Fractions

Q1: Find the sum of:

i. \(\frac{2}{7} + \frac{3}{7}\)

Step 1: Denominators are same, so add numerators. \[ \frac{2}{7} + \frac{3}{7} = \frac{2+3}{7} = \frac{5}{7} \] Answer: \(\frac{5}{7}\)

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

Step 1: Denominators are same, so add numerators. \[ \frac{5}{8} + \frac{1}{8} = \frac{5+1}{8} = \frac{6}{8} \] Step 2: Simplify the fraction. \[ \frac{6}{8} = \frac{3}{4} \] Answer: \(\frac{3}{4}\)

iii. \(\frac{7}{9} + \frac{4}{9}\)

Step 1: Denominators are same, so add numerators. \[ \frac{7}{9} + \frac{4}{9} = \frac{7+4}{9} = \frac{11}{9} \] Step 2: Convert improper fraction to mixed number. \[ \frac{11}{9} = 1 \frac{2}{9} \] Answer: \(1 \frac{2}{9}\)

iv. \(1\frac{5}{6} + \frac{1}{6}\)

Step 1: Convert mixed number to improper fraction. \[ 1 \frac{5}{6} = \frac{6 \times 1 + 5}{6} = \frac{11}{6} \] Step 2: Add fractions with same denominator. \[ \frac{11}{6} + \frac{1}{6} = \frac{11+1}{6} = \frac{12}{6} = 2 \] Answer: 2

Q2: Find the sum of:

i. \(\frac{5}{14} + \frac{3}{14} + \frac{1}{14}\)

Step 1: Denominators are same, so add numerators. \[ \frac{5}{14} + \frac{3}{14} + \frac{1}{14} = \frac{5 + 3 + 1}{14} = \frac{9}{14} \] Answer: \(\frac{9}{14}\)

ii. \(\frac{7}{12} + \frac{5}{12} + \frac{11}{12}\)

Step 1: Denominators are same, so add numerators. \[ \frac{7}{12} + \frac{5}{12} + \frac{11}{12} = \frac{7 + 5 + 11}{12} = \frac{23}{12} \] Step 2: Convert improper fraction to mixed number. \[ \frac{23}{12} = 1 \frac{11}{12} \] Answer: \(1 \frac{11}{12}\)

iii. \(1 \frac{7}{8} + \frac{3}{8} + 1 \frac{5}{8}\)

Step 1: Convert mixed numbers to improper fractions. \[ 1 \frac{7}{8} = \frac{8 \times 1 + 7}{8} = \frac{15}{8} \\ 1 \frac{5}{8} = \frac{8 \times 1 + 5}{8} = \frac{13}{8} \] Step 2: Add all fractions (denominators are same). \[ \frac{15}{8} + \frac{3}{8} + \frac{13}{8} = \frac{15 + 3 + 13}{8} = \frac{31}{8} \] Step 3: Convert improper fraction to mixed number. \[ \frac{31}{8} = 3 \frac{7}{8} \] Answer: \(3 \frac{7}{8}\)

Q3: Find the sum of:

i. \(\frac{4}{9} + \frac{5}{6}\)

Step 1: Find the LCM of denominators 9 and 6.
LCM (9,6) = 18
Step 2: Convert fractions to equivalent fractions with denominator 18. \[ \frac{4}{9} = \frac{4 \times 2}{9 \times 2} = \frac{8}{18} \\ \frac{5}{6} = \frac{5 \times 3}{6 \times 3} = \frac{15}{18} \] Step 3: Add the fractions. \[ \frac{8}{18} + \frac{15}{18} = \frac{8 + 15}{18} = \frac{23}{18} \] Step 4: Convert improper fraction to mixed number. \[ \frac{23}{18} = 1 \frac{5}{18} \] Answer: \(1 \frac{5}{18}\)

ii. \(\frac{5}{12} + \frac{9}{16}\)

Step 1: Find the LCM of denominators 12 and 16.
LCM (12,16) = 48
Step 2: Convert fractions to equivalent fractions with denominator 48. \[ \frac{5}{12} = \frac{5 \times 4}{12 \times 4} = \frac{20}{48} \\ \frac{9}{16} = \frac{9 \times 3}{16 \times 3} = \frac{27}{48} \] Step 3: Add the fractions. \[ \frac{20}{48} + \frac{27}{48} = \frac{20 + 27}{48} = \frac{47}{48} \] Answer: \(\frac{47}{48}\)

iii. \(\frac{7}{12} + \frac{13}{18}\)

