Fractions

Q1: Find the product: \(\frac{5}{7} \times \frac{2}{3}\)

Step 1: Write the multiplication: \[ \frac{5}{7} \times \frac{2}{3} \]Step 2: Multiply numerators and denominators: \[ \text{Numerator} = 5 \times 2 = 10 \\ \text{Denominator} = 7 \times 3 = 21 \]So, \[ \frac{5}{7} \times \frac{2}{3} = \frac{10}{21} \]Answer: \(\frac{10}{21}\)

Q2: Find the product: \(\frac{3}{5} \times \frac{4}{9}\)

Step 1: Write the multiplication: \[ \frac{3}{5} \times \frac{4}{9} \]Step 2: Simplify by cancelling any common factors between numerators and denominators.
– Numerator 3 and denominator 9 have a common factor 3: \[ 3 \div 3 = 1, \quad 9 \div 3 = 3 \]No other common factors.
So, multiplication becomes: \[ \frac{1}{5} \times \frac{4}{3} \]Step 3: Multiply numerators and denominators: \[ \frac{1 \times 4}{5 \times 3} = \frac{4}{15} \]Answer: \(\frac{4}{15}\)

Q3: Find the product: \(\frac{12}{35} \times \frac{14}{27}\)

Step 1: Write the multiplication: \[ \frac{12}{35} \times \frac{14}{27} \]Step 2: Simplify by cancelling common factors between numerators and denominators before multiplying.
– Numerator 12 and denominator 27: \[ 12 = 2^2 \times 3, \quad 27 = 3^3 \]Common factor: 3 \[ 12 \div 3 = 4, \quad 27 \div 3 = 9 \]– Numerator 14 and denominator 35: \[ 14 = 2 \times 7, \quad 35 = 5 \times 7 \]Common factor: 7 \[ 14 \div 7 = 2, \quad 35 \div 7 = 5 \]After simplification, multiplication becomes: \[ \frac{4}{5} \times \frac{2}{9} \]Step 3: Multiply numerators and denominators: \[ \frac{4 \times 2}{5 \times 9} = \frac{8}{45} \]Answer: \(\frac{8}{45}\)

Q4: Find the product: \(\frac{30}{51} \times \frac{17}{25}\)

Step 1: Write the multiplication of fractions. \[ \frac{30}{51} \times \frac{17}{25} \]Step 2: Simplify by dividing numerator and denominator with common factors before multiplying.
– Numerator of first fraction = 30
– Denominator of second fraction = 25
Common factors: 5 \[ 30 \div 5 = 6, \quad 25 \div 5 = 5 \]– Numerator of second fraction = 17
– Denominator of first fraction = 51
Check common factors between 17 and 51.
17 is a prime number and 51 = 3 × 17
So, \[ 17 \div 17 = 1, \quad 51 \div 17 = 3 \]After simplification, fractions become: \[ \frac{6}{3} \times \frac{1}{5} \]Step 3: Simplify \(\frac{6}{3}\) \[ \frac{6}{3} = 2 \]So multiplication becomes: \[ 2 \times \frac{1}{5} = \frac{2}{5} \]Answer: \(\frac{2}{5}\)

Q5: Find the product: \(\frac{9}{16} \times \frac{14}{15}\)

Step 1: Write the multiplication: \[ \frac{9}{16} \times \frac{14}{15} \]Step 2: Simplify before multiplying to make calculation easier:
– Numerator 9 and denominator 15 share a common factor 3: \[ 9 \div 3 = 3, \quad 15 \div 3 = 5 \]– Numerator 14 and denominator 16 share a common factor 2: \[ 14 \div 2 = 7, \quad 16 \div 2 = 8 \]Now the multiplication becomes: \[ \frac{3}{8} \times \frac{7}{5} \]Step 3: Multiply numerators and denominators: \[ \text{Numerator} = 3 \times 7 = 21 \\ \text{Denominator} = 8 \times 5 = 40 \]So, \[ \frac{9}{16} \times \frac{14}{15} = \frac{21}{40} \]Answer: \(\frac{21}{40}\)

Q6: Find the product: \(2\frac{11}{27} \times 18\)

Step 1: Convert the mixed fraction \(2\frac{11}{27}\) to an improper fraction: \[ 2\frac{11}{27} = \frac{2 \times 27 + 11}{27} = \frac{54 + 11}{27} = \frac{65}{27} \]Step 2: Write the multiplication as: \[ \frac{65}{27} \times 18 \]Step 3: Express 18 as a fraction: \[ 18 = \frac{18}{1} \]So, \[ \frac{65}{27} \times \frac{18}{1} \]Step 4: Simplify before multiplying:
– Numerator 18 and denominator 27 share a common factor 9: \[ 18 \div 9 = 2, \quad 27 \div 9 = 3 \]Now the multiplication becomes: \[ \frac{65}{3} \times \frac{2}{1} \]Step 5: Multiply numerators and denominators: \[ \text{Numerator} = 65 \times 2 = 130 \\ \text{Denominator} = 3 \times 1 = 3 \]So, \[ 2\frac{11}{27} \times 18 = \frac{130}{3} \]Step 6: Convert the improper fraction \(\frac{130}{3}\) back to a mixed number: \[ 130 \div 3 = 43 \text{ remainder } 1 \]Thus, \[ \frac{130}{3} = 43 \frac{1}{3} \]Answer: \(43 \frac{1}{3}\)

