Fractions

Q1: Reduce to a single fraction:

i. \(\frac{1}{2} + \frac{2}{3}\)

Step 1: Find the LCM of 2 and 3, which is 6.
Step 2: Convert each fraction to have a denominator of 6:
\(\frac{1}{2} = \frac{3}{6}, \quad \frac{2}{3} = \frac{4}{6}\)
Step 3: Add the fractions: \(\frac{3}{6} + \frac{4}{6} = \frac{7}{6}\)
Answer: \(\frac{7}{6} = 1\frac{1}{6}\)

ii. \(\frac{3}{5} – \frac{1}{10}\)

Step 1: Find the LCM of 5 and 10, which is 10.
Step 2: Convert each fraction to have a denominator of 10:
\(\frac{3}{5} = \frac{6}{10}, \quad \frac{1}{10} = \frac{1}{10}\)
Step 3: Subtract the fractions: \(\frac{6}{10} – \frac{1}{10} = \frac{5}{10} = \frac{1}{2}\)
Answer: \(\frac{1}{2}\)

iii. \(\frac{2}{3} – \frac{1}{6}\)

Step 1: Find the LCM of 3 and 6, which is 6.
Step 2: Convert each fraction to have a denominator of 6:
\(\frac{2}{3} = \frac{4}{6}, \quad \frac{1}{6} = \frac{1}{6}\)
Step 3: Subtract the fractions: \(\frac{4}{6} – \frac{1}{6} = \frac{3}{6} = \frac{1}{2}\)
Answer: \(\frac{1}{2}\)

iv. \(1\frac{1}{3} + 2\frac{1}{4}\)

Step 1: Convert the mixed fractions to improper fractions:
\(1\frac{1}{3} = \frac{4}{3}, \quad 2\frac{1}{4} = \frac{9}{4}\)
Step 2: Find the LCM of 3 and 4, which is 12.
Step 3: Convert the fractions to have a denominator of 12:
\(\frac{4}{3} = \frac{16}{12}, \quad \frac{9}{4} = \frac{27}{12}\)
Step 4: Add the fractions: \(\frac{16}{12} + \frac{27}{12} = \frac{43}{12}\)
Answer: \(\frac{43}{12} = 3\frac{7}{12}\)

v. \(\frac{1}{4} + \frac{5}{6} – \frac{1}{12}\)

Step 1: Find the LCM of 4, 6, and 12, which is 12.
Step 2: Convert each fraction to have a denominator of 12:
\(\frac{1}{4} = \frac{3}{12}, \quad \frac{5}{6} = \frac{10}{12}, \quad \frac{1}{12} = \frac{1}{12}\)
Step 3: Add and subtract the fractions: \(\frac{3}{12} + \frac{10}{12} – \frac{1}{12} = \frac{12}{12} = 1\)
Answer: \(1\)

vi. \(\frac{2}{3} – \frac{3}{5} + 3 – \frac{1}{5}\)

Step 1: Convert the whole number 3 to a fraction: \(3 = \frac{15}{5}\)
Step 2: Find the LCM of 3, 5, and 5, which is 15.
Step 3: Convert each fraction to have a denominator of 15:
\(\frac{2}{3} = \frac{10}{15}, \quad \frac{3}{5} = \frac{9}{15}, \quad 3 = \frac{45}{15}, \quad \frac{1}{5} = \frac{3}{15}\)
Step 4: Perform the addition and subtraction: \(\frac{10}{15} – \frac{9}{15} + \frac{45}{15} – \frac{3}{15} = \frac{43}{15}\)
Answer: \(\frac{43}{15} = 2\frac{13}{15}\)

vii. \(\frac{2}{3} – \frac{1}{5} + \frac{1}{10}\)

Step 1: Find the LCM of 3, 5, and 10, which is 30.
Step 2: Convert each fraction to have a denominator of 30:
\(\frac{2}{3} = \frac{20}{30}, \quad \frac{1}{5} = \frac{6}{30}, \quad \frac{1}{10} = \frac{3}{30}\)
Step 3: Add and subtract the fractions: \(\frac{20}{30} – \frac{6}{30} + \frac{3}{30} = \frac{17}{30}\)
Answer: \(\frac{17}{30}\)

viii. \(2\frac{1}{2} + 2\frac{1}{3} – 1\frac{1}{4}\)

Step 1: Convert the mixed fractions to improper fractions:
\(2\frac{1}{2} = \frac{5}{2}, \quad 2\frac{1}{3} = \frac{7}{3}, \quad 1\frac{1}{4} = \frac{5}{4}\)
Step 2: Find the LCM of 2, 3, and 4, which is 12.
Step 3: Convert each fraction to have a denominator of 12:
\(\frac{5}{2} = \frac{30}{12}, \quad \frac{7}{3} = \frac{28}{12}, \quad \frac{5}{4} = \frac{15}{12}\)
Step 4: Perform the addition and subtraction: \(\frac{30}{12} + \frac{28}{12} – \frac{15}{12} = \frac{43}{12}\)
Answer: \(\frac{43}{12} = 3\frac{7}{12}\)

ix. \(2\frac{5}{8} – 2\frac{1}{6} + 4\frac{3}{4}\)

