Fractions

Q1: Point out the proper and improper fractions from the following:

i. \(\frac{7}{9}\)

Since \(7 < 9\), it is a proper fraction.

ii. \(\frac{16}{11}\)

Since \(16 > 11\), it is an improper fraction.

iii. \(\frac{18}{25}\)

Since \(18 < 25\), it is a proper fraction.

iv. \(\frac{10}{10}\)

Since numerator = denominator, it is an improper fraction.

v. \(\frac{37}{23}\)

Since \(37 > 23\), it is an improper fraction.

vi. \(\frac{21}{8}\)

Since \(21 > 8\), it is an improper fraction.

vii. \(\frac{56}{57}\)

Since \(56 < 57\), it is a proper fraction.

viii. \(\frac{137}{105}\)

Since \(137 > 105\), it is an improper fraction.

ix. \(\frac{2}{1}\)

Since \(2 > 1\), it is an improper fraction.

x. \(\frac{100}{101}\)

Since \(100 < 101\), it is a proper fraction.

Answer:
Proper fractions: \(\frac{7}{9}, \frac{18}{25}, \frac{56}{57}, \frac{100}{101}\)Improper fractions: \(\frac{16}{11}, \frac{10}{10}, \frac{37}{23}, \frac{21}{8}, \frac{137}{105}, \frac{2}{1}\)

Q2: Convert each of the following mixed fractions into an improper fraction:

Step 1: Recall the formula to convert mixed fraction to improper fraction: \[ \text{Improper fraction} = \frac{(\text{Whole number} \times \text{Denominator}) + \text{Numerator}}{\text{Denominator}} \]

i. \(8\frac{3}{4}\)

\[ = \frac{(8 \times 4) + 3}{4} = \frac{32 + 3}{4} = \frac{35}{4} \] Answer: \(\frac{35}{4}\)

ii. \(5\frac{11}{13}\)

\[ = \frac{(5 \times 13) + 11}{13} = \frac{65 + 11}{13} = \frac{76}{13} \] Answer: \(\frac{76}{13}\)

iii. \(10\frac{7}{9}\)

\[ = \frac{(10 \times 9) + 7}{9} = \frac{90 + 7}{9} = \frac{97}{9} \] Answer: \(\frac{97}{9}\)

iv. \(33\frac{1}{3}\)

\[ = \frac{(33 \times 3) + 1}{3} = \frac{99 + 1}{3} = \frac{100}{3} \] Answer: \(\frac{100}{3}\)

v. \(9\frac{7}{16}\)

\[ = \frac{(9 \times 16) + 7}{16} = \frac{144 + 7}{16} = \frac{151}{16} \] Answer: \(\frac{151}{16}\)

Q3: Convert each of the following improper fractions into a mixed fraction:

i. \(\frac{31}{5}\)

Divide 31 by 5: \[ 31 \div 5 = 6 \text{ quotient}, \quad 31 – (6 \times 5) = 1 \text{ remainder} \] So, \[ \frac{31}{5} = 6\frac{1}{5} \] Answer: \(6\frac{1}{5}\)

ii. \(\frac{80}{7}\)

Divide 80 by 7: \[ 80 \div 7 = 11 \text{ quotient}, \quad 80 – (11 \times 7) = 3 \text{ remainder} \] So, \[ \frac{80}{7} = 11\frac{3}{7} \] Answer: \(11\frac{3}{7}\)

iii. \(\frac{107}{3}\)

Divide 107 by 3: \[ 107 \div 3 = 35 \text{ quotient}, \quad 107 – (35 \times 3) = 2 \text{ remainder} \] So, \[ \frac{107}{3} = 35\frac{2}{3} \] Answer: \(35\frac{2}{3}\)

iv. \(\frac{115}{13}\)

Divide 115 by 13: \[ 115 \div 13 = 8 \text{ quotient}, \quad 115 – (8 \times 13) = 11 \text{ remainder} \] So, \[ \frac{115}{13} = 8\frac{11}{13} \] Answer: \(8\frac{11}{13}\)

v. \(\frac{200}{9}\)

Divide 200 by 9: \[ 200 \div 9 = 22 \text{ quotient}, \quad 200 – (22 \times 9) = 2 \text{ remainder} \] So, \[ \frac{200}{9} = 22\frac{2}{9} \] Answer: \(22\frac{2}{9}\)

Q4: Convert each of the following sets of unlike fractions into that fractions:

i. \(\frac{4}{5}, \frac{7}{10}, \frac{11}{15}, \frac{13}{20}\)

