Fractions

Q1: Find:

i. \(\frac{1}{8}\) of 40

Step 1: Multiply the fraction by the whole number: \[ \frac{1}{8} \times 40 = \frac{1 \times 40}{8} = \frac{40}{8} \] Step 2: Simplify the fraction: \[ \frac{40}{8} = 5 \] Answer: \(5\)

ii. \(\frac{4}{11}\) of \(4\frac{2}{5}\)

Step 1: Convert mixed fraction to improper fraction: \[ 4\frac{2}{5} = \frac{(4 \times 5) + 2}{5} = \frac{20 + 2}{5} = \frac{22}{5} \] Step 2: Multiply: \[ \frac{4}{11} \times \frac{22}{5} = \frac{4 \times 22}{11 \times 5} = \frac{88}{55} \] Step 3: Simplify: \[ \frac{88}{55} = \frac{8}{5} = 1\frac{3}{5} \] Answer: \(1\frac{3}{5}\)

iii. \(1\frac{3}{5}\) of \(6\frac{1}{4}\)

Step 1: Convert both mixed fractions to improper fractions: \[ 1\frac{3}{5} = \frac{(1 \times 5) + 3}{5} = \frac{8}{5}, \\ 6\frac{1}{4} = \frac{(6 \times 4) + 1}{4} = \frac{25}{4} \] Step 2: Multiply: \[ \frac{8}{5} \times \frac{25}{4} = \frac{8 \times 25}{5 \times 4} = \frac{200}{20} = 10 \] Answer: 10

iv. \(\frac{3}{4}\) of ₹1

Step 1: Multiply: \[ \frac{3}{4} \times 1 = \frac{3}{4} \] Answer: ₹\(\frac{3}{4}\) = ₹0.75

v. \(\frac{5}{8}\) of 1 km

Step 1: Multiply: \[ \frac{5}{8} \times 1 = \frac{5}{8} = 0.625 \] Answer: \(\frac{5}{8}\) km = 625 m

vi. \(\frac{5}{12}\) of 1 hour

Step 1: Multiply: \[ \frac{5}{12} \times 1 = \frac{5}{12} \\ = \frac{5 \times \ 60}{12} = 25 \] Answer: \(\frac{5}{12}\) hour = 25 minutes

Q2: Simplify: \(1\frac{1}{6}\div1\frac{5}{9}\times3\frac{1}{3}\)

Step 1: \[ 1\frac{1}{6} = \frac{7}{6}, \\ 1\frac{5}{9} = \frac{14}{9}, \\ 3\frac{1}{3} = \frac{10}{3} \]Step 2: Division: \[ \frac{7}{6} \div \frac{14}{9} = \frac{7}{6} \times \frac{9}{14} \]Step 3: Multiply: \[ \frac{7}{6} \times \frac{9}{14} = \frac{7 \times 9}{6 \times 14} = \frac{63}{84} \]Step 4: Simplify: \[ \frac{63}{84} = \frac{3}{4} \]Step 5: \[ \frac{3}{4} \times \frac{10}{3} = \frac{3 \times 10}{4 \times 3} = \frac{30}{12} \]Step 6: Simplify: \[ \frac{30}{12} = \frac{5}{2} = 2\frac{1}{2} \]Answer: \(2\frac{1}{2}\)

Q3: Simplify: \(1\frac{1}{3} \div \left(\frac{3}{7} \text{ of } 2\frac{5}{8}\right) + 1\frac{1}{9}\)

Step 1: \[ 1\frac{1}{3} = \frac{4}{3}, \\ 2\frac{5}{8} = \frac{21}{8}, \\ 1\frac{1}{9} = \frac{10}{9} \]Step 2: Multiply: \[ \frac{3}{7} \text{ of } \frac{21}{8} = \frac{3}{7} \times \frac{21}{8} = \frac{63}{56} \]Step 3: \[ \frac{4}{3} \div \frac{63}{56} = \frac{4}{3} \times \frac{56}{63} = \frac{224}{189} \]Step 4: Simplify \( \frac{224}{189} \): GCD of 224 and 189 is 7 \[ \frac{224 \div 7}{189 \div 7} = \frac{32}{27} = 1\frac{5}{27} \]Step 5: Convert \( \frac{10}{9} \) to denominator 27: \[ \frac{10}{9} = \frac{30}{27} \]Step 6: Add: \[ \frac{32}{27} + \frac{30}{27} = \frac{62}{27} = 2\frac{8}{27} \]Answer: \(2\frac{8}{27}\)

Q4: Simplify: \(6\frac{1}{5} \div \left(3\frac{1}{10} \text{ of } 2\frac{1}{2}\right) \div \frac{1}{4}\)