Step 1: Find the LCM of denominators 12 and 18.
LCM (12,18) = 36
Step 2: Convert fractions to equivalent fractions with denominator 36. \[ \frac{7}{12} = \frac{7 \times 3}{12 \times 3} = \frac{21}{36} \\ \frac{13}{18} = \frac{13 \times 2}{18 \times 2} = \frac{26}{36} \] Step 3: Add the fractions. \[ \frac{21}{36} + \frac{26}{36} = \frac{21 + 26}{36} = \frac{47}{36} \] Step 4: Convert improper fraction to mixed number. \[ \frac{47}{36} = 1 \frac{11}{36} \] Answer: \(1 \frac{11}{36}\)

Q4: Find the sum of:

i. \(\frac{3}{10} + \frac{8}{15} + \frac{7}{20}\)

Step 1: Find the LCM of denominators 10, 15, and 20.
LCM (10,15,20) = 60
Step 2: Convert each fraction to have denominator 60. \[ \frac{3}{10} = \frac{3 \times 6}{10 \times 6} = \frac{18}{60} \\ \frac{8}{15} = \frac{8 \times 4}{15 \times 4} = \frac{32}{60} \\ \frac{7}{20} = \frac{7 \times 3}{20 \times 3} = \frac{21}{60} \] Step 3: Add the fractions. \[ \frac{18}{60} + \frac{32}{60} + \frac{21}{60} = \frac{18 + 32 + 21}{60} = \frac{71}{60} \] Step 4: Convert improper fraction to mixed number. \[ \frac{71}{60} = 1 \frac{11}{60} \] Answer: \(1 \frac{11}{60}\)

ii. \(\frac{7}{8} + \frac{9}{16} + \frac{17}{24}\)

Step 1: Find the LCM of denominators 8, 16, and 24.
LCM (8,16,24) = 48
Step 2: Convert each fraction to have denominator 48. \[ \frac{7}{8} = \frac{7 \times 6}{8 \times 6} = \frac{42}{48} \\ \frac{9}{16} = \frac{9 \times 3}{16 \times 3} = \frac{27}{48} \\ \frac{17}{24} = \frac{17 \times 2}{24 \times 2} = \frac{34}{48} \] Step 3: Add the fractions. \[ \frac{42}{48} + \frac{27}{48} + \frac{34}{48} = \frac{42 + 27 + 34}{48} = \frac{103}{48} \] Step 4: Convert improper fraction to mixed number. \[ \frac{103}{48} = 2 \frac{7}{48} \] Answer: \(2 \frac{7}{48}\)

iii. \(\frac{5}{6} + \frac{8}{9} + \frac{11}{18} + \frac{13}{27}\)

Step 1: Find the LCM of denominators 6, 9, 18, and 27.
LCM (6,9,18,27) = 54
Step 2: Convert each fraction to have denominator 54. \[ \frac{5}{6} = \frac{5 \times 9}{6 \times 9} = \frac{45}{54} \\ \frac{8}{9} = \frac{8 \times 6}{9 \times 6} = \frac{48}{54} \\ \frac{11}{18} = \frac{11 \times 3}{18 \times 3} = \frac{33}{54} \\ \frac{13}{27} = \frac{13 \times 2}{27 \times 2} = \frac{26}{54} \] Step 3: Add the fractions. \[ \frac{45}{54} + \frac{48}{54} + \frac{33}{54} + \frac{26}{54} \\ = \frac{45 + 48 + 33 + 26}{54} = \frac{152}{54} \] Step 4: Simplify the fraction. \[ \frac{152}{54} = \frac{76}{27} \] Step 5: Convert improper fraction to mixed number. \[ \frac{76}{27} = 2 \frac{22}{27} \] Answer: \(2 \frac{22}{27}\)

Q5: Find the sum of:

i. \(4\frac{1}{6} + 2\frac{5}{8} + 3\frac{7}{12}\)

Step 1: Convert mixed fractions to improper fractions. \[ 4\frac{1}{6} = \frac{4 \times 6 + 1}{6} = \frac{25}{6} \\ 2\frac{5}{8} = \frac{2 \times 8 + 5}{8} = \frac{21}{8} \\ 3\frac{7}{12} = \frac{3 \times 12 + 7}{12} = \frac{43}{12} \] Step 2: Find LCM of denominators 6, 8, and 12.
LCM (6,8,12) = 24
Step 3: Convert fractions to denominator 24. \[ \frac{25}{6} = \frac{25 \times 4}{6 \times 4} = \frac{100}{24} \\ \frac{21}{8} = \frac{21 \times 3}{8 \times 3} = \frac{63}{24} \\ \frac{43}{12} = \frac{43 \times 2}{12 \times 2} = \frac{86}{24} \] Step 4: Add the fractions. \[ \frac{100}{24} + \frac{63}{24} + \frac{86}{24} = \frac{249}{24} \] Step 5: Convert improper fraction to mixed number. \[ \frac{249}{24} = 10 \frac{9}{24} = 10 \frac{3}{8} \] Answer: \(10 \frac{3}{8}\)

ii. \(1\frac{1}{2} + 2\frac{2}{3} + 3\frac{3}{4} + 4\frac{4}{5}\)