LCM of 5, 10, 15, 20 = 60 \[ \frac{4}{5} = \frac{4 \times 12}{5 \times 12} = \frac{48}{60}\\ \frac{7}{10} = \frac{7 \times 6}{10 \times 6} = \frac{42}{60}\\ \frac{11}{15} = \frac{11 \times 4}{15 \times 4} = \frac{44}{60}\\ \frac{13}{20} = \frac{13 \times 3}{20 \times 3} = \frac{39}{60} \] Answer: \(\frac{48}{60}, \frac{42}{60}, \frac{44}{60}, \frac{39}{60}\)

ii. \(\frac{2}{3}, \frac{1}{4}, \frac{5}{6}, \frac{7}{8}, \frac{11}{12}\)

LCM of 3, 4, 6, 8, 12 = 24 \[ \frac{2}{3} = \frac{16}{24}\\ \frac{1}{4} = \frac{6}{24}\\ \frac{5}{6} = \frac{20}{24}\\ \frac{7}{8} = \frac{21}{24}\\ \frac{11}{12} = \frac{22}{24} \] Answer: \(\frac{16}{24}, \frac{6}{24}, \frac{20}{24}, \frac{21}{24}, \frac{22}{24}\)

iii. \(\frac{1}{3}, \frac{3}{4}, \frac{5}{12}, \frac{9}{16}, \frac{17}{24}\)

LCM of 3, 4, 12, 16, 24 = 48 \[ \frac{1}{3} = \frac{16}{48}\\ \frac{3}{4} = \frac{36}{48}\\ \frac{5}{12} = \frac{20}{48}\\ \frac{9}{16} = \frac{27}{48}\\ \frac{17}{24} = \frac{34}{48} \] Answer: \(\frac{16}{48}, \frac{36}{48}, \frac{20}{48}, \frac{27}{48}, \frac{34}{48}\)

iv. \(\frac{2}{3}, \frac{1}{6}, \frac{5}{9}, \frac{7}{12}, \frac{13}{18}\)

LCM of 3, 6, 9, 12, 18 = 36 \[ \frac{2}{3} = \frac{24}{36}\\ \frac{1}{6} = \frac{6}{36}\\ \frac{5}{9} = \frac{20}{36}\\ \frac{7}{12} = \frac{21}{36}\\ \frac{13}{18} = \frac{26}{36} \] Answer: \(\frac{24}{36}, \frac{6}{36}, \frac{20}{36}, \frac{21}{36}, \frac{26}{36}\)

v. \(\frac{1}{2}, \frac{4}{7}, \frac{9}{14}, \frac{11}{21}, \frac{37}{42}\)

LCM of 2, 7, 14, 21, 42 = 42 \[ \frac{1}{2} = \frac{21}{42}\\ \frac{4}{7} = \frac{24}{42}\\ \frac{9}{14} = \frac{27}{42}\\ \frac{11}{21} = \frac{22}{42}\\ \frac{37}{42} = \frac{37}{42} \] Answer: \(\frac{21}{42}, \frac{24}{42}, \frac{27}{42}, \frac{22}{42}, \frac{37}{42}\)

vi. \(\frac{2}{7}, \frac{5}{8}, \frac{11}{14}, \frac{9}{16}, \frac{3}{4}\)

LCM of 7, 8, 14, 16, 4 = 112 \[ \frac{2}{7} = \frac{32}{112},\\ \frac{5}{8} = \frac{70}{112},\\ \frac{11}{14} = \frac{88}{112}\\ \frac{9}{16} = \frac{63}{112}\\ \frac{3}{4} = \frac{84}{112} \] Answer: \(\frac{32}{112}, \frac{70}{112}, \frac{88}{112}, \frac{63}{112}, \frac{84}{112}\)

Q5: Fill in the placeholders with > or < :

i. \(\frac{7}{9}\) ___ \(\frac{5}{9}\)

Step 1: Same denominator. Compare numerators.
7 > 5
Answer: \(\frac{7}{9} \gt \frac{5}{9}\)

ii. \(\frac{9}{13}\) ___ \(\frac{11}{13}\)

Step 1: Same denominator. Compare numerators.
9 < 11
Answer: \(\frac{9}{13} \lt \frac{11}{13}\)

iii. \(\frac{3}{5}\) ___ \(\frac{3}{4}\)

Step 1: Convert to same denominator (LCM = 20):
\(\frac{3}{5} = \frac{12}{20}, \quad \frac{3}{4} = \frac{15}{20}\)
12 < 15
Answer: \(\frac{3}{5} \lt \frac{3}{4}\)

iv. \(\frac{7}{9}\) ___ \(\frac{7}{11}\)