Q7: Find the product: \(7\frac{1}{12} \times \frac{8}{17}\)

Step 1: Convert the mixed fraction \(7\frac{1}{12}\) to an improper fraction: \[ 7\frac{1}{12} = \frac{7 \times 12 + 1}{12} = \frac{84 + 1}{12} = \frac{85}{12} \]Step 2: Write the multiplication as: \[ \frac{85}{12} \times \frac{8}{17} \]Step 3: Simplify before multiplying:
– Numerator 85 and denominator 17 share a common factor 17: \[ 85 \div 17 = 5, \quad 17 \div 17 = 1 \]– Numerator 8 and denominator 12 share a common factor 4: \[ 8 \div 4 = 2, \quad 12 \div 4 = 3 \]Now the multiplication becomes: \[ \frac{5}{3} \times \frac{2}{1} \]Step 4: Multiply numerators and denominators: \[ \text{Numerator} = 5 \times 2 = 10 \\ \text{Denominator} = 3 \times 1 = 3 \]So, \[ 7\frac{1}{12} \times \frac{8}{17} = \frac{10}{3} \]Step 5: Convert the improper fraction \(\frac{10}{3}\) back to a mixed number: \[ 10 \div 3 = 3 \text{ remainder } 1 \]Thus, \[ \frac{10}{3} = 3 \frac{1}{3} \]Answer: \(3 \frac{1}{3}\)

Q8: Find the product: \(8\frac{1}{21} \times 1\frac{1}{13}\)

Step 1: Convert the mixed fractions to improper fractions: \[ 8\frac{1}{21} = \frac{8 \times 21 + 1}{21} = \frac{168 + 1}{21} = \frac{169}{21} \\ 1\frac{1}{13} = \frac{1 \times 13 + 1}{13} = \frac{13 + 1}{13} = \frac{14}{13} \]Step 2: Write the multiplication: \[ \frac{169}{21} \times \frac{14}{13} \]Step 3: Simplify before multiplying:
– Numerator 169 and denominator 13 share a common factor 13: \[ 169 \div 13 = 13, \quad 13 \div 13 = 1 \]– Numerator 14 and denominator 21 share a common factor 7: \[ 14 \div 7 = 2, \quad 21 \div 7 = 3 \]Now the multiplication becomes: \[ \frac{13}{3} \times \frac{2}{1} \]Step 4: Multiply numerators and denominators: \[ \text{Numerator} = 13 \times 2 = 26 \\ \text{Denominator} = 3 \times 1 = 3 \]So, \[ 8\frac{1}{21} \times 1\frac{1}{13} = \frac{26}{3} \]Step 5: Convert the improper fraction \(\frac{26}{3}\) to a mixed number: \[ 26 \div 3 = 8 \text{ remainder } 2 \]Thus, \[ \frac{26}{3} = 8 \frac{2}{3} \]Answer: \(8 \frac{2}{3}\)

Q9: Find the product: \(7\frac{7}{11} \times 6\frac{3}{16}\)

Step 1: Convert the mixed fractions to improper fractions: \[ 7\frac{7}{11} = \frac{7 \times 11 + 7}{11} = \frac{77 + 7}{11} = \frac{84}{11} \\ 6\frac{3}{16} = \frac{6 \times 16 + 3}{16} = \frac{96 + 3}{16} = \frac{99}{16} \]Step 2: Write the multiplication: \[ \frac{84}{11} \times \frac{99}{16} \]Step 3: Simplify before multiplying:
– Numerator 84 and denominator 16 share a common factor 4: \[ 84 \div 4 = 21, \quad 16 \div 4 = 4 \]– Numerator 99 and denominator 11 share a common factor 11: \[ 99 \div 11 = 9, \quad 11 \div 11 = 1 \]Now the multiplication becomes: \[ \frac{21}{1} \times \frac{9}{4} \]Step 4: Multiply numerators and denominators: \[ \text{Numerator} = 21 \times 9 = 189 \\ \text{Denominator} = 1 \times 4 = 4 \]So, \[ 7\frac{7}{11} \times 6\frac{3}{16} = \frac{189}{4} \]Step 5: Convert the improper fraction \(\frac{189}{4}\) to a mixed number: \[ 189 \div 4 = 47 \text{ remainder } 1 \]Thus, \[ \frac{189}{4} = 47 \frac{1}{4} \]Answer: \(47 \frac{1}{4}\)