Step 1: Convert mixed fractions to improper fractions. \[ 1\frac{1}{2} = \frac{1 \times 2 + 1}{2} = \frac{3}{2} \\ 2\frac{2}{3} = \frac{2 \times 3 + 2}{3} = \frac{8}{3} \\ 3\frac{3}{4} = \frac{3 \times 4 + 3}{4} = \frac{15}{4} \\ 4\frac{4}{5} = \frac{4 \times 5 + 4}{5} = \frac{24}{5} \] Step 2: Find LCM of denominators 2, 3, 4, and 5.
LCM (2,3,4,5) = 60
Step 3: Convert fractions to denominator 60. \[ \frac{3}{2} = \frac{3 \times 30}{2 \times 30} = \frac{90}{60} \\ \frac{8}{3} = \frac{8 \times 20}{3 \times 20} = \frac{160}{60} \\ \frac{15}{4} = \frac{15 \times 15}{4 \times 15} = \frac{225}{60} \\ \frac{24}{5} = \frac{24 \times 12}{5 \times 12} = \frac{288}{60} \] Step 4: Add the fractions. \[ \frac{90}{60} + \frac{160}{60} + \frac{225}{60} + \frac{288}{60} = \frac{763}{60} \] Step 5: Convert improper fraction to mixed number. \[ \frac{763}{60} = 12 \frac{43}{60} \] Answer: \(12 \frac{43}{60}\)

iii. \(3\frac{1}{3} + 2\frac{2}{9} + 4\frac{1}{2} + 1\frac{13}{18}\)

Step 1: Convert mixed fractions to improper fractions. \[ 3\frac{1}{3} = \frac{3 \times 3 + 1}{3} = \frac{10}{3} \\ 2\frac{2}{9} = \frac{2 \times 9 + 2}{9} = \frac{20}{9} \\ 4\frac{1}{2} = \frac{4 \times 2 + 1}{2} = \frac{9}{2} \\ 1\frac{13}{18} = \frac{1 \times 18 + 13}{18} = \frac{31}{18} \] Step 2: Find LCM of denominators 3, 9, 2, and 18.
LCM (3,9,2,18) = 18
Step 3: Convert fractions to denominator 18. \[ \frac{10}{3} = \frac{10 \times 6}{3 \times 6} = \frac{60}{18} \\ \frac{20}{9} = \frac{20 \times 2}{9 \times 2} = \frac{40}{18} \\ \frac{9}{2} = \frac{9 \times 9}{2 \times 9} = \frac{81}{18} \\ \frac{31}{18} = \frac{31}{18} \] Step 4: Add the fractions. \[ \frac{60}{18} + \frac{40}{18} + \frac{81}{18} + \frac{31}{18} = \frac{212}{18} \] Step 5: Simplify the fraction. \[ \frac{212}{18} = \frac{106}{9} \] Step 6: Convert improper fraction to mixed number. \[ \frac{106}{9} = 11 \frac{7}{9} \] Answer: \(11 \frac{7}{9}\)

Q6: Find the difference:

i. \(\frac{9}{11} – \frac{5}{11}\)

Step 1: Since denominators are the same, subtract numerators. \[ \frac{9}{11} – \frac{5}{11} = \frac{9 – 5}{11} = \frac{4}{11} \] Answer: \(\frac{4}{11}\)

ii. \(8\frac{5}{7} – \frac{6}{7}\)

Step 1: Convert mixed fraction to improper fraction. \[ 8\frac{5}{7} = \frac{8 \times 7 + 5}{7} = \frac{61}{7} \] Step 2: Subtract the fractions (denominator same). \[ \frac{61}{7} – \frac{6}{7} = \frac{61 – 6}{7} = \frac{55}{7} \] Step 3: Convert improper fraction to mixed number. \[ \frac{55}{7} = 7 \frac{6}{7} \] Answer: \(7 \frac{6}{7}\)

iii. \(3\frac{3}{8} – 1\frac{7}{8}\)