Step 1: Convert to same denominator (LCM = 99):
\(\frac{7}{9} = \frac{77}{99}, \quad \frac{7}{11} = \frac{63}{99}\)
77 > 63
Answer: \(\frac{7}{9} \gt \frac{7}{11}\)

v. \(\frac{5}{8}\) ___ \(\frac{5}{6}\)

Step 1: LCM = 24:
\(\frac{5}{8} = \frac{15}{24}, \quad \frac{5}{6} = \frac{20}{24}\)
15 < 20
Answer: \(\frac{5}{8} \lt \frac{5}{6}\)

vi. \(\frac{7}{9}\) ___ \(\frac{6}{11}\)

Step 1: LCM = 99:
\(\frac{7}{9} = \frac{77}{99}, \quad \frac{6}{11} = \frac{54}{99}\)
77 > 54
Answer: \(\frac{7}{9} \gt \frac{6}{11}\)

vii. \(\frac{6}{5}\) ___ \(\frac{5}{4}\)

Step 1: Convert to same denominator (LCM = 20):
\(\frac{6}{5} = \frac{24}{20}, \quad \frac{5}{4} = \frac{25}{20}\)
24 < 25
Answer: \(\frac{6}{5} \lt \frac{5}{4}\)

viii. \(\frac{7}{11}\) ___ \(\frac{8}{13}\)

Step 1: Cross-multiply: \(7 \times 13 = 91, \ 8 \times 11 = 88\)
91 > 88
Answer: \(\frac{7}{11} \gt \frac{8}{13}\)

ix. \(\frac{10}{13}\) ___ \(\frac{13}{16}\)

Step 1: Cross-multiply: \(10 \times 16 = 160, \ 13 \times 13 = 169\)
160 < 169
Answer: \(\frac{10}{13} \lt \frac{13}{16}\)

x. \(\frac{2}{9}\) ___ \(\frac{3}{14}\)

Step 1: Cross-multiply: \(2 \times 14 = 28, \ 3 \times 9 = 27\)
28 > 27
Answer: \(\frac{2}{9} \gt \frac{3}{14}\)

xi. \(\frac{7}{12}\) ___ \(\frac{5}{9}\)

Step 1: Cross-multiply: \(7 \times 9 = 63, \ 5 \times 12 = 60\)
63 > 60
Answer: \(\frac{7}{12} \gt \frac{5}{9}\)

\(\frac{15}{19}\) ___ \(\frac{3}{4}\)

Step 1: Cross-multiply: \(15 \times 4 = 60, \ 3 \times 19 = 57\)
60 > 57
Answer: \(\frac{15}{19} \gt \frac{3}{4}\)

Q6: Arrange the following fractions in ascending order:

i. \(\frac{3}{11},\frac{9}{11},\frac{4}{11},\frac{5}{11},\frac{1}{11}\)

Step: All fractions have the same denominator (11). When denominator is same, compare **numerators**:
⇒ \(1 < 3 < 4 < 5 < 9\) So, order of fractions:
Answer: \(\frac{1}{11},\frac{3}{11},\frac{4}{11},\frac{5}{11},\frac{9}{11}\)

ii. \(\frac{2}{13},\frac{2}{9},\frac{2}{15},\frac{2}{7},\frac{2}{5}\)

Step: All fractions have same numerator (2). When numerator is same, **larger the denominator, smaller the fraction**:
⇒ \(15 > 13 > 9 > 7 > 5\)
So the order is:
Answer: \(\frac{2}{15},\frac{2}{13},\frac{2}{9},\frac{2}{7},\frac{2}{5}\)

iii. \(\frac{2}{3},\frac{5}{6},\frac{7}{9},\frac{11}{12},\frac{13}{18}\)

Step: Denominators are different, numerators are different — make denominators same (LCM of 3, 6, 9, 12, 18 = 36)
Convert to like denominators:

• \(\frac{2}{3} = \frac{24}{36}\)
• \(\frac{5}{6} = \frac{30}{36}\)
• \(\frac{7}{9} = \frac{28}{36}\)
• \(\frac{11}{12} = \frac{33}{36}\)
• \(\frac{13}{18} = \frac{26}{36}\)

⇒ \(24 < 26 < 28 < 30 < 33\)
Answer: \(\frac{2}{3},\frac{13}{18},\frac{7}{9},\frac{5}{6},\frac{11}{12}\)

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

Step: Different numerators and denominators — make denominators same (LCM of 3, 4, 6, 8, 12 = 24)
Convert to like denominators:

• \(\frac{2}{3} = \frac{16}{24}\)
• \(\frac{1}{4} = \frac{6}{24}\)
• \(\frac{5}{6} = \frac{20}{24}\)
• \(\frac{3}{8} = \frac{9}{24}\)
• \(\frac{7}{12} = \frac{14}{24}\)

⇒ \(6 < 9 < 14 < 16 < 20\)
Answer: \(\frac{1}{4},\frac{3}{8},\frac{7}{12},\frac{2}{3},\frac{5}{6}\)

Q7: Arrange the following fractions in descending order:

i. \(\frac{5}{12},\ \frac{1}{12},\ \frac{7}{12},\ \frac{11}{12},\ \frac{9}{12}\)

Step 1: All fractions have the same denominator (12).
Step 2: When denominators are same, compare the numerators.
Numerators are: 5, 1, 7, 11, 9
Step 3: Arrange numerators in descending order: 11, 9, 7, 5, 1
Answer: \(\frac{11}{12} > \frac{9}{12} > \frac{7}{12} > \frac{5}{12} > \frac{1}{12}\)

ii. \(\frac{4}{7},\ \frac{4}{3},\ \frac{4}{9},\ \frac{4}{5},\ \frac{4}{11}\)

Step 1: All fractions have the same numerator (4).
Step 2: When numerators are same, the fraction with the smallest denominator is the greatest.
Denominators are: 7, 3, 9, 5, 11
Order of denominators in ascending: 3, 5, 7, 9, 11
Answer: \(\frac{4}{3} > \frac{4}{5} > \frac{4}{7} > \frac{4}{9} > \frac{4}{11}\)

iii. \(\frac{2}{3},\ \frac{5}{6},\ \frac{7}{9},\ \frac{3}{4},\ \frac{1}{2}\)

Step 1: Convert all fractions to have a common denominator.
LCM of 3, 6, 9, 4, 2 = 36 \[ \frac{2}{3} = \frac{24}{36},\quad \frac{5}{6} = \frac{30}{36},\quad \frac{7}{9} = \frac{28}{36},\quad \frac{3}{4} = \frac{27}{36},\quad \frac{1}{2} = \frac{18}{36} \] Step 2: Arrange: 30, 28, 27, 24, 18
Answer: \(\frac{5}{6} > \frac{7}{9} > \frac{3}{4} > \frac{2}{3} > \frac{1}{2}\)

iv. \(\frac{2}{3},\ \frac{3}{5},\ \frac{7}{10},\ \frac{8}{15},\ \frac{11}{20}\)

Step 1: Convert all fractions to have a common denominator.
LCM of 3, 5, 10, 15, 20 = 60 \[ \frac{2}{3} = \frac{40}{60},\quad \frac{3}{5} = \frac{36}{60},\quad \frac{7}{10} = \frac{42}{60},\quad \frac{8}{15} = \frac{32}{60},\quad \frac{11}{20} = \frac{33}{60} \] Step 2: Arrange: 42, 40, 36, 33, 32
Answer: \(\frac{7}{10} > \frac{2}{3} > \frac{3}{5} > \frac{11}{20} > \frac{8}{15}\)

v. \(\frac{17}{32},\ \frac{7}{12},\ \frac{19}{48},\ \frac{13}{24},\ \frac{9}{16}\)

Step 1: Convert all fractions to have a common denominator.
LCM of 32, 12, 48, 24, 16 = 96 \[ \frac{17}{32} = \frac{51}{96},\quad \frac{7}{12} = \frac{56}{96},\quad \frac{19}{48} = \frac{38}{96},\quad \frac{13}{24} = \frac{52}{96},\quad \frac{9}{16} = \frac{54}{96} \] Step 2: Arrange: 56, 54, 52, 51, 38
Answer: \(\frac{7}{12} > \frac{9}{16} > \frac{13}{24} > \frac{17}{32} > \frac{19}{48}\)

vi. \(\frac{5}{6},\ \frac{7}{9},\ \frac{17}{24},\ \frac{3}{4},\ \frac{23}{36}\)

Step 1: Convert all fractions to a common denominator.
LCM of 6, 9, 24, 4, 36 = 72 \[ \frac{5}{6} = \frac{60}{72},\quad \frac{7}{9} = \frac{56}{72},\quad \frac{17}{24} = \frac{51}{72},\quad \frac{3}{4} = \frac{54}{72},\quad \frac{23}{36} = \frac{46}{72} \] Step 2: Arrange: 60, 56, 54, 51, 46
Answer: \(\frac{5}{6} > \frac{7}{9} > \frac{3}{4} > \frac{17}{24} > \frac{23}{36}\)