Step 1: Convert the mixed fractions to improper fractions:
\(2\frac{5}{8} = \frac{21}{8}, \quad 2\frac{1}{6} = \frac{13}{6}, \quad 4\frac{3}{4} = \frac{19}{4}\)
Step 2: Find the LCM of 8, 6, and 4, which is 24.
Step 3: Convert each fraction to have a denominator of 24:
\(\frac{21}{8} = \frac{63}{24}, \quad \frac{13}{6} = \frac{52}{24}, \quad \frac{19}{4} = \frac{114}{24}\)
Step 4: Perform the subtraction and addition: \(\frac{63}{24} – \frac{52}{24} + \frac{114}{24} = \frac{125}{24}\)
Answer: \(\frac{125}{24} = 5\frac{5}{24}\)

Q2: Simplify:

i. \(\frac{3}{4} \times 6\)

Step 1: Multiply the fraction by the whole number:
\(\frac{3}{4} \times 6 = \frac{3 \times 6}{4} = \frac{18}{4} = \frac{9}{2}\)
Answer: \(\frac{9}{2} = 4\frac{1}{2}\)

ii. \(\frac{2}{3} \times 15\)

Step 1: Multiply the fraction by the whole number:
\(\frac{2}{3} \times 15 = \frac{2 \times 15}{3} = \frac{30}{3} = 10\)
Answer: \(10\)

iii. \(\frac{3}{4} \times \frac{1}{2}\)

Step 1: Multiply the fractions:
\(\frac{3}{4} \times \frac{1}{2} = \frac{3 \times 1}{4 \times 2} = \frac{3}{8}\)
Answer: \(\frac{3}{8}\)

iv. \(\frac{9}{12} \times \frac{4}{7}\)

Step 1: Multiply the fractions:
\(\frac{9}{12} \times \frac{4}{7} = \frac{9 \times 4}{12 \times 7} = \frac{36}{84}\)
Step 2: Simplify the fraction:
\(\frac{36}{84} = \frac{3}{7}\)
Answer: \(\frac{3}{7}\)

v. \(45 \times 2\frac{1}{3}\)

Step 1: Convert the mixed number to an improper fraction:
\(2\frac{1}{3} = \frac{7}{3}\)
Step 2: Multiply the whole number by the fraction:
\(45 \times \frac{7}{3} = \frac{45 \times 7}{3} = \frac{315}{3} = 105\)
Answer: \(105\)

vi. \(36 \times 3\frac{1}{4}\)

Step 1: Convert the mixed number to an improper fraction:
\(3\frac{1}{4} = \frac{13}{4}\)
Step 2: Multiply the whole number by the fraction:
\(36 \times \frac{13}{4} = \frac{36 \times 13}{4} = \frac{468}{4} = 117\)
Answer: \(117\)

vii. \(2 \div \frac{1}{3}\)

Step 1: Divide by a fraction by multiplying by its reciprocal:
\(2 \div \frac{1}{3} = 2 \times \frac{3}{1} = 6\)
Answer: \(6\)

viii. \(3 \div \frac{2}{5}\)

Step 1: Divide by a fraction by multiplying by its reciprocal:
\(3 \div \frac{2}{5} = 3 \times \frac{5}{2} = \frac{15}{2}\)
Answer: \(\frac{15}{2} = 7\frac{1}{2}\)

ix. \(1 \div \frac{3}{5}\)

Step 1: Divide by a fraction by multiplying by its reciprocal:
\(1 \div \frac{3}{5} = 1 \times \frac{5}{3} = \frac{5}{3}\)
Answer: \(\frac{5}{3} = 1\frac{2}{3}\)

x. \(\frac{1}{3} \div \frac{1}{4}\)

Step 1: Divide by a fraction by multiplying by its reciprocal:
\(\frac{1}{3} \div \frac{1}{4} = \frac{1}{3} \times \frac{4}{1} = \frac{4}{3}\)
Answer: \(\frac{4}{3} = 1\frac{1}{3}\)

xi. \(-\frac{5}{8} \div \frac{3}{4}\)

Step 1: Divide by a fraction by multiplying by its reciprocal:
\(-\frac{5}{8} \div \frac{3}{4} = -\frac{5}{8} \times \frac{4}{3} = -\frac{20}{24} = -\frac{5}{6}\)
Answer: \(-\frac{5}{6}\)

xii. \(3\frac{3}{7} \div 1\frac{1}{14}\)