Step 1: Convert mixed fractions to improper fractions. \[ 3\frac{3}{8} = \frac{3 \times 8 + 3}{8} = \frac{27}{8} \\ 1\frac{7}{8} = \frac{1 \times 8 + 7}{8} = \frac{15}{8} \] Step 2: Subtract the fractions (denominators are same). \[ \frac{27}{8} – \frac{15}{8} = \frac{27 – 15}{8} = \frac{12}{8} \] Step 3: Simplify the fraction. \[ \frac{12}{8} = \frac{3}{2} = 1 \frac{1}{2} \] Answer: \(1 \frac{1}{2}\)

iv. \(\frac{7}{4} – \frac{3}{4}\)

Step 1: Subtract the numerators as denominators are same. \[ \frac{7}{4} – \frac{3}{4} = \frac{7 – 3}{4} = \frac{4}{4} = 1 \] Answer: \(1\)

Q7: Find the difference:

i. \(\frac{7}{15} – \frac{9}{20}\)

Step 1: Find LCM of 15 and 20.
LCM(15, 20) = 60 \[ \frac{7}{15} = \frac{28}{60},\quad \frac{9}{20} = \frac{27}{60} \] Step 2: Subtract the numerators. \[ \frac{28}{60} – \frac{27}{60} = \frac{1}{60} \] Answer: \(\frac{1}{60}\)

ii. \(4\frac{1}{7} – 2\frac{1}{14}\)

Step 1: Convert to improper fractions. \[ 4\frac{1}{7} = \frac{29}{7},\quad 2\frac{1}{14} = \frac{29}{14} \] Step 2: Find LCM of 7 and 14 = 14 \[ \frac{29}{7} = \frac{58}{14},\quad \frac{29}{14} = \frac{29}{14} \\ \frac{58}{14} – \frac{29}{14} = \frac{29}{14} \] Step 3: Convert to mixed number. \[ \frac{29}{14} = 2 \frac{1}{14} \] Answer: \(2 \frac{1}{14}\)

iii. \(1\frac{7}{8} – \frac{5}{12}\)

Step 1: Convert mixed to improper fraction. \[ 1\frac{7}{8} = \frac{15}{8} \] Step 2: Find LCM of 8 and 12 = 24 \[ \frac{15}{8} = \frac{45}{24},\quad \frac{5}{12} = \frac{10}{24} \] Step 3: Subtract: \[ \frac{45}{24} – \frac{10}{24} = \frac{35}{24} \] Step 4: Convert to mixed number. \[ \frac{35}{24} = 1 \frac{11}{24} \] Answer: \(1 \frac{11}{24}\)

iv. \(5 – 2\frac{3}{4}\)

Step 1: Convert mixed number to improper fraction. \[ 2\frac{3}{4} = \frac{11}{4} \\ 5 = \frac{20}{4} \] Step 2: Subtract: \[ \frac{20}{4} – \frac{11}{4} = \frac{9}{4} = 2 \frac{1}{4} \] Answer: \(2 \frac{1}{4}\)

Q8: Find the difference:

i. \(5\frac{1}{8} – 4\frac{1}{12}\)

Step 1: Convert to improper fractions. \[ 5\frac{1}{8} = \frac{41}{8},\quad 4\frac{1}{12} = \frac{49}{12} \] Step 2: Find LCM of 8 and 12 = 24 \[ \frac{41}{8} = \frac{123}{24},\quad \frac{49}{12} = \frac{98}{24} \] Step 3: Subtract: \[ \frac{123}{24} – \frac{98}{24} = \frac{25}{24} = 1 \frac{1}{24} \] Answer: \(1 \frac{1}{24}\)

ii. \(6\frac{1}{6} – 3\frac{7}{10}\)

Step 1: Convert to improper fractions. \[ 6\frac{1}{6} = \frac{37}{6},\quad 3\frac{7}{10} = \frac{37}{10} \] Step 2: Find LCM of 6 and 10 = 30 \[ \frac{37}{6} = \frac{185}{30},\quad \frac{37}{10} = \frac{111}{30} \] Step 3: Subtract: \[ \frac{185}{30} – \frac{111}{30} = \frac{74}{30} \] Step 4: Simplify: \[ \frac{74}{30} = \frac{37}{15} = 2 \frac{7}{15} \] Answer: \(2 \frac{7}{15}\)

iii. \(12 – 6\frac{5}{8}\)

Step 1: Convert mixed number to improper fraction. \[ 6\frac{5}{8} = \frac{53}{8},\quad 12 = \frac{96}{8} \] Step 2: Subtract: \[ \frac{96}{8} – \frac{53}{8} = \frac{43}{8} = 5 \frac{3}{8} \] Answer: \(5 \frac{3}{8}\)