Q10: Find the product: \(11\frac{1}{4} \times 6\frac{7}{9}\)

Step 1: Convert mixed fractions to improper fractions: \[ 11\frac{1}{4} = \frac{11 \times 4 + 1}{4} = \frac{44 + 1}{4} = \frac{45}{4} \\ 6\frac{7}{9} = \frac{6 \times 9 + 7}{9} = \frac{54 + 7}{9} = \frac{61}{9} \]Step 2: Multiply the improper fractions: \[ \frac{45}{4} \times \frac{61}{9} \]Step 3: Simplify before multiplication:
– 45 and 9 share a common factor 9: \[ 45 \div 9 = 5, \quad 9 \div 9 = 1 \]– No other common factors between numerator and denominator.
So multiplication becomes: \[ \frac{5}{4} \times \frac{61}{1} = \frac{5 \times 61}{4 \times 1} = \frac{305}{4} \]Step 4: Convert \(\frac{305}{4}\) to a mixed number: \[ 305 \div 4 = 76 \text{ remainder } 1 \]So, \[ \frac{305}{4} = 76 \frac{1}{4} \]Answer: \(76 \frac{1}{4}\)

Q11: Find the product: \(1\frac{3}{4} \times 2\frac{1}{7} \times 4\frac{4}{5}\)

Step 1: Convert mixed fractions to improper fractions: \[ 1\frac{3}{4} = \frac{1 \times 4 + 3}{4} = \frac{7}{4} \\ 2\frac{1}{7} = \frac{2 \times 7 + 1}{7} = \frac{15}{7} \\ 4\frac{4}{5} = \frac{4 \times 5 + 4}{5} = \frac{24}{5} \]Step 2: Multiply the three improper fractions: \[ \frac{7}{4} \times \frac{15}{7} \times \frac{24}{5} \]Step 3: Simplify before multiplication:
– Cancel \(7\) in numerator and denominator: \[ \frac{7}{4} \times \frac{15}{7} \times \frac{24}{5} \]– Cancel 15 and 5 (since \(15 = 3 \times 5\)): \[ \frac{1}{4} \times \frac{3 \times \ 5}{1} \times \frac{24}{5} = \frac{1}{4} \times 3 \times 24 \]– Cancel 24 and 4 (since \(24 = 6 \times 4\)): \[ \frac{1}{4} \times 3 \times \frac{6 \times \ 4}{1} = 1 \times 3 \times 6 = 18 \]Answer: 18

Q12: Find the product: \(3\frac{1}{6} \times 2\frac{3}{4} \times 2\frac{4}{11}\)

Step 1: Convert mixed fractions to improper fractions: \[ 3\frac{1}{6} = \frac{3 \times 6 + 1}{6} = \frac{19}{6} \\ 2\frac{3}{4} = \frac{2 \times 4 + 3}{4} = \frac{11}{4} \\ 2\frac{4}{11} = \frac{2 \times 11 + 4}{11} = \frac{26}{11} \]Step 2: Multiply the three improper fractions: \[ \frac{19}{6} \times \frac{11}{4} \times \frac{26}{11} \]Step 3: Simplify before multiplication:
– Cancel 11 in numerator and denominator: \[ \frac{19}{6} \times \frac{11}{4} \times \frac{26}{11} = \frac{19}{6} \times \frac{1}{4} \times 26 \]Step 4: Multiply the numerators and denominators: \[ \frac{19 \times 1 \times 26}{6 \times 4 \times 1} = \frac{494}{24} \]Step 5: Simplify the fraction \(\frac{494}{24}\):– Find the GCD of 494 and 24, which is 2. \[ \frac{494 \div 2}{24 \div 2} = \frac{247}{12} \]Step 6: Convert \(\frac{247}{12}\) to a mixed fraction: \[ 247 \div 12 = 20 \text{ remainder } 7 \]So, \[ \frac{247}{12} = 20\frac{7}{12} \]Answer: \(20\frac{7}{12}\)

Q13: Find the reciprocal of:

i. \(\frac{5}{9}\)

Step 1: Recall the reciprocal of a fraction \(\frac{a}{b}\) is \(\frac{b}{a}\).
Here, reciprocal of \(\frac{5}{9}\) is \(\frac{9}{5}\).
Answer: \(\frac{9}{5}\)

ii. \(\frac{7}{13}\)

Step 1: Reciprocal of \(\frac{7}{13}\) is \(\frac{13}{7}\).
Answer: \(\frac{13}{7}\)

iii. \(8\) (Whole number)

Step 1: Whole number \(8\) can be written as \(\frac{8}{1}\).
Step 2: Reciprocal of \(\frac{8}{1}\) is \(\frac{1}{8}\).
Answer: \(\frac{1}{8}\)

iv. \(\frac{1}{5}\)