Step 1: Convert the mixed numbers to improper fractions:
\(3\frac{3}{7} = \frac{24}{7}, \quad 1\frac{1}{14} = \frac{15}{14}\)
Step 2: Divide by a fraction by multiplying by its reciprocal:
\(\frac{24}{7} \div \frac{15}{14} = \frac{24}{7} \times \frac{14}{15} = \frac{336}{105} = \frac{16}{5}\)
Answer: \(\frac{16}{5} = 3\frac{1}{5}\)

xiii. \(3\frac{3}{4} \times 1\frac{1}{5} \times \frac{20}{21}\)

Step 1: Convert the mixed numbers to improper fractions:
\(3\frac{3}{4} = \frac{15}{4}, \quad 1\frac{1}{5} = \frac{6}{5}\)
Step 2: Multiply the fractions:
\(\frac{15}{4} \times \frac{6}{5} = \frac{90}{20} = \frac{9}{2}\)
Step 3: Multiply by \(\frac{20}{21}\):
\(\frac{9}{2} \times \frac{20}{21} = \frac{180}{42} = \frac{30}{7}\)
Answer: \(\frac{30}{7} = 4\frac{2}{7}\)

Q3: Subtract:

i. \(2\ from\ \frac{2}{3}\)

Step 1: Subtract the whole number from the fraction:
\(\frac{2}{3} – 2 = \frac{2}{3} – \frac{6}{3} = \frac{2 – 6}{3} = \frac{-4}{3}\)
Answer: \(\frac{-4}{3} = -1\frac{1}{3}\)

ii. \(\frac{1}{8}\ from\ \frac{5}{8}\)

Step 1: Subtract the fractions with equal denominators:
\(\frac{5}{8} – \frac{1}{8} = \frac{5 – 1}{8} = \frac{4}{8} = \frac{1}{2}\)
Answer: \(\frac{1}{2}\)

iii. \(-\frac{2}{5}\ from\ \frac{2}{5}\)

Step 1: Subtract the fractions:
\(\frac{2}{5} – (-\frac{2}{5}) = \frac{2}{5} + \frac{2}{5} = \frac{4}{5}\)
Answer: \(\frac{4}{5}\)

iv. \(-\frac{3}{7}\ from\ \frac{3}{7}\)

Step 1: Subtract the fractions:
\(\frac{3}{7} – (-\frac{3}{7}) = \frac{3}{7} + \frac{3}{7} = \frac{6}{7}\)
Answer: \(\frac{6}{7}\)

v. \(0\ from\ -\frac{4}{5}\)

Step 1: Subtract 0 from the fraction:
\(-\frac{4}{5} – 0 = -\frac{4}{5}\)
Answer: \(-\frac{4}{5}\)

vi. \(\frac{2}{9}\ from\ \frac{4}{5}\)

Step 1: Make the denominators equal by finding the LCM of 9 and 5, which is 45:
\(\frac{4}{5} = \frac{36}{45}, \quad \frac{2}{9} = \frac{10}{45}\)
Step 2: Subtract the fractions with equal denominators:
\(\frac{36}{45} – \frac{10}{45} = \frac{36 – 10}{45} = \frac{26}{45}\)
Answer: \(\frac{26}{45}\)

vii. \(-\frac{4}{7}\ from\ -\frac{6}{11}\)

Step 1: Make the denominators equal by finding the LCM of 7 and 11, which is 77:
\(-\frac{4}{7} = -\frac{44}{77}, \quad -\frac{6}{11} = -\frac{42}{77}\)
Step 2: Subtract the fractions with equal denominators:
\(-\frac{42}{77} – (-\frac{44}{77}) = -\frac{42}{77} + \frac{44}{77} = \frac{-42 + 44}{77} = \frac{2}{77}\)
Answer: \(\frac{2}{77}\)

Q4: Find the value of:

i. \(\frac{1}{2}\ of\ 10\) kg

Step 1: Multiply the fraction by the given value:
\(\frac{1}{2} \times 10 = \frac{10}{2} = 5\) kg
Answer: 5 kg

ii. \(\frac{3}{5}\ of\ 1\) hours

Step 1: Multiply the fraction by the given value:
\(\frac{3}{5} \times 1 = \frac{3}{5}\) hours \(= \frac{3 \times 60}{5} = 36\) minutes
Answer: \(\frac{3}{5}\) hours or \(\frac{3 \times 60}{5} = 36\) minutes

iii. \(\frac{4}{7}\ of\ 2\frac{1}{3}\) kg

Step 1: Convert the mixed fraction to an improper fraction:
\(2\frac{1}{3} = \frac{7}{3}\)
Step 2: Multiply the fraction by the given value:
\(\frac{4}{7} \times \frac{7}{3} = \frac{28}{21} = \frac{4}{3}\) kg
Answer: \(\frac{4}{3}\) kg or 1\(\frac{1}{3}\) kg

iv. \(3\frac{1}{2}\ times\ of\ 2\) metre

Step 1: Convert the mixed fraction to an improper fraction:
\(3\frac{1}{2} = \frac{7}{2}\)
Step 2: Multiply the fraction by the given value:
\(\frac{7}{2} \times 2 = \frac{14}{2} = 7\) metres
Answer: 7 metres

v. \(\frac{1}{2}\ of\ 2\frac{2}{3}\)