Q9: Simplify: \(\frac{4}{9} – \frac{5}{12} + \frac{1}{4}\)

Step 1: Find the LCM of the denominators 9, 12, and 4.
LCM of 9, 12, and 4 = 36
Step 2: Convert all fractions to have a denominator of 36: \[ \frac{4}{9} = \frac{4 \times 4}{9 \times 4} = \frac{16}{36} \\ \frac{5}{12} = \frac{5 \times 3}{12 \times 3} = \frac{15}{36} \\ \frac{1}{4} = \frac{1 \times 9}{4 \times 9} = \frac{9}{36} \]Step 3: Now perform the operations: \[ \frac{16}{36} – \frac{15}{36} + \frac{9}{36} = \frac{(16 – 15 + 9)}{36} = \frac{10}{36} \]Step 4: Simplify the result: \[ \frac{10}{36} = \frac{5}{18} \]Answer: \(\frac{5}{18}\)

Q10: Simplify: \(6\frac{2}{3} + 4\frac{1}{6} – 2\frac{2}{9}\)

Step 1: Convert all mixed fractions into improper fractions: \[ 6\frac{2}{3} = \frac{20}{3} \\ 4\frac{1}{6} = \frac{25}{6} \\ 2\frac{2}{9} = \frac{20}{9} \]Step 2: Find LCM of denominators: 3, 6, and 9.
LCM = 18
Step 3: Convert each fraction to have denominator 18: \[ \frac{20}{3} = \frac{120}{18} \\ \frac{25}{6} = \frac{75}{18} \\ \frac{20}{9} = \frac{40}{18} \]Step 4: Now perform the operations: \[ \frac{120}{18} + \frac{75}{18} – \frac{40}{18} = \frac{(120 + 75 – 40)}{18} = \frac{155}{18} \]Step 5: Convert to mixed fraction: \[ \frac{155}{18} = 8\frac{11}{18} \]Answer: \(8\frac{11}{18}\)

Q11: Simplify: \(9 – 4\frac{1}{2} – 2\frac{1}{5}\)

Step 1: Convert mixed fractions into improper fractions: \[ 4\frac{1}{2} = \frac{9}{2} \\ 2\frac{1}{5} = \frac{11}{5} \] Step 2: Convert 9 into a fraction with LCM of 2 and 5 = 10. \[ 9 = \frac{90}{10},\quad \frac{9}{2} = \frac{45}{10},\quad \frac{11}{5} = \frac{22}{10} \]Step 3: Perform the subtraction: \[ \frac{90}{10} – \frac{45}{10} – \frac{22}{10} = \frac{(90 – 45 – 22)}{10} = \frac{23}{10} \]Step 4: Convert to mixed number: \[ \frac{23}{10} = 2\frac{3}{10} \]Answer: \(2\frac{3}{10}\)

Q12: Simplify: \(10\frac{3}{4} – 5\frac{1}{8} – 4\frac{5}{12}\)

Step 1: Convert all mixed numbers into improper fractions: \[ 10\frac{3}{4} = \frac{43}{4} \\ 5\frac{1}{8} = \frac{41}{8} \\ 4\frac{5}{12} = \frac{53}{12} \] Step 2: Find LCM of denominators 4, 8, and 12.
LCM (4, 8, 12) = 24
Convert all fractions to have denominator 24:\[ \frac{43}{4} = \frac{258}{24},\quad \frac{41}{8} = \frac{123}{24},\quad \frac{53}{12} = \frac{106}{24} \]Step 3: Perform the subtraction:\[ \frac{258}{24} – \frac{123}{24} – \frac{106}{24} = \frac{(258 – 123 – 106)}{24} = \frac{29}{24} \]Step 4: Convert to mixed number:\[ \frac{29}{24} = 1\frac{5}{24} \]Answer: \(1\frac{5}{24}\)

Q13: Simplify: \(6\frac{4}{5} – 3\frac{4}{15} + 4\frac{3}{10}\)

Step 1: Convert all mixed numbers into improper fractions: \[6\frac{4}{5} = \frac{34}{5} \\ 3\frac{4}{15} = \frac{49}{15} \\ 4\frac{3}{10} = \frac{43}{10} \]Step 2: Find LCM of denominators 5, 15, and 10.
LCM (5, 15, 10) = 30
Convert all fractions to have denominator 30:\[ \frac{34}{5} = \frac{204}{30},\quad \frac{49}{15} = \frac{98}{30},\quad \frac{43}{10} = \frac{129}{30} \]Step 3: Perform the operations:\[ \frac{204}{30} – \frac{98}{30} + \frac{129}{30} = \frac{(204 – 98 + 129)}{30} = \frac{235}{30} \]Step 4: Simplify the result:\[ \frac{235}{30} = 7\frac{25}{30} = 7\frac{5}{6} \]Answer: \(7\frac{5}{6}\)

