Exercise: 3-B
Q1: For each pair, given below, state whether it forms like fractions or unlike fractions:
i. \(\frac{5}{8}\) and \(\frac{7}{8}\)
Step 1: Like fractions are fractions that have the same denominator. Both fractions \(\frac{5}{8}\) and \(\frac{7}{8}\) have the same denominator, which is 8.
Answer: Like Fractions.
ii. \(\frac{8}{15}\) and \(\frac{8}{21}\)
Step 1: Like fractions must have the same denominator. Here, the denominators are 15 and 21, which are different.
Answer: Unlike Fractions.
iii. \(\frac{4}{9}\) and \(\frac{9}{4}\)
Step 1: To form like fractions, the fractions must have the same denominator. The denominators are 9 and 4, which are different.
Answer: Unlike Fractions.
Q2: Convert given fractions into tractions with equal denominators:
i. \(\frac{5}{6}\) and \(\frac{7}{9}\)
Step 1: To convert these fractions into like fractions, we need to find the Least Common Denominator (LCD). The denominators are 6 and 9.
The LCM of 6 and 9 is 18.
Step 2: Convert each fraction to have the denominator 18:
For \(\frac{5}{6}\), multiply both the numerator and denominator by 3: \( \frac{5 \times 3}{6 \times 3} = \frac{15}{18} \).
For \(\frac{7}{9}\), multiply both the numerator and denominator by 2: \( \frac{7 \times 2}{9 \times 2} = \frac{14}{18} \).
Answer: \(\frac{15}{18}\) and \(\frac{14}{18}\).
ii. \(\frac{2}{3}, \frac{5}{6}\) and \(\frac{7}{12}\)
Step 1: Find the Least Common Denominator (LCD) of the fractions. The denominators are 3, 6, and 12.
The LCM of 3, 6, and 12 is 12.
Step 2: Convert each fraction to have the denominator 12:
For \(\frac{2}{3}\), multiply both the numerator and denominator by 4: \( \frac{2 \times 4}{3 \times 4} = \frac{8}{12} \).
For \(\frac{5}{6}\), multiply both the numerator and denominator by 2: \( \frac{5 \times 2}{6 \times 2} = \frac{10}{12} \).
For \(\frac{7}{12}\), the denominator is already 12, so it remains the same: \( \frac{7}{12} \).
Answer: \(\frac{8}{12}\), \(\frac{10}{12}\), and \(\frac{7}{12}\).
iii. \(\frac{4}{5}, \frac{17}{20}, \frac{23}{40}\) and \(\frac{11}{16}\)
Step 1: Find the Least Common Denominator (LCD) of the fractions. The denominators are 5, 20, 40, and 16.
The LCM of 5, 20, 40, and 16 is 80.
Step 2: Convert each fraction to have the denominator 160:
For \(\frac{4}{5}\), multiply both the numerator and denominator by 32: \( \frac{4 \times 16}{5 \times 16} = \frac{64}{80} \).
For \(\frac{17}{20}\), multiply both the numerator and denominator by 8: \( \frac{17 \times 4}{20 \times 4} = \frac{68}{80} \).
For \(\frac{23}{40}\), multiply both the numerator and denominator by 4: \( \frac{23 \times 2}{40 \times 2} = \frac{46}{80} \).
For \(\frac{11}{16}\), multiply both the numerator and denominator by 10: \( \frac{11 \times 5}{16 \times 5} = \frac{55}{80} \).
Answer: \(\frac{64}{80}\), \(\frac{68}{80}\), \(\frac{46}{80}\), and \(\frac{55}{80}\).
Q3: Convert given fractions into tractions with equal numerators:
i. \(\frac{8}{9}\) and \(\frac{12}{17}\)
Step 1: To convert these fractions into fractions with equal numerators, we need to find the Least Common Multiple (LCM) of the numerators 8 and 12.
The LCM of 8 and 12 is 24.
Step 2: Convert each fraction to have the numerator 24:
For \(\frac{8}{9}\), multiply both the numerator and denominator by 3: \( \frac{8 \times 3}{9 \times 3} = \frac{24}{27} \).
For \(\frac{12}{17}\), multiply both the numerator and denominator by 2: \( \frac{12 \times 2}{17 \times 2} = \frac{24}{34} \).
Answer: \(\frac{24}{27}\) and \(\frac{24}{34}\).
ii. \(\frac{6}{13}, \frac{15}{23}\) and \(\frac{12}{17}\)
Step 1: Find the Least Common Multiple (LCM) of the numerators 6, 15, and 12.
The LCM of 6, 15, and 12 is 60.
Step 2: Convert each fraction to have the numerator 60:
For \(\frac{6}{13}\), multiply both the numerator and denominator by 10: \( \frac{6 \times 10}{13 \times 10} = \frac{60}{130} \).
For \(\frac{15}{23}\), multiply both the numerator and denominator by 4: \( \frac{15 \times 4}{23 \times 4} = \frac{60}{92} \).
For \(\frac{12}{17}\), multiply both the numerator and denominator by 5: \( \frac{12 \times 5}{17 \times 5} = \frac{60}{85} \).
Answer: \(\frac{60}{130}\), \(\frac{60}{92}\), and \(\frac{60}{85}\).
iii. \(\frac{15}{19}, \frac{25}{28}, \frac{9}{11}\) and \(\frac{45}{47}\)
Step 1: Find the Least Common Multiple (LCM) of the numerators 15, 25, 9, and 45.
The LCM of 15, 25, 9, and 45 is 225.
Step 2: Convert each fraction to have the numerator 225:
For \(\frac{15}{19}\), multiply both the numerator and denominator by 15: \( \frac{15 \times 15}{19 \times 15} = \frac{225}{285} \).
For \(\frac{25}{28}\), multiply both the numerator and denominator by 9: \( \frac{25 \times 9}{28 \times 9} = \frac{225}{252} \).
For \(\frac{9}{11}\), multiply both the numerator and denominator by 25: \( \frac{9 \times 25}{11 \times 25} = \frac{225}{275} \).
For \(\frac{45}{47}\), multiply both the numerator and denominator by 5: \( \frac{45 \times 5}{47 \times 5} = \frac{225}{235} \).
Answer: \(\frac{225}{285}\), \(\frac{225}{252}\), \(\frac{225}{275}\), and \(\frac{225}{235}\).
Q4: Compare the given fractions by making the denominators equal:
i. \(\frac{2}{5}\) and \(\frac{4}{9}\)
Step 1: To compare these fractions, we need to find the Least Common Denominator (LCD). The denominators are 5 and 9.
The LCM of 5 and 9 is 45.
Step 2: Convert each fraction to have the denominator 45:
For \(\frac{2}{5}\), multiply both the numerator and denominator by 9: \( \frac{2 \times 9}{5 \times 9} = \frac{18}{45} \).
For \(\frac{4}{9}\), multiply both the numerator and denominator by 5: \( \frac{4 \times 5}{9 \times 5} = \frac{20}{45} \).
Step 3: Compare the numerators: 18 and 20.
Since \( \frac{18}{45} < \frac{20}{45} \), we can conclude that: \(\frac{2}{5} < \frac{4}{9}\).
Answer: \(\frac{2}{5} < \frac{4}{9}\).
ii. \(\frac{5}{7}\) and \(\frac{8}{11}\)
Step 1: Find the Least Common Denominator (LCD) of the fractions. The denominators are 7 and 11.
The LCM of 7 and 11 is 77.
Step 2: Convert each fraction to have the denominator 77:
For \(\frac{5}{7}\), multiply both the numerator and denominator by 11: \( \frac{5 \times 11}{7 \times 11} = \frac{55}{77} \).
For \(\frac{8}{11}\), multiply both the numerator and denominator by 7: \( \frac{8 \times 7}{11 \times 7} = \frac{56}{77} \).
Step 3: Compare the numerators: 55 and 56.
Since \( \frac{55}{77} < \frac{56}{77} \), we can conclude that: \(\frac{5}{7} < \frac{8}{11}\).
Answer: \(\frac{5}{7} < \frac{8}{11}\).
iii. \(\frac{7}{15}\) and \(\frac{9}{20}\)
Step 1: Find the Least Common Denominator (LCD) of the fractions. The denominators are 15 and 20.
The LCM of 15 and 20 is 60.
Step 2: Convert each fraction to have the denominator 60:
For \(\frac{7}{15}\), multiply both the numerator and denominator by 4: \( \frac{7 \times 4}{15 \times 4} = \frac{28}{60} \).
For \(\frac{9}{20}\), multiply both the numerator and denominator by 3: \( \frac{9 \times 3}{20 \times 3} = \frac{27}{60} \).
Step 3: Compare the numerators: 28 and 27.
Since \( \frac{28}{60} > \frac{27}{60} \), we can conclude that: \(\frac{7}{15} > \frac{9}{20}\).
Answer: \(\frac{7}{15} > \frac{9}{20}\).
Q5: Compare the given fractions by making the numerators equal:
i. \(\frac{4}{9}\) and \(\frac{2}{5}\)
Step 1: To compare these fractions, we need to make the numerators equal. The denominators are 9 and 5.
We need to find a common denominator for the fractions. The LCM of 9 and 5 is 45.
Step 2: Convert each fraction to have the numerator equal:
For \(\frac{4}{9}\), multiply both the numerator and denominator by 5: \( \frac{4 \times 5}{9 \times 5} = \frac{20}{45} \).
For \(\frac{2}{5}\), multiply both the numerator and denominator by 9: \( \frac{2 \times 9}{5 \times 9} = \frac{18}{45} \).
Step 3: Compare the numerators: 20 and 18.
Since \( \frac{20}{45} > \frac{18}{45} \), we can conclude that: \(\frac{4}{9} > \frac{2}{5}\).
Answer: \(\frac{4}{9} > \frac{2}{5}\).
ii. \(\frac{5}{12}\) and \(\frac{8}{19}\)
Step 1: To compare these fractions, we need to make the numerators equal. The denominators are 12 and 19.
We need to find a common denominator for the fractions. The LCM of 12 and 19 is 228.
Step 2: Convert each fraction to have the numerator equal:
For \(\frac{5}{12}\), multiply both the numerator and denominator by 19: \( \frac{5 \times 19}{12 \times 19} = \frac{95}{228} \).
For \(\frac{8}{19}\), multiply both the numerator and denominator by 12: \( \frac{8 \times 12}{19 \times 12} = \frac{96}{228} \).
Step 3: Compare the numerators: 95 and 96.
Since \( \frac{95}{228} < \frac{96}{228} \), we can conclude that: \(\frac{5}{12} < \frac{8}{19}\).
Answer: \(\frac{5}{12} < \frac{8}{19}\).
iii. \(\frac{5}{7}\) and \(\frac{9}{14}\)
Step 1: To compare these fractions, we need to make the numerators equal. The denominators are 7 and 14.
We need to find a common denominator for the fractions. The LCM of 7 and 14 is 14.
Step 2: Convert each fraction to have the numerator equal:
For \(\frac{5}{7}\), multiply both the numerator and denominator by 2: \( \frac{5 \times 2}{7 \times 2} = \frac{10}{14} \).
For \(\frac{9}{14}\), it already has the denominator 14, so the fraction remains \( \frac{9}{14} \).
Step 3: Compare the numerators: 10 and 9.
Since \( \frac{10}{14} > \frac{9}{14} \), we can conclude that: \(\frac{5}{7} > \frac{9}{14}\).
Answer: \(\frac{5}{7} > \frac{9}{14}\).
Q6: Compare the given fractions by cross multiplication method:
i. \(\frac{2}{5}\) and \(\frac{4}{9}\)
Step 1: To compare \(\frac{2}{5}\) and \(\frac{4}{9}\) using cross multiplication, we multiply the numerator of the first fraction by the denominator of the second fraction, and vice versa.
Cross multiplication: \(2 \times 9 = 18\) and \(5 \times 4 = 20\).
Step 2: Now, compare the results: \(18\) and \(20\).
Since \(18 < 20\), we conclude that: \(\frac{2}{5} < \frac{4}{9}\).
Answer: \(\frac{2}{5} < \frac{4}{9}\).
ii. \(\frac{3}{8}\) and \(\frac{6}{11}\)
Step 1: To compare \(\frac{3}{8}\) and \(\frac{6}{11}\) using cross multiplication, we multiply the numerator of the first fraction by the denominator of the second fraction, and vice versa.
Cross multiplication: \(3 \times 11 = 33\) and \(8 \times 6 = 48\).
Step 2: Now, compare the results: \