Step 1: Convert the mixed fraction to an improper fraction:
\(2\frac{2}{3} = \frac{8}{3}\)
Step 2: Multiply the fraction by the given value:
\(\frac{1}{2} \times \frac{8}{3} = \frac{8}{6} = \frac{4}{3}\)
Answer: \(\frac{4}{3}\) or 1\(\frac{1}{3}\)

vi. \(\frac{5}{11}\ of\ \frac{4}{5}\ of\ 22\) kg

Step 1: First, calculate \(\frac{4}{5}\ of\ 22\):
\(\frac{4}{5} \times 22 = \frac{88}{5} = 17\frac{3}{5}\) kg
Step 2: Multiply \(\frac{5}{11}\) by 17\(\frac{3}{5}\):
\(\frac{5}{11} \times 17\frac{3}{5} = \frac{5}{11} \times \frac{88}{5} = \frac{440}{55} = 8\) kg
Answer: 8 kg

Q5: Simplify and reduce to a simple fraction:

i. \(\frac{3}{3\frac{3}{4}}\)

Step 1: Convert the mixed fraction in the denominator to an improper fraction:
\(3\frac{3}{4} = \frac{15}{4}\)
Step 2: Simplify the division:
\(\frac{3}{\frac{15}{4}} = 3 \times \frac{4}{15} = \frac{12}{15}\)
Step 3: Reduce to lowest terms:
\(\frac{12}{15} = \frac{4}{5}\)
Answer: \(\frac{4}{5}\)

ii. \(\frac{\frac{3}{5}}{7}\)

Step 1: Convert the division into multiplication by the reciprocal:
\(\frac{3}{5} \div 7 = \frac{3}{5} \times \frac{1}{7} = \frac{3}{35}\)
Answer: \(\frac{3}{35}\)

iii. \(\frac{3}{\frac{5}{7}}\)

Step 1: Convert the division into multiplication by the reciprocal:
\(\frac{3}{\frac{5}{7}} = 3 \times \frac{7}{5} = \frac{21}{5}\)
Answer: \(\frac{21}{5}\) or \(4\frac{1}{5}\)

iv. \(\frac{2\frac{1}{5}}{1\frac{1}{10}}\)

Step 1: Convert the mixed fractions to improper fractions:
\(2\frac{1}{5} = \frac{11}{5},\ 1\frac{1}{10} = \frac{11}{10}\)
Step 2: Divide the fractions:
\(\frac{11}{5} \div \frac{11}{10} = \frac{11}{5} \times \frac{10}{11} = \frac{110}{55} = 2\)
Answer: 2

v. \(\frac{2}{5}\ of\ \frac{6}{11}\times1\frac{1}{4}\)

Step 1: Convert the mixed fraction to an improper fraction:
\(1\frac{1}{4} = \frac{5}{4}\)
Step 2: Multiply the fractions:
\(\frac{2}{5} \times \frac{6}{11} = \frac{12}{55}\)
\(\frac{12}{55} \times \frac{5}{4} = \frac{60}{220} = \frac{3}{11}\)
Answer: \(\frac{3}{11}\)

vi. \(2\frac{1}{4}\div\frac{1}{7}\times\frac{1}{3}\)

Step 1: Convert the mixed fraction to an improper fraction:
\(2\frac{1}{4} = \frac{9}{4}\)
Step 2: Divide and multiply the fractions:
\(\frac{9}{4} \div \frac{1}{7} = \frac{9}{4} \times \frac{7}{1} = \frac{63}{4}\)
\(\frac{63}{4} \times \frac{1}{3} = \frac{63}{12} = \frac{21}{4}\)
Answer: \(\frac{21}{4}\) or \(5\frac{1}{4}\)

vii. \(\frac{1}{3}\times4\frac{2}{3}\div3\frac{1}{2}\times\frac{1}{2}\)

Step 1: Convert the mixed fractions to improper fractions:
\(4\frac{2}{3} = \frac{14}{3},\ 3\frac{1}{2} = \frac{7}{2}\)
Step 2: Perform the operations step by step:
\(\frac{1}{3} \times \frac{14}{3} = \frac{14}{9}\)
\(\frac{14}{9} \div \frac{7}{2} = \frac{14}{9} \times \frac{2}{7} = \frac{28}{63} = \frac{4}{9}\)
\(\frac{4}{9} \times \frac{1}{2} = \frac{4}{18} = \frac{2}{9}\)
Answer: \(\frac{2}{9}\)

viii. \(\frac{2}{3}\times1\frac{1}{4}\div\frac{3}{7}\ of\ 2\frac{5}{8}\)