Q14: Simplify: \(4\frac{5}{12}+3\frac{11}{18}-2\frac{7}{24}\)

Step 1: Convert mixed numbers into improper fractions: \[ 4\frac{5}{12} = \frac{53}{12} \\ 3\frac{11}{18} = \frac{65}{18} \\ 2\frac{7}{24} = \frac{55}{24} \]Step 2: Find LCM of 12, 18, and 24.
LCM (12, 18, 24) = 72
Convert each fraction to have denominator 72: \[ \frac{53}{12} = \frac{318}{72} \\ \frac{65}{18} = \frac{260}{72} \\ \frac{55}{24} = \frac{165}{72} \]Step 3: Add and subtract the fractions:\[ \frac{318}{72} + \frac{260}{72} – \frac{165}{72} = \frac{(318 + 260 – 165)}{72} = \frac{413}{72} \]Step 4: Convert to mixed fraction:\[ \frac{413}{72} = 5\frac{53}{72} \]Answer: \(5\frac{53}{72}\)

Q15: Subtract the sum of \(9\frac{3}{4}\) and \(3\frac{5}{6}\) from \(15\frac{7}{12}\).

Step 1: Convert all mixed numbers into improper fractions: \[ 9\frac{3}{4} = \frac{39}{4} \\ 3\frac{5}{6} = \frac{23}{6} \\ 15\frac{7}{12} = \frac{187}{12} \]Step 2: Find the sum of the first two fractions:
We find LCM of 4 and 6:
LCM (4, 6) = 12
Convert both fractions to have denominator 12: \[ \frac{39}{4} = \frac{117}{12} \\ \frac{23}{6} = \frac{46}{12} \\ \frac{117}{12} + \frac{46}{12} = \frac{163}{12} \]Step 3: Subtract this sum from \(\frac{187}{12}\):\[ \frac{187}{12} – \frac{163}{12} = \frac{24}{12} = 2 \]Answer: 2

Q16: Subtract the sum of \(2\frac{5}{12}\) and \(3\frac{3}{4}\) from the sum of \(7\frac{1}{3}\) and \(5\frac{1}{6}\).

Step 1: Convert all mixed fractions to improper fractions. \[ 2\frac{5}{12} = \frac{29}{12} \\ 3\frac{3}{4} = \frac{15}{4} \\ 7\frac{1}{3} = \frac{22}{3} \\ 5\frac{1}{6} = \frac{31}{6} \] Step 2: Find the sum of \(2\frac{5}{12}\) and \(3\frac{3}{4}\)
We need LCM of 12 and 4 = 12
\[ \frac{15}{4} = \frac{45}{12} \\ \frac{29}{12} + \frac{45}{12} = \frac{74}{12} \]Step 3: Find the sum of \(7\frac{1}{3}\) and \(5\frac{1}{6}\)
LCM of 3 and 6 = 6 \[ \frac{22}{3} = \frac{44}{6} \\ \frac{44}{6} + \frac{31}{6} = \frac{75}{6} \]Step 4: Subtract the two results:
First, convert both to a common denominator. LCM of 6 and 12 is 12.\[ \frac{75}{6} = \frac{150}{12} \\ \frac{74}{12} \text{ remains the same} \\ \frac{150}{12} – \frac{74}{12} = \frac{76}{12} \]Reduce the fraction:\[ \frac{76}{12} = \frac{19}{3} = 6\frac{1}{3} \]Answer: \(6\frac{1}{3}\)

Q17: What should be added to \(9\frac{4}{7}\) to get 16?

Step 1: Convert the mixed fraction \(9\frac{4}{7}\) to an improper fraction:\[ 9\frac{4}{7} = \frac{(9 \times 7) + 4}{7} = \frac{63 + 4}{7} = \frac{67}{7} \]Step 2: Let the number to be added be \(x\). According to the problem:\[ 9\frac{4}{7} + x = 16 \]Convert 16 to a fraction with denominator 7 for easier calculation:\[ 16 = \frac{16 \times 7}{7} = \frac{112}{7} \]Step 3: Substitute values:\[ \frac{67}{7} + x = \frac{112}{7} \]Step 4: Solve for \(x\):\[ x = \frac{112}{7} – \frac{67}{7} = \frac{112 – 67}{7} = \frac{45}{7} \]Convert \(\frac{45}{7}\) back to mixed fraction:\[ \frac{45}{7} = 6\frac{3}{7} \]Answer: \(6\frac{3}{7}\)

Q18: What must be subtracted from \(9\frac{1}{6}\) to get \(6\frac{1}{9}\)?