Step 1: Reciprocal of \(\frac{1}{5}\) is \(\frac{5}{1} = 5\).
Answer: 5

v. \(3\frac{2}{5}\) (Mixed fraction)

Step 1: Convert mixed fraction to improper fraction: \[ 3\frac{2}{5} = \frac{3 \times 5 + 2}{5} = \frac{15 + 2}{5} = \frac{17}{5} \] Step 2: Reciprocal of \(\frac{17}{5}\) is \(\frac{5}{17}\).
Answer: \(\frac{5}{17}\)

vi. \(8\frac{1}{9}\) (Mixed fraction)

Step 1: Convert mixed fraction to improper fraction: \[ 8\frac{1}{9} = \frac{8 \times 9 + 1}{9} = \frac{72 + 1}{9} = \frac{73}{9} \] Step 2: Reciprocal of \(\frac{73}{9}\) is \(\frac{9}{73}\).
Answer: \(\frac{9}{73}\)

vii. \(5\frac{3}{8}\) (Mixed fraction)

Step 1: Convert mixed fraction to improper fraction: \[ 5\frac{3}{8} = \frac{5 \times 8 + 3}{8} = \frac{40 + 3}{8} = \frac{43}{8} \] Step 2: Reciprocal of \(\frac{43}{8}\) is \(\frac{8}{43}\).
Answer: \(\frac{8}{43}\)

viii. \(23\) (Whole number)

Step 1: Write \(23\) as \(\frac{23}{1}\).
Step 2: Reciprocal of \(\frac{23}{1}\) is \(\frac{1}{23}\).
Answer: \(\frac{1}{23}\)

Q14: Divide: \(\frac{13}{21} \div \frac{2}{7}\)

Step 1: Recall the rule: To divide two fractions, multiply the first fraction by the reciprocal of the second. \[ \frac{13}{21} \div \frac{2}{7} = \frac{13}{21} \times \frac{7}{2} \]Step 2: Multiply the numerators and the denominators: \[ = \frac{13 \times 7}{21 \times 2} = \frac{91}{42} \]Step 3: Simplify the fraction by dividing both numerator and denominator by the common factor 7: \[ \frac{91 \div 7}{42 \div 7} = \frac{13}{6} \]Answer: \(\frac{13}{6}\) or \(2\frac{1}{6}\)

Q15: Divide: \(3\frac{1}{8} \div 5\)

Step 1: Convert the mixed number \(3\frac{1}{8}\) into an improper fraction. \[ 3\frac{1}{8} = \frac{(3 \times 8 + 1)}{8} = \frac{25}{8} \]Step 2: Divide \(\frac{25}{8}\) by 5. Dividing by 5 is the same as multiplying by \(\frac{1}{5}\): \[ \frac{25}{8} \div 5 = \frac{25}{8} \times \frac{1}{5} \]Step 3: Multiply the numerators and the denominators: \[ = \frac{25 \times 1}{8 \times 5} = \frac{25}{40} \]Step 4: Simplify the fraction \(\frac{25}{40}\) by dividing numerator and denominator by 5: \[ \frac{25 \div 5}{40 \div 5} = \frac{5}{8} \]Answer: \(\frac{5}{8}\)

Q16: Divide: \(1 \div 2\frac{3}{5}\)

Step 1: Convert the mixed number \(2\frac{3}{5}\) into an improper fraction. \[ 2\frac{3}{5} = \frac{(2 \times 5 + 3)}{5} = \frac{13}{5} \]Step 2: Now divide 1 by \(\frac{13}{5}\). Dividing by a fraction is the same as multiplying by its reciprocal: \[ 1 \div \frac{13}{5} = 1 \times \frac{5}{13} \]Step 3: Multiply the numerators and the denominators: \[ = \frac{1 \times 5}{1 \times 13} = \frac{5}{13} \]Answer: \(\frac{5}{13}\)

Q17: Divide: \(7\frac{5}{9} \div 34\)

Step 1: Convert the mixed number \(7\frac{5}{9}\) into an improper fraction. \[ 7\frac{5}{9} = \frac{(7 \times 9 + 5)}{9} = \frac{63 + 5}{9} = \frac{68}{9} \]Step 2: Express 34 as a fraction: \[ 34 = \frac{34}{1} \]Step 3: Divide \(\frac{68}{9}\) by \(\frac{34}{1}\) by multiplying \(\frac{68}{9}\) by the reciprocal of \(\frac{34}{1}\): \[ \frac{68}{9} \div \frac{34}{1} = \frac{68}{9} \times \frac{1}{34} \]Step 4: Simplify before multiplying: \[ \frac{68}{9} \times \frac{1}{34} = \frac{68 \div 34}{9 \times 1} = \frac{2}{9} \]Answer: \(\frac{2}{9}\)