Step 1: Convert the mixed fractions to improper fractions:
\(1\frac{1}{4} = \frac{5}{4},\ 2\frac{5}{8} = \frac{21}{8}\)
Step 2: Perform the operations step by step:
\(\frac{3}{7}\ of\ 2\frac{5}{8} = \frac{3}{7} \times \frac{21}{8} = \frac{63}{56} = \frac{9}{8}\)
\(1\frac{1}{4} \div \frac{9}{8} = \frac{5}{4} \div \frac{9}{8} = \frac{5}{4} \times \frac{8}{9} = \frac{40}{36} = \frac{10}{9}\)
\(\frac{2}{3} \times \frac{10}{9} = \frac{20}{27}\)
Answer: \(\frac{20}{27}\)

ix. \(0\div\frac{8}{11}\)

Step 1: Any number divided by a non-zero number is 0.
Answer: 0

x. \(\frac{4}{5}\div\frac{7}{15}\ of\ \frac{8}{9}\)

Step 1: Multiply the fractions:
\(\frac{7}{15} \ of\frac{8}{9} = \frac{7}{15} \times \frac{8}{9} = \frac{56}{135}\)
Step 2: Divide the fractions:
\(\frac{4}{5} \div \frac{56}{135} = \frac{4}{5} \times \frac{135}{56} = \frac{540}{280} = \frac{27}{14}\)
Answer: \(\frac{27}{14}\) or \(1\frac{13}{14}\)

xi. \(\frac{4}{5}\div\frac{7}{15}\times\frac{8}{9}\)

Step 1: Divide the fractions:
\(\frac{4}{5} \div \frac{7}{15} = \frac{4}{5} \times \frac{15}{7} = \frac{60}{35} = \frac{12}{7}\)
Step 2: Multiply by \(\frac{8}{9}\):
\(\frac{12}{7} \times \frac{8}{9} = \frac{96}{63} = \frac{32}{21}\)
Answer: \(\frac{32}{21}\) or \(1\frac{11}{21}\)

xii. \(\frac{4}{5}\ of\ \frac{7}{15}\div\frac{8}{9}\)

Step 1: Find \(\frac{4}{5}\) of \(\frac{7}{15}\):
\(\frac{4}{5} \times \frac{7}{15} = \frac{28}{75}\)
Step 2: Divide by \(\frac{8}{9}\):
\(\frac{28}{75} \div \frac{8}{9} = \frac{28}{75} \times \frac{9}{8} = \frac{252}{600} = \frac{21}{50}\)
Answer: \(\frac{21}{50}\)

xiii. \(\frac{1}{2}\ of\ \frac{3}{4}\times\frac{1}{2}\div\frac{2}{3}\)

Step 1: Find \(\frac{1}{2}\) of \(\frac{3}{4}\):
\(\frac{1}{2} \times \frac{3}{4} = \frac{3}{8}\)
Step 2: Multiply by \(\frac{1}{2}\):
\(\frac{3}{8} \times \frac{1}{2} = \frac{3}{16}\)
Step 3: Divide by \(\frac{2}{3}\):
\(\frac{3}{16} \div \frac{2}{3} = \frac{3}{16} \times \frac{3}{2} = \frac{9}{32}\)
Answer: \(\frac{9}{32}\)

Q6: A bought \(3\frac{3}{4}\) kg of wheat and \(2\frac{1}{2}\) kg of rice. Find the total weight of wheat and rice bought by A.

Step 1: Convert the mixed fractions to improper fractions:
\(3\frac{3}{4} = \frac{12+3}{4} = \frac{15}{4}\)
\(2\frac{1}{2} = \frac{4+1}{2} = \frac{5}{2}\)
Step 2: To add the fractions, we need a common denominator. The least common denominator (LCD) of 4 and 2 is 4.
Convert \(\frac{5}{2}\) to an equivalent fraction with denominator 4:
\[ \frac{5}{2} = \frac{5 \times 2}{2 \times 2} = \frac{10}{4} \]Step 3: Now add the two fractions:
\[ \frac{15}{4} + \frac{10}{4} = \frac{15+10}{4} = \frac{25}{4} \]Step 4: Convert the improper fraction \(\frac{25}{4}\) back to a mixed fraction:
\[ \frac{25}{4} = 6 \frac{1}{4} \]Answer: The total weight of wheat and rice bought by A is \(6\frac{1}{4}\) kg.

Q7: Which is greater, \(\frac{3}{5}\) or \(\frac{7}{10}\) and by how much?