Step 1: Convert the mixed fractions into improper fractions:\[ 9\frac{1}{6} = \frac{(9 \times 6) + 1}{6} = \frac{54 + 1}{6} = \frac{55}{6} \\ 6\frac{1}{9} = \frac{(6 \times 9) + 1}{9} = \frac{54 + 1}{9} = \frac{55}{9} \]Step 2: Let the number to be subtracted be \(x\). According to the problem:\[ 9\frac{1}{6} – x = 6\frac{1}{9} \]Substitute the improper fractions:\[ \frac{55}{6} – x = \frac{55}{9} \]Step 3: Solve for \(x\):\[ x = \frac{55}{6} – \frac{55}{9} \]To subtract, find the common denominator \(= 18\):\[ \frac{55}{6} = \frac{55 \times 3}{18} = \frac{165}{18} \\ \frac{55}{9} = \frac{55 \times 2}{18} = \frac{110}{18} \\ x = \frac{165}{18} – \frac{110}{18} = \frac{55}{18} \]Step 4: Convert \(\frac{55}{18}\) to a mixed fraction:\[ \frac{55}{18} = 3\frac{1}{18} \]Answer: \(3\frac{1}{18}\)

Q19: Of \(\frac{17}{20}\) and \(\frac{21}{25}\), which is greater and by how much?

Step 1: Find a common denominator to compare the two fractions.The denominators are 20 and 25. The Least Common Denominator (LCD) is:
LCD = LCM (20, 25) = 100
Step 2: Convert both fractions to equivalent fractions with denominator 100:\[ \frac{17}{20} = \frac{17 \times 5}{20 \times 5} = \frac{85}{100} \\ \frac{21}{25} = \frac{21 \times 4}{25 \times 4} = \frac{84}{100} \]Step 3: Compare the numerators:\[ 85 > 84 \]So, \(\frac{17}{20}\) is greater.
Step 4: Find how much greater:\[ \frac{17}{20} – \frac{21}{25} = \frac{85}{100} – \frac{84}{100} = \frac{1}{100} \]Answer: \(\frac{17}{20}\) is greater than \(\frac{21}{25}\) by \(\frac{1}{100}\).

Q20: The sum of two fractions is \(14\frac{5}{12}\). If one of them is \(7\frac{2}{3}\), find the other.

Step 1: Convert the mixed fractions to improper fractions.\[ 14\frac{5}{12} = \frac{14 \times 12 + 5}{12} = \frac{168 + 5}{12} = \frac{173}{12} \\ 7\frac{2}{3} = \frac{7 \times 3 + 2}{3} = \frac{21 + 2}{3} = \frac{23}{3} \]Step 2: Let the other fraction be \(x\). According to the problem:\[ x + \frac{23}{3} = \frac{173}{12} \]Step 3: Subtract \(\frac{23}{3}\) from both sides to find \(x\):\[ x = \frac{173}{12} – \frac{23}{3} \]Step 4: Find a common denominator for subtraction:\[ \text{LCD of } 12 \text{ and } 3 = 12 \]Convert \(\frac{23}{3}\) to denominator 12:\[ \frac{23}{3} = \frac{23 \times 4}{3 \times 4} = \frac{92}{12} \]Step 5: Subtract the fractions:\[ x = \frac{173}{12} – \frac{92}{12} = \frac{173 – 92}{12} = \frac{81}{12} \]Step 6: Simplify the fraction \(\frac{81}{12}\):
Divide numerator and denominator by 3:\[ \frac{81}{12} = \frac{27}{4} \]Convert improper fraction to mixed fraction:\[ \frac{27}{4} = 6\frac{3}{4} \]Answer: The other fraction is \(6\frac{3}{4}\).

Q21: From a piece of wire \(12\frac{3}{4}\) m long, a small piece of length \(3\frac{5}{6}\) has been cut off. What is the length of the remaining piece?