Q18: Divide: \(70 \div 8\frac{2}{5}\)

i. Division of a whole number by a mixed number

Step 1: Convert the mixed number \(8\frac{2}{5}\) into an improper fraction. \[ 8\frac{2}{5} = \frac{(8 \times 5 + 2)}{5} = \frac{40 + 2}{5} = \frac{42}{5} \]Step 2: Express 70 as a fraction: \[ 70 = \frac{70}{1} \]Step 3: Divide \(\frac{70}{1}\) by \(\frac{42}{5}\) by multiplying \(\frac{70}{1}\) by the reciprocal of \(\frac{42}{5}\): \[ \frac{70}{1} \div \frac{42}{5} = \frac{70}{1} \times \frac{5}{42} \]Step 4: Simplify before multiplying: \[ = \frac{70 \times 5}{1 \times 42} = \frac{350}{42} \]Simplify numerator and denominator by 14: \[ \frac{350 \div 14}{42 \div 14} = \frac{25}{3} \]Step 5: Convert the improper fraction \(\frac{25}{3}\) to a mixed number: \[ \frac{25}{3} = 8 \frac{1}{3} \]Answer: \(8 \frac{1}{3}\)

Q19: Divide: \(4\frac{1}{2} \div 6\frac{1}{2}\)

Step 1: Convert both mixed numbers into improper fractions. \[ 4\frac{1}{2} = \frac{(4 \times 2) + 1}{2} = \frac{8 + 1}{2} = \frac{9}{2} \\ 6\frac{1}{2} = \frac{(6 \times 2) + 1}{2} = \frac{12 + 1}{2} = \frac{13}{2} \]Step 2: Divide \(\frac{9}{2}\) by \(\frac{13}{2}\) by multiplying \(\frac{9}{2}\) by the reciprocal of \(\frac{13}{2}\): \[ \frac{9}{2} \div \frac{13}{2} = \frac{9}{2} \times \frac{2}{13} \]Step 3: Simplify before multiplying: \[ = \frac{9 \times 2}{2 \times 13} = \frac{18}{26} \]Simplify numerator and denominator by 2: \[ \frac{18 \div 2}{26 \div 2} = \frac{9}{13} \]Answer: \(\frac{9}{13}\)

Q20: Divide: \(5\frac{7}{10} \div 3\frac{1}{6}\)

Step 1: Convert both mixed numbers into improper fractions. \[ 5\frac{7}{10} = \frac{(5 \times 10) + 7}{10} = \frac{50 + 7}{10} = \frac{57}{10} \\ 3\frac{1}{6} = \frac{(3 \times 6) + 1}{6} = \frac{18 + 1}{6} = \frac{19}{6} \]Step 2: Divide \(\frac{57}{10}\) by \(\frac{19}{6}\) by multiplying \(\frac{57}{10}\) by the reciprocal of \(\frac{19}{6}\): \[ \frac{57}{10} \div \frac{19}{6} = \frac{57}{10} \times \frac{6}{19} \]Step 3: Multiply the numerators and denominators: \[ = \frac{57 \times 6}{10 \times 19} = \frac{342}{190} \]Step 4: Simplify the fraction \(\frac{342}{190}\):
Divide numerator and denominator by 2: \[ \frac{342 \div 38}{190 \div 38} = \frac{9}{5} \]Step 5: Convert the improper fraction to a mixed number: \[ \frac{9}{5} = 1\frac{4}{5} \]Answer: \(1\frac{4}{5}\)

Q21: Divide: \(10\frac{5}{7} \div 1\frac{11}{14}\)

Step 1: Convert the mixed numbers into improper fractions. \[ 10\frac{5}{7} = \frac{(10 \times 7) + 5}{7} = \frac{70 + 5}{7} = \frac{75}{7} \\ 1\frac{11}{14} = \frac{(1 \times 14) + 11}{14} = \frac{14 + 11}{14} = \frac{25}{14} \]Step 2: Divide \(\frac{75}{7}\) by \(\frac{25}{14}\) by multiplying by the reciprocal: \[ \frac{75}{7} \div \frac{25}{14} = \frac{75}{7} \times \frac{14}{25} \]Step 3: Multiply numerators and denominators: \[ = \frac{75 \times 14}{7 \times 25} = \frac{1050}{175} \]Step 4: Simplify the fraction:
Divide numerator and denominator by 25: \[ \frac{1050 \div 25}{175 \div 25} = \frac{42}{7} \]Divide numerator and denominator by 7: \[ \frac{42 \div 7}{7 \div 7} = \frac{6}{1} = 6 \]Answer: 6

Q22: Divide: \(15\frac{8}{9} \div 3\frac{2}{3}\)