Step 1: To compare \(\frac{3}{5}\) and \(\frac{7}{10}\), we need a common denominator. The least common denominator (LCD) of 5 and 10 is 10.
Step 2: Convert \(\frac{3}{5}\) to an equivalent fraction with denominator 10:
\[ \frac{3}{5} = \frac{3 \times 2}{5 \times 2} = \frac{6}{10} \]Step 3: Now compare the two fractions: \(\frac{6}{10}\) and \(\frac{7}{10}\). Since the denominators are the same, we can directly compare the numerators.
Clearly, \(\frac{7}{10} > \frac{6}{10}\).Step 4: To find by how much \(\frac{7}{10}\) is greater than \(\frac{3}{5}\), subtract \(\frac{6}{10}\) from \(\frac{7}{10}\):
\[ \frac{7}{10} – \frac{6}{10} = \frac{1}{10} \]Answer: \(\frac{7}{10}\) is greater than \(\frac{3}{5}\) by \(\frac{1}{10}\).

Q8: What number should be added to \(8\frac{2}{3}\) to get \(12\frac{5}{6}\)?

Step 1: To solve this, we need to subtract \(8\frac{2}{3}\) from \(12\frac{5}{6}\). The number to be added to \(8\frac{2}{3}\) is the difference between these two mixed fractions.
Step 2: Convert both mixed fractions to improper fractions.
For \(8\frac{2}{3}\): \[ 8\frac{2}{3} = \frac{8 \times 3 + 2}{3} = \frac{24 + 2}{3} = \frac{26}{3} \]For \(12\frac{5}{6}\): \[ 12\frac{5}{6} = \frac{12 \times 6 + 5}{6} = \frac{72 + 5}{6} = \frac{77}{6} \]Step 3: Subtract \(\frac{26}{3}\) from \(\frac{77}{6}\). To do this, we need a common denominator. The least common denominator (LCD) of 3 and 6 is 6.
Convert \(\frac{26}{3}\) to a fraction with denominator 6: \[ \frac{26}{3} = \frac{26 \times 2}{3 \times 2} = \frac{52}{6} \]Step 4: Now subtract the fractions: \[ \frac{77}{6} – \frac{52}{6} = \frac{77 – 52}{6} = \frac{25}{6} \]Step 5: Convert the improper fraction \(\frac{25}{6}\) back to a mixed fraction: \[ \frac{25}{6} = 4\frac{1}{6} \]Answer: The number that should be added to \(8\frac{2}{3}\) to get \(12\frac{5}{6}\) is \(4\frac{1}{6}\).

Q9: What should be subtracted from \(8\frac{3}{4}\) to get \(2\frac{2}{3}\)?

Step 1: To find what should be subtracted, we need to subtract \(2\frac{2}{3}\) from \(8\frac{3}{4}\). \[ \text{What to subtract} = 8\frac{3}{4} – 2\frac{2}{3} \]Step 2: Convert the mixed fractions to improper fractions.For \(8\frac{3}{4}\): \[ 8\frac{3}{4} = \frac{8 \times 4 + 3}{4} = \frac{32 + 3}{4} = \frac{35}{4} \]For \(2\frac{2}{3}\): \[ 2\frac{2}{3} = \frac{2 \times 3 + 2}{3} = \frac{6 + 2}{3} = \frac{8}{3} \]Step 3: Subtract the fractions: \[ \frac{35}{4} – \frac{8}{3} \] To subtract these fractions, we need to find a common denominator. The least common denominator (LCD) of 4 and 3 is 12.Convert both fractions: \[ \frac{35}{4} = \frac{35 \times 3}{4 \times 3} = \frac{105}{12} \] \[ \frac{8}{3} = \frac{8 \times 4}{3 \times 4} = \frac{32}{12} \]Now subtract the fractions: \[ \frac{105}{12} – \frac{32}{12} = \frac{105 – 32}{12} = \frac{73}{12} \]Step 4: Convert the improper fraction \(\frac{73}{12}\) back to a mixed fraction: \[ \frac{73}{12} = 6\frac{1}{12} \]Answer: The number that should be subtracted from \(8\frac{3}{4}\) to get \(2\frac{2}{3}\) is \(6\frac{1}{12}\).

Q10: A rectangular field is \(16\frac{1}{2}\) m long and \(12\frac{2}{5}\) m wide. Find the perimeter of the field.

Step 1: The perimeter \(P\) of a rectangle is given by the formula: \[ P = 2 \times ( \text{length} + \text{width} ) \]Step 2: Convert the mixed fractions for length and width into improper fractions.For the length \(16\frac{1}{2}\): \[ 16\frac{1}{2} = \frac{16 \times 2 + 1}{2} = \frac{32 + 1}{2} = \frac{33}{2} \]For the width \(12\frac{2}{5}\): \[ 12\frac{2}{5} = \frac{12 \times 5 + 2}{5} = \frac{60 + 2}{5} = \frac{62}{5} \]Step 3: Add the length and the width: \[ \frac{33}{2} + \frac{62}{5} \] To add these fractions, we need to find a common denominator. The least common denominator (LCD) of 2 and 5 is 10.Convert both fractions: \[ \frac{33}{2} = \frac{33 \times 5}{2 \times 5} = \frac{165}{10} \] \[ \frac{62}{5} = \frac{62 \times 2}{5 \times 2} = \frac{124}{10} \]Now add the fractions: \[ \frac{165}{10} + \frac{124}{10} = \frac{165 + 124}{10} = \frac{289}{10} \]Step 4: Multiply the sum by 2 to find the perimeter: \[ P = 2 \times \frac{289}{10} = \frac{578}{10} = 57\frac{8}{10} = 57\frac{4}{5} \]Answer: The perimeter of the field is \(57\frac{4}{5}\) meters.