Step 1: Convert the mixed fractions to improper fractions.\[ 12\frac{3}{4} = \frac{12 \times 4 + 3}{4} = \frac{48 + 3}{4} = \frac{51}{4} \\ 3\frac{5}{6} = \frac{3 \times 6 + 5}{6} = \frac{18 + 5}{6} = \frac{23}{6} \]Step 2: Subtract the cut piece from the total length:\[ \text{Remaining length} = \frac{51}{4} – \frac{23}{6} \]Step 3: Find the least common denominator (LCD) of 4 and 6, which is 12.
Convert the fractions:\[ \frac{51}{4} = \frac{51 \times 3}{4 \times 3} = \frac{153}{12} \\ \frac{23}{6} = \frac{23 \times 2}{6 \times 2} = \frac{46}{12} \]Step 4: Subtract the fractions:\[ \frac{153}{12} – \frac{46}{12} = \frac{153 – 46}{12} = \frac{107}{12} \]Step 5: Convert \(\frac{107}{12}\) to a mixed fraction:\[ \frac{107}{12} = 8 \frac{11}{12} \]Answer: The length of the remaining piece of wire is \(8\frac{11}{12}\) meters.

Q22: Three boxes weigh \(9\frac{1}{2}\) kg, \(14\frac{1}{5}\) kg and \(18\frac{3}{4}\) kg respectively. A porter carriers all the three boxes. What is the total weight carried by the porter?

Step 1: Convert the mixed fractions to improper fractions.\[ 9\frac{1}{2} = \frac{9 \times 2 + 1}{2} = \frac{18 + 1}{2} = \frac{19}{2} \\ 14\frac{1}{5} = \frac{14 \times 5 + 1}{5} = \frac{70 + 1}{5} = \frac{71}{5} \\ 18\frac{3}{4} = \frac{18 \times 4 + 3}{4} = \frac{72 + 3}{4} = \frac{75}{4} \]Step 2: Find the least common denominator (LCD) of 2, 5, and 4.
– Prime factors:
– 2 = 2
– 5 = 5
– 4 = \(2^2\)
LCD = \(2^2 \times 5 = 4 \times 5 = 20\)
Step 3: Convert all fractions to denominator 20.\[ \frac{19}{2} = \frac{19 \times 10}{2 \times 10} = \frac{190}{20} \\ \frac{71}{5} = \frac{71 \times 4}{5 \times 4} = \frac{284}{20} \\ \frac{75}{4} = \frac{75 \times 5}{4 \times 5} = \frac{375}{20} \\ \]Step 4: Add the fractions.\[ \frac{190}{20} + \frac{284}{20} + \frac{375}{20} = \frac{190 + 284 + 375}{20} = \frac{849}{20} \]Step 5: Convert \(\frac{849}{20}\) to a mixed fraction.\[ \frac{849}{20} = 42 \frac{9}{20} \]Answer: The total weight carried by the porter is \(42\frac{9}{20}\) kilograms.

Q23: On one day, a labourer earned ₹125. Out of this money, he spent ₹\(68\frac{1}{2}\) on food, ₹\(20\frac{3}{4}\) on tea and ₹\(16\frac{2}{5}\) on other eatables. How much does he save on that day?

Step 1: Convert the mixed fractions to improper fractions.\[ 68\frac{1}{2} = \frac{68 \times 2 + 1}{2} = \frac{136 + 1}{2} = \frac{137}{2} \\ 20\frac{3}{4} = \frac{20 \times 4 + 3}{4} = \frac{80 + 3}{4} = \frac{83}{4} \\ 16\frac{2}{5} = \frac{16 \times 5 + 2}{5} = \frac{80 + 2}{5} = \frac{82}{5} \]Step 2: Find the least common denominator (LCD) of 2, 4, and 5.
– Prime factors:
– 2 = 2
– 4 = \(2^2\)
– 5 = 5
LCD = \(2^2 \times 5 = 4 \times 5 = 20\)
Step 3: Convert all fractions to denominator 20.\[ \frac{137}{2} = \frac{137 \times 10}{2 \times 10} = \frac{1370}{20} \\ \frac{83}{4} = \frac{83 \times 5}{4 \times 5} = \frac{415}{20} \\ \frac{82}{5} = \frac{82 \times 4}{5 \times 4} = \frac{328}{20} \]Step 4: Add the expenditures.\[ \frac{1370}{20} + \frac{415}{20} + \frac{328}{20} = \frac{1370 + 415 + 328}{20} = \frac{2113}{20} \]Step 5: Convert ₹125 into fraction with denominator 20.\[ 125 = \frac{125 \times 20}{20} = \frac{2500}{20} \]Step 6: Calculate savings.\[ \text{Savings} = \frac{2500}{20} – \frac{2113}{20} = \frac{2500 – 2113}{20} = \frac{387}{20} \]Step 7: Convert \(\frac{387}{20}\) into mixed fraction.\[ \frac{387}{20} = 19 \frac{7}{20} \]Answer: The labourer saves ₹\(19\frac{7}{20}\) on that day.