Step 1: Convert the mixed numbers into improper fractions. \[ 15\frac{8}{9} = \frac{(15 \times 9) + 8}{9} = \frac{135 + 8}{9} = \frac{143}{9} \\ 3\frac{2}{3} = \frac{(3 \times 3) + 2}{3} = \frac{9 + 2}{3} = \frac{11}{3} \]Step 2: Divide \(\frac{143}{9}\) by \(\frac{11}{3}\) by multiplying by the reciprocal: \[ \frac{143}{9} \div \frac{11}{3} = \frac{143}{9} \times \frac{3}{11} \]Step 3: Multiply numerators and denominators: \[ = \frac{143 \times 3}{9 \times 11} = \frac{429}{99} \]Step 4: Simplify the fraction:
Divide numerator and denominator by 11: \[ \frac{429 \div 11}{99 \div 11} = \frac{39}{9} \]Simplify further by dividing numerator and denominator by 3: \[ \frac{39 \div 3}{9 \div 3} = \frac{13}{3} \]Step 5: Convert improper fraction back to mixed number: \[ \frac{13}{3} = 4\frac{1}{3} \]Answer: \(4\frac{1}{3}\)

Q23: Divide: \(9\frac{4}{5} \div 3\frac{23}{25}\)

i. Division of two mixed numbers

Step 1: Convert the mixed numbers into improper fractions. \[ 9\frac{4}{5} = \frac{(9 \times 5) + 4}{5} = \frac{45 + 4}{5} = \frac{49}{5} \\ 3\frac{23}{25} = \frac{(3 \times 25) + 23}{25} = \frac{75 + 23}{25} = \frac{98}{25} \]Step 2: Divide \(\frac{49}{5}\) by \(\frac{98}{25}\) by multiplying by the reciprocal: \[ \frac{49}{5} \div \frac{98}{25} = \frac{49}{5} \times \frac{25}{98} \]Step 3: Multiply numerators and denominators: \[ = \frac{49 \times 25}{5 \times 98} = \frac{1225}{490} \]Step 4: Simplify the fraction:
Divide numerator and denominator by 35: \[ \frac{1225 \div 35}{490 \div 35} = \frac{35}{14} \]Simplify further by dividing numerator and denominator by 7: \[ \frac{35 \div 7}{14 \div 7} = \frac{5}{2} \]Step 5: Convert improper fraction back to mixed number: \[ \frac{5}{2} = 2\frac{1}{2} \]Answer: \(2\frac{1}{2}\)

Q24: Divide: \(2\frac{17}{38} \div 1\frac{12}{19}\)

Step 1: Convert the mixed numbers into improper fractions. \[ 2\frac{17}{38} = \frac{(2 \times 38) + 17}{38} = \frac{76 + 17}{38} = \frac{93}{38} \\ 1\frac{12}{19} = \frac{(1 \times 19) + 12}{19} = \frac{19 + 12}{19} = \frac{31}{19} \]Step 2: Divide \(\frac{93}{38}\) by \(\frac{31}{19}\) by multiplying by the reciprocal: \[ \frac{93}{38} \div \frac{31}{19} = \frac{93}{38} \times \frac{19}{31} \]Step 3: Multiply numerators and denominators: \[ = \frac{93 \times 19}{38 \times 31} = \frac{1767}{1178} \]Step 4: Simplify the fraction if possible:
Check GCD of 1767 and 1178.
Since no common factors other than 1, fraction is in simplest form.
Step 5: Convert the improper fraction to a mixed number:
Divide 1767 by 1178: \[ 1767 \div 1178 = 1 \text{ remainder } 589 \] So, \[ \frac{1767}{1178} = 1\frac{589}{1178} \]Simplify the fractional part:
Divide numerator and denominator by 589: \[ \frac{589}{1178} = \frac{1}{2} \]Answer: \(1\frac{1}{2}\)

Q25: Divide: \(8\frac{7}{25} \div 3\frac{1}{15}\)

Step 1: Convert the mixed numbers into improper fractions. \[ 8\frac{7}{25} = \frac{(8 \times 25) + 7}{25} = \frac{200 + 7}{25} = \frac{207}{25} \\ 3\frac{1}{15} = \frac{(3 \times 15) + 1}{15} = \frac{45 + 1}{15} = \frac{46}{15} \]Step 2: Divide \(\frac{207}{25}\) by \(\frac{46}{15}\) by multiplying by the reciprocal: \[ \frac{207}{25} \div \frac{46}{15} = \frac{207}{25} \times \frac{15}{46} \]Step 3: Simplify before multiplying: \[ \frac{207}{25} \times \frac{15}{46} = \frac{207 \times 15}{25 \times 46} \] Check for common factors:
– \(15\) and \(25\) share factor \(5\): \[ 15 = 3 \times 5, \quad 25 = 5 \times 5 \] Cancel one 5: \[ \frac{207 \times 3}{5 \times 46} \]– \(207\) and \(46\) share factor \(23\) since: \[ 207 = 9 \times 23, \quad 46 = 2 \times 23 \] Cancel 23: \[ \frac{9 \times 3}{5 \times 2} = \frac{27}{10} \]Step 4: Convert the improper fraction \(\frac{27}{10}\) into a mixed number: \[ 27 \div 10 = 2 \text{ remainder } 7 \] So, \[ \frac{27}{10} = 2\frac{7}{10} \]Answer: \(2\frac{7}{10}\)

Q26: The cost of 1 litre of milk is ₹\(42\frac{3}{5}\), Find the cost of \(12\frac{1}{2}\) litres of milk.