Q11: Sugar costs ₹ \(37\frac{1}{2}\) per kg. Find the cost of \(8\frac{3}{4}\) kg sugar.

Step 1: Convert the mixed numbers to improper fractions.Cost per kg: \[ ₹37\frac{1}{2} = \frac{75}{2} \]Weight of sugar: \[ 8\frac{3}{4} = \frac{35}{4} \]Step 2: Multiply the two improper fractions: \[ \text{Total cost} = \frac{75}{2} \times \frac{35}{4} \]Multiply numerators and denominators: \[ = \frac{75 \times 35}{2 \times 4} = \frac{2625}{8} \]Step 3: Convert the improper fraction to a mixed number: \[ \frac{2625}{8} = 328 \frac{1}{8} \]Answer: ₹ \(328 \frac{1}{8}\) or ₹ 328.125

Q12: A cycle runs \(31\frac{1}{4}\) km consuming 1 litte of petrol. How much distance will it run consuming \(1\frac{3}{5}\) litre of petrol?

Step 1: Convert the mixed numbers to improper fractions.Distance per litre: \[ 31\frac{1}{4} = \frac{125}{4} \]Petrol used: \[ 1\frac{3}{5} = \frac{8}{5} \]Step 2: Multiply the two values to get total distance: \[ \text{Total distance} = \frac{125}{4} \times \frac{8}{5} \]Multiply numerators and denominators: \[ = \frac{125 \times 8}{4 \times 5} = \frac{1000}{20} \]Simplify: \[ = 50 \text{ km} \]Answer: 50 km

Q13: A rectangular park has length = \(23\frac{2}{5}\) m and breadth = \(16\frac{2}{3}\) m. Find the area of the park.

Step 1: Convert mixed numbers to improper fractions.Length: \[ 23\frac{2}{5} = \frac{117}{5} \]Breadth: \[ 16\frac{2}{3} = \frac{50}{3} \]Step 2: Use the formula for area of a rectangle: \[ \text{Area} = \text{Length} \times \text{Breadth} \]Substitute the values: \[ \text{Area} = \frac{117}{5} \times \frac{50}{3} \]Step 3: Multiply numerators and denominators: \[ = \frac{117 \times 50}{5 \times 3} = \frac{5850}{15} \]Step 4: Simplify the fraction: \[ = 390 \text{ m}^2 \]Answer: 390 m²

Q14: Each of 40 identical boxes weighs \(4\frac{4}{5}\) kg. Find the total weight of all the boxes.

Step 1: Convert the mixed number to an improper fraction. \[ 4\frac{4}{5} = \frac{24}{5} \]Step 2: Multiply the weight of one box by the total number of boxes. \[ \text{Total weight} = 40 \times \frac{24}{5} \]Step 3: Multiply: \[ = \frac{40 \times 24}{5} = \frac{960}{5} \]Step 4: Simplify: \[ = 192 \text{ kg} \]Answer: 192 kg

Q15: Out of 24 kg of wheat, \(\frac{5}{6}\)th of wheat is consumed. Find how much wheat is still left?

Step 1: Total wheat available = 24 kgStep 2: Fraction of wheat consumed = \(\frac{5}{6}\)Step 3: Find the quantity consumed:
\[ \frac{5}{6} \times 24 = \frac{120}{6} = 20 \text{ kg} \]Step 4: Subtract consumed wheat from total:
\[ \text{Wheat left} = 24 – 20 = 4 \text{ kg} \]Answer: 4 kg

Q16: A rod of \(2\frac{2}{5}\) metre is divided into five equal parts. Find the length of each part so obtained.