Step 1: \[ 6\frac{1}{5} = \frac{31}{5}, \\ 3\frac{1}{10} = \frac{31}{10}, \\ 2\frac{1}{2} = \frac{5}{2} \]Step 2: \[ \frac{31}{10} \text{ of } \frac{5}{2} = \frac{31}{10} \times \frac{5}{2} = \frac{155}{20} \]Step 3: Simplify \( \frac{155}{20} \): GCD of 155 and 20 is 5 \[ \frac{155 \div 5}{20 \div 5} = \frac{31}{4} \]Step 4: First division: \[ \frac{31}{5} \div \frac{31}{4} = \frac{31}{5} \times \frac{4}{31} = \frac{4}{5} \]Step 5: Second division: \[ \frac{4}{5} \div \frac{1}{4} = \frac{4}{5} \times \frac{4}{1} = \frac{16}{5} = 3\frac{1}{5} \]Answer: \(3\frac{1}{5}\)

Q5: Simplify: \(3\frac{2}{3} – \frac{3}{11} \text{ of } 2\frac{3}{4} \div 1\frac{1}{4} \times 1\frac{2}{3} + \frac{1}{3}\)

Step 1: \[ 3\frac{2}{3} = \frac{11}{3},\\ 2\frac{3}{4} = \frac{11}{4},\\ 1\frac{1}{4} = \frac{5}{4},\\ 1\frac{2}{3} = \frac{5}{3} \]Step 2: \[ \frac{3}{11} \text{ of } \frac{11}{4} = \frac{3}{11} \times \frac{11}{4} = \frac{33}{44} = \frac{3}{4} \]Step 3: Now divide by \( \frac{5}{4} \): \[ \frac{3}{4} \div \frac{5}{4} = \frac{3}{4} \times \frac{4}{5} = \frac{12}{20} = \frac{3}{5} \]Step 4: Multiply by \( \frac{5}{3} \): \[ \frac{3}{5} \times \frac{5}{3} = \frac{15}{15} = 1 \]Step 5: \[ \frac{11}{3} – 1 + \frac{1}{3} \]Step 6: \[ \frac{11}{3} – 1 = \frac{11}{3} – \frac{3}{3} = \frac{8}{3} \]Step 7: \[ \frac{8}{3} + \frac{1}{3} = \frac{9}{3} = 3 \]Answer: 3

Q6: Simplify: \(\left(2\frac{2}{7}\ \text{ of } 15\frac{3}{4}\right)\times2\frac{1}{4}\div\left(\frac{4}{7}\ \text{ of } 2\frac{5}{8}\right)\)

Step 1: \[ 2\frac{2}{7} = \frac{16}{7},\\ 15\frac{3}{4} = \frac{63}{4},\\ 2\frac{1}{4} = \frac{9}{4},\\ 2\frac{5}{8} = \frac{21}{8} \]Step 2: \[ \frac{16}{7} \text{ of } \frac{63}{4} = \frac{16}{7} \times \frac{63}{4} = \frac{1008}{28} = 36 \]Step 3: \[ \frac{4}{7} \text{ of } \frac{21}{8} = \frac{4}{7} \times \frac{21}{8} = \frac{84}{56} = \frac{3}{2} \]Step 4: \[ (36) \times \frac{9}{4} \div \frac{3}{2} \]

iv. First multiply, then divide:

Step 5: \[ 36 \times \frac{9}{4} = \frac{324}{4} = 81 \]Step 6: \[ 81 \div \frac{3}{2} = 81 \times \frac{2}{3} = \frac{162}{3} = 54 \]Answer: 54

Q7: Simplify: \(1\div\frac{4}{7}-\frac{1}{3}\ \text{of}\ 3\frac{3}{4}+\frac{1}{2}\div3\)

Step 1: Convert all mixed numbers to improper fractions: \[ 3\frac{3}{4} = \frac{15}{4} \]Step 2: Simplify each part of the expression: \[ \frac{1}{3} \text{ of } \frac{15}{4} = \frac{1}{3} \times \frac{15}{4} = \frac{15}{12} = \frac{5}{4} \]Step 3: \[ 1 \div \frac{4}{7} = 1 \times \frac{7}{4} = \frac{7}{4} \]Step 4: \[ \frac{1}{2} \div 3 = \frac{1}{2} \times \frac{1}{3} = \frac{1}{6} \]Step 5: Put it all together: \[ \frac{7}{4} – \frac{5}{4} + \frac{1}{6} \]Step 6: \[ \left(\frac{7}{4} – \frac{5}{4}\right) = \frac{2}{4} = \frac{1}{2} \]Step 7: \[ \frac{1}{2} + \frac{1}{6} = \frac{3}{6} + \frac{1}{6} = \frac{4}{6} = \frac{2}{3} \]Answer: \(\frac{2}{3}\)