Step 1: Convert the mixed numbers into improper fractions.
Cost of 1 litre milk = ₹\(42\frac{3}{5}\) = ₹\(\frac{(42 \times 5) + 3}{5} = \frac{210 + 3}{5} = \frac{213}{5}\)
Quantity = \(12\frac{1}{2} = \frac{(12 \times 2) + 1}{2} = \frac{24 + 1}{2} = \frac{25}{2}\) litresStep 2: Multiply cost per litre by the quantity: \[ \text{Total cost} = \frac{213}{5} \times \frac{25}{2} = \frac{213 \times 25}{5 \times 2} \]Step 3: Simplify the fraction: \[ \frac{213 \times 25}{5 \times 2} = \frac{213 \times 5 \times 5}{5 \times 2} \] Cancel \(5\) from numerator and denominator: \[ = \frac{213 \times 5}{2} = \frac{1065}{2} \]Step 4: Convert the improper fraction to mixed number: \[ 1065 \div 2 = 532 \text{ remainder } 1 \] So, \[ \frac{1065}{2} = 532\frac{1}{2} \]Answer: ₹\(532\frac{1}{2}\) or ₹532.50

Q27: The cost of 1 litre of petrol is ₹\(65\frac{3}{4}\). Find the cost of 36 litres of petrol.

Step 1: Convert the mixed number into an improper fraction.
Cost of 1 litre petrol = \[ \text{₹ } 65\frac{3}{4} = \text{₹ } \frac{(65 \times 4) + 3}{4} = \frac{260 + 3}{4} = \frac{263}{4} \]Step 2: Multiply the cost per litre by the number of litres: \[ \text{Total cost} = \frac{263}{4} \times 36 = \frac{263 \times 36}{4} \]Step 3: Simplify the fraction:
Divide numerator and denominator by 4: \[ = \frac{263 \times 9}{1} = 263 \times 9 = 2367 \]Answer: ₹2367

Q28: The cost of \(3\frac{1}{2}\) metres of cloth is ₹\(547\frac{3}{4}\). Find the cost of 1 metre of cloth.

Step 1: Convert mixed numbers into improper fractions.
Length of cloth = \[ 3\frac{1}{2} = \frac{(3 \times 2) + 1}{2} = \frac{7}{2} \text{ metres} \] Cost of \(3\frac{1}{2}\) metres = ₹\(547\frac{3}{4} = \frac{(547 \times 4) + 3}{4} = \frac{2188 + 3}{4} = \frac{2191}{4}\)
Step 2: Cost of 1 metre = \(\frac{\text{Total cost}}{\text{Length}}\) \[ = \frac{\frac{2191}{4}}{\frac{7}{2}} = \frac{2191}{4} \times \frac{2}{7} = \frac{2191 \times 2}{4 \times 7} = \frac{4382}{28} \]Step 3: Simplify the fraction:
Divide numerator and denominator by 2: \[ = \frac{2191}{14} \]Step 4: Convert improper fraction into mixed number: Divide 2191 by 14: \(2191 \div 14 = 156\) remainder \(7\)
So, \[ \frac{2191}{14} = 156 \frac{7}{14} = 156 \frac{1}{2} \]Answer: ₹156\frac{1}{2}

Q29: Tanvy cuts 54 m of Cloth into pieces, each of length \(3\frac{3}{8}\) m. How many pieces does she get?

Step 1: Convert the mixed fraction into an improper fraction.
Length of each piece = \(3\frac{3}{8} = \frac{(3 \times 8) + 3}{8} = \frac{24 + 3}{8} = \frac{27}{8}\) metres
Step 2: Total length of cloth = 54 metres (whole number can be written as fraction with denominator 1): \[ 54 = \frac{54}{1} \]Step 3: Number of pieces = \(\frac{\text{Total length}}{\text{Length of each piece}} = \frac{54}{1} \div \frac{27}{8} = \frac{54}{1} \times \frac{8}{27}\) \[ = \frac{54 \times 8}{1 \times 27} = \frac{432}{27} \]Step 4: Simplify the fraction:
Divide numerator and denominator by 27: \[ = \frac{432 \div 27}{27 \div 27} = \frac{16}{1} = 16 \]Answer: 16 pieces

Q30: A cord of length \(126\frac{1}{2}\) m has been cut into 46 pieces of equal length. What is the length of each piece?