Step 1: Total length of the rod = \(2\frac{2}{5}\) metres Convert to improper fraction: \[ 2\frac{2}{5} = \frac{(2 \times 5) + 2}{5} = \frac{10 + 2}{5} = \frac{12}{5} \]Step 2: Divide the rod into 5 equal parts: \[ \text{Length of each part} = \frac{12}{5} \div 5 = \frac{12}{5} \times \frac{1}{5} = \frac{12}{25} \]Step 3: Convert to decimal or mixed fraction (optional): \[ \frac{12}{25} = 0.48 \text{ metre} \]Answer: \(\frac{12}{25}\) metre or 0.48 metre

Q17: 1f A = \(3\frac{3}{8}\) and B = \(6\frac{5}{8}\), find:

i. A ÷ B

Step 1: Convert A and B to improper fractions A = \(3\frac{3}{8} = \frac{(3×8)+3}{8} = \frac{24+3}{8} = \frac{27}{8}\) B = \(6\frac{5}{8} = \frac{(6×8)+5}{8} = \frac{48+5}{8} = \frac{53}{8}\)Step 2: Calculate A ÷ B \[ \frac{27}{8} \div \frac{53}{8} = \frac{27}{8} \times \frac{8}{53} = \frac{216}{424} \] Now reduce the fraction: \[ \frac{216}{424} = \frac{54}{106} = \frac{27}{53} \]Answer: A ÷ B = \(\frac{27}{53}\)

ii. B ÷ A

Step 1: Calculate B ÷ A \[ \frac{53}{8} \div \frac{27}{8} = \frac{53}{8} \times \frac{8}{27} = \frac{424}{216} \] Now reduce the fraction: \[ \frac{424}{216} = \frac{106}{54} = \frac{53}{27} \]Answer: B ÷ A = \(\frac{53}{27}\) or \(1\frac{26}{27}\)

Q18: Cost of \(3\frac{5}{7}\) litres of oil is ₹ \(83\frac{1}{2}\). Find the cost of one litre oil.

Step 1: Convert the mixed numbers into improper fractions.Cost = ₹ \(83\frac{1}{2} = \frac{(83×2)+1}{2} = \frac{166+1}{2} = \frac{167}{2}\) Quantity = \(3\frac{5}{7} = \frac{(3×7)+5}{7} = \frac{21+5}{7} = \frac{26}{7}\)Step 2: To find the cost of 1 litre oil, divide total cost by total litres: \[ \text{Cost of 1 litre} = \frac{167}{2} \div \frac{26}{7} = \frac{167}{2} \times \frac{7}{26} \]Step 3: Multiply the numerators and denominators: \[ = \frac{167×7}{2×26} = \frac{1169}{52} \]Step 4: Convert the improper fraction into a mixed number: \[ 1169 ÷ 52 = 22 \text{ remainder } 25 \] \[ \Rightarrow \frac{1169}{52} = 22\frac{25}{52} \]Answer: ₹ \(22\frac{25}{52}\) per litre

Q19: The product of two numbers is \(20\frac{5}{7}\). If one of these numbers is \(6\frac{2}{3}\), find the other.

Step 1: Convert the mixed numbers into improper fractions. \[ 20\frac{5}{7} = \frac{(20 \times 7) + 5}{7} = \frac{140 + 5}{7} = \frac{145}{7} \] \[ 6\frac{2}{3} = \frac{(6 \times 3) + 2}{3} = \frac{18 + 2}{3} = \frac{20}{3} \]Step 2: Let the other number be \(x\). Then: \[ x \times \frac{20}{3} = \frac{145}{7} \]Step 3: Solve for \(x\) by dividing both sides by \(\frac{20}{3}\): \[ x = \frac{145}{7} \div \frac{20}{3} = \frac{145}{7} \times \frac{3}{20} \]Step 4: Multiply the fractions: \[ x = \frac{145 \times 3}{7 \times 20} = \frac{435}{140} \]Step 5: Simplify the fraction: \[ \frac{435}{140} = \frac{87}{28} \quad \text{(Divide numerator and denominator by 5)} \]Step 6: Convert to mixed number: \[ 87 ÷ 28 = 3 \text{ remainder } 3 \Rightarrow \frac{87}{28} = 3\frac{3}{28} \]Answer: \(3\frac{3}{28}\)

Q20: By what number should \(5\frac{5}{6}\) be multiplied to get \(3\frac{1}{3}\)?

Step 1: Convert the mixed numbers into improper fractions. \[ 5\frac{5}{6} = \frac{(5 \times 6) + 5}{6} = \frac{30 + 5}{6} = \frac{35}{6} \] \[ 3\frac{1}{3} = \frac{(3 \times 3) + 1}{3} = \frac{9 + 1}{3} = \frac{10}{3} \]Step 2: Let the required number be \(x\). Then, \[ \frac{35}{6} \times x = \frac{10}{3} \]Step 3: Solve for \(x\) by dividing both sides by \(\frac{35}{6}\): \[ x = \frac{10}{3} \div \frac{35}{6} = \frac{10}{3} \times \frac{6}{35} \]Step 4: Multiply the numerators and denominators: \[ x = \frac{10 \times 6}{3 \times 35} = \frac{60}{105} \]Step 5: Simplify the fraction: \[ \frac{60}{105} = \frac{12}{21} = \frac{4}{7} \quad \text{(Divide by 3, then by 3 again)} \]Answer: \(\frac{4}{7}\)