Q8: Simplify: \(\left(\left(\frac{2}{3}+\frac{4}{9}\right)\ of\ \frac{3}{5}\right)\div1\frac{2}{3}\times1\frac{1}{4}-\frac{1}{3}\)

Step 1: First, add \(\frac{2}{3}+\frac{4}{9}\): \[ \frac{2}{3} = \frac{6}{9},\\ \frac{6}{9} + \frac{4}{9} = \frac{10}{9} \]Step 2: Find the product with \(\frac{3}{5}\): \[ \frac{10}{9} \times \frac{3}{5} = \frac{30}{45} = \frac{2}{3} \]Step 3: Convert mixed numbers to improper fractions: \[ 1\frac{2}{3} = \frac{5}{3}, \\ 1\frac{1}{4} = \frac{5}{4} \]Step 4: Continue with division and multiplication: \[ \frac{2}{3} \div \frac{5}{3} = \frac{2}{3} \times \frac{3}{5} = \frac{6}{15} = \frac{2}{5} \]Step 5: \[ \frac{2}{5} \times \frac{5}{4} = \frac{10}{20} = \frac{1}{2} \]Step 6: Subtract \(\frac{1}{3}\): \[ \frac{1}{2} – \frac{1}{3} = \frac{3}{6} – \frac{2}{6} = \frac{1}{6} \]Answer: \(\frac{1}{6}\)

Q9: Simplify: \(\left(\frac{14}{15}\div1\frac{1}{6}+\frac{7}{10}\right)\times\frac{3}{4}\)

Step 1: Convert mixed number to improper fraction: \[ 1\frac{1}{6} = \frac{7}{6} \]Step 2: Divide \(\frac{14}{15}\div\frac{7}{6}\): \[ \frac{14}{15} \div \frac{7}{6} = \frac{14}{15} \times \frac{6}{7} = \frac{84}{105} = \frac{4}{5} \]Step 3: Add \(\frac{4}{5} + \frac{7}{10}\): \[ \frac{4}{5} = \frac{8}{10},\\ \frac{8}{10} + \frac{7}{10} = \frac{15}{10} = \frac{3}{2} \]Step 4: Multiply with \(\frac{3}{4}\): \[ \frac{3}{2} \times \frac{3}{4} = \frac{9}{8} = 1\frac{1}{8} \]Answer: \(1\frac{1}{8}\)

Q10: Simplify: \(\frac{1}{3}\left(2\frac{1}{2}+3\frac{1}{3}\right)\div\frac{2}{9}\left(3\frac{1}{8}-1\frac{1}{12}\right)\)

i.

Step 1: Convert mixed numbers to improper fractions: \[ 2\frac{1}{2} = \frac{5}{2}, \\ 3\frac{1}{3} = \frac{10}{3}, \\ 3\frac{1}{8} = \frac{25}{8}, \\ 1\frac{1}{12} = \frac{13}{12} \]Step 2: Evaluate the expressions in parentheses: \[ \frac{5}{2} + \frac{10}{3} = \frac{15}{6} + \frac{20}{6} = \frac{35}{6} \\ \frac{25}{8} – \frac{13}{12} = \text{LCM of 8 and 12 is 24} \\ \frac{75}{24} – \frac{26}{24} = \frac{49}{24} \]Step 3: Multiply with outside fractions: \[ \frac{1}{3} \times \frac{35}{6} = \frac{35}{18} \\ \frac{2}{9} \times \frac{49}{24} = \frac{98}{216} = \frac{49}{108} \]Step 4: Divide the two results: \[ \frac{35}{18} \div \frac{49}{108} = \frac{35}{18} \times \frac{108}{49} \\ = \frac{3780}{882} = \frac{210}{49} = \frac{30}{7} = 4\frac{2}{7} \]Answer: \(4\frac{2}{7}\)

Q11: Simplify: \(\left(\frac{1}{4}-\frac{1}{9}\right)\div\left(\frac{1}{2}+\frac{1}{4}\div\frac{1}{3}\right)\)

Step 1: Solve the expression in the numerator: \[ \frac{1}{4} – \frac{1}{9} = \text{LCM of 4 and 9 is 36} \\ = \frac{9}{36} – \frac{4}{36} = \frac{5}{36} \]Step 2: Solve the expression inside the denominator: \[ \frac{1}{4} \div \frac{1}{3} = \frac{1}{4} \times \frac{3}{1} = \frac{3}{4} \\ \text{Now, } \frac{1}{2} + \frac{3}{4} = \frac{2}{4} + \frac{3}{4} = \frac{5}{4} \]Step 3: Divide the numerator by the denominator: \[ \frac{5}{36} \div \frac{5}{4} = \frac{5}{36} \times \frac{4}{5} = \frac{20}{180} = \frac{1}{9} \]Answer: \(\frac{1}{9}\)

Q12: Simplify: \(3\frac{7}{8} – \left\{ 1\frac{3}{8} \div \left( 2\frac{4}{5} – 1\frac{7}{10} \right) \right\}\)

Step 1: Solve the expression inside the parentheses first:
Convert mixed fractions to improper fractions: \[ 2\frac{4}{5} = \frac{14}{5}, \\ 1\frac{7}{10} = \frac{17}{10} \]Step 2: Find the difference: \[ 2\frac{4}{5} – 1\frac{7}{10} = \frac{14}{5} – \frac{17}{10} \] LCM of 5 and 10 is 10: \[ = \frac{28}{10} – \frac{17}{10} = \frac{11}{10} \]Step 3: Now solve the division inside the braces:
Convert \(1\frac{3}{8}\) to improper fraction: \[ 1\frac{3}{8} = \frac{11}{8} \]Step 4: Divide: \[ \frac{11}{8} \div \frac{11}{10} = \frac{11}{8} \times \frac{10}{11} = \frac{10}{8} = \frac{5}{4} = 1\frac{1}{4} \]Step 5: Finally, subtract this from \(3\frac{7}{8}\):
Convert \(3\frac{7}{8}\) to improper fraction: \[ 3\frac{7}{8} = \frac{31}{8} \]Step 6: Subtract: \[ \frac{31}{8} – \frac{5}{4} = \frac{31}{8} – \frac{10}{8} = \frac{21}{8} = 2\frac{5}{8} \]Answer: \(2\frac{5}{8}\)

Q13: Simplify: \(3 \div \left[ 3 \times \left\{ 3 – \left( 3 – \frac{1}{4} \right) \right\} \right]\)

Step 1: Solve the innermost parentheses first:
Calculate \(3 – \frac{1}{4}\): \[ 3 – \frac{1}{4} = \frac{12}{4} – \frac{1}{4} = \frac{11}{4} \]Step 2: Substitute and solve the expression inside the curly braces \(\{\}\):
Calculate: \[ 3 – \frac{11}{4} = \frac{12}{4} – \frac{11}{4} = \frac{1}{4} \]Step 3: Now multiply by 3 (inside the square brackets \([]\)): \[ 3 \times \frac{1}{4} = \frac{3}{4} \]Step 4: Finally, divide 3 by the result obtained: \[ 3 \div \frac{3}{4} = 3 \times \frac{4}{3} = 4 \]Answer: 4

Q14: Simplify: \(5\frac{1}{3} – \left[ 2\frac{1}{3} \div \left\{ \frac{3}{4} – \frac{1}{2} \times \left( \frac{7}{10} – \frac{3}{5} \right) \right\} \right]\)

Step 1: Solve inside the innermost parentheses \(\left(\frac{7}{10} – \frac{3}{5}\right)\):
Convert \(\frac{3}{5}\) to denominator 10: \[ \frac{3}{5} = \frac{6}{10} \] Calculate: \[ \frac{7}{10} – \frac{6}{10} = \frac{1}{10} \]Step 2: Multiply \(\frac{1}{2}\) by the result \(\frac{1}{10}\): \[ \frac{1}{2} \times \frac{1}{10} = \frac{1}{20} \]Step 3: Subtract from \(\frac{3}{4}\):
Convert \(\frac{3}{4}\) to denominator 20: \[ \frac{3}{4} = \frac{15}{20} \] Calculate: \[ \frac{15}{20} – \frac{1}{20} = \frac{14}{20} = \frac{7}{10} \]Step 4: Convert mixed numbers to improper fractions: \[ 5\frac{1}{3} = \frac{16}{3}, \\ 2\frac{1}{3} = \frac{7}{3} \]Step 5: Divide \(\frac{7}{3}\) by \(\frac{7}{10}\): \[ \frac{7}{3} \div \frac{7}{10} = \frac{7}{3} \times \frac{10}{7} = \frac{10}{3} \]Step 6: Subtract the result from \(\frac{16}{3}\): \[ \frac{16}{3} – \frac{10}{3} = \frac{6}{3} = 2 \]Answer: 2