Step 1: Convert the mixed fraction into an improper fraction.
Total length of the cord: \[ 126\frac{1}{2} = \frac{(126 \times 2) + 1}{2} = \frac{252 + 1}{2} = \frac{253}{2} \text{ metres} \]Step 2: Number of pieces = 46
Step 3: Length of each piece = \(\frac{\text{Total length}}{\text{Number of pieces}} = \frac{253}{2} \div 46 = \frac{253}{2} \times \frac{1}{46}\) \[ = \frac{253 \times 1}{2 \times 46} = \frac{253}{92}// = \frac{253 \div 23}{92 \div 23} = \frac{11}{4} \]Step 4: Simplify the fraction if possible.
Convert the improper fraction to a mixed fraction: \[ \frac{11}{4} = 2\frac{3}{4} \]Answer: Each piece is \(2\frac{3}{4}\) metres long.

Q31: A car travels \(283\frac{1}{2}\) km in \(4\frac{2}{3}\) hours. How far does it go in 1 hour?

Step 1: Convert mixed fractions to improper fractions.
Distance travelled = \(283\frac{1}{2}\) km = \(\frac{(283 \times 2) + 1}{2} = \frac{566 + 1}{2} = \frac{567}{2}\) km
Time taken = \(4\frac{2}{3}\) hours = \(\frac{(4 \times 3) + 2}{3} = \frac{12 + 2}{3} = \frac{14}{3}\) hours
Step 2: Distance travelled in 1 hour = \(\frac{\text{Total distance}}{\text{Total time}} = \frac{567}{2} \div \frac{14}{3}\) \[ = \frac{567}{2} \times \frac{3}{14} = \frac{567 \times 3}{2 \times 14} = \frac{1701}{28} \]Step 3: Simplify \(\frac{1701}{28}\)
Since 1701 and 28 have no common factors other than 1, convert to mixed fraction: \[ 1701 \div 28 = 60 \text{ remainder } 21 \] So, \[ \frac{1701}{28} = 60\frac{21}{28} \]Step 4: Simplify the fraction \(\frac{21}{28}\) by dividing numerator and denominator by 7: \[ \frac{21}{28} = \frac{3}{4} \]Answer: The car travels \(60\frac{3}{4}\) km in 1 hour.

Q32: The area of a rectangular plot of land is \(46\frac{2}{5}\) sq.m. If its length is \(7\frac{1}{4}\) m, find itd breadth.

Step 1: Convert mixed fractions to improper fractions.
Area \(= 46\frac{2}{5} = \frac{(46 \times 5) + 2}{5} = \frac{230 + 2}{5} = \frac{232}{5}\) sq.m.
Length \(= 7\frac{1}{4} = \frac{(7 \times 4) + 1}{4} = \frac{28 + 1}{4} = \frac{29}{4}\) m
Step 2: Using the formula: \[ \text{Breadth} = \frac{\text{Area}}{\text{Length}} = \frac{\frac{232}{5}}{\frac{29}{4}} = \frac{232}{5} \times \frac{4}{29} = \frac{232 \times 4}{5 \times 29} = \frac{928}{145} \]Step 3: Simplify the fraction \(\frac{928}{145}\) to a mixed fraction:
Divide 928 by 145: \[ 928 \div 145 = 6 \text{ remainder } 58 \] So, \[ \frac{928}{145} = 6\frac{58}{145} \]Step 4: Check if \(\frac{58}{145}\) can be simplified.
Factors of 58: 1, 2, 29, 58
Factors of 145: 1, 5, 29, 145
Common factor = 29
Divide numerator and denominator by 29: \[ \frac{58}{145} = \frac{2}{5} \]Answer: The breadth of the rectangular plot is \(6\frac{2}{5}\) meters.

Q33: The product of two fractions is \(15\frac{3}{4}\). If one of them is \(4\frac{1}{2}\), find the other.

Step 1: Convert mixed fractions into improper fractions.
Product \(= 15\frac{3}{4} = \frac{(15 \times 4) + 3}{4} = \frac{60 + 3}{4} = \frac{63}{4}\)
One fraction \(= 4\frac{1}{2} = \frac{(4 \times 2) + 1}{2} = \frac{8 + 1}{2} = \frac{9}{2}\)
Step 2: Use the product formula: \[ \text{Product} = \text{First fraction} \times \text{Second fraction} \\ \Rightarrow \frac{63}{4} = \frac{9}{2} \times x \]Step 3: Solve for \(x\): \[ x = \frac{63}{4} \div \frac{9}{2} = \frac{63}{4} \times \frac{2}{9} = \frac{63 \times 2}{4 \times 9} = \frac{126}{36} = \frac{7}{2} \]Step 4: Convert \(\frac{7}{2}\) into a mixed fraction: \[ 7 \div 2 = 3 \text{ remainder } 1 \Rightarrow \frac{7}{2} = 3\frac{1}{2} \]Answer: The other fraction is \(3\frac{1}{2}\).