenty-fourths.
(vii) \( \frac{3}{13}\div\left(\frac{-4}{65}\right) \)
\( = \frac{3}{13}\times\frac{65}{-4} \)
\( = \frac{3 \times 65}{13 \times (-4)} \)
\( = \frac{3 \times (5 \times 13)}{13 \times (-4)} \)
\( = \frac{3 \times 5}{-4} \)
\( = \frac{15}{-4} \)
\( = -3\frac{3}{4} \) The final value obtained is negative three and three-fourths.
In simple words: To divide fractions, flip the second fraction upside down (find its reciprocal) and then multiply it by the first fraction. Remember your rules for multiplying with negative numbers: two negatives make a positive, one negative makes a negative. Always simplify your answer if possible.

Exam Tip: Remember to always convert mixed numbers to improper fractions before performing division or multiplication operations, and simplify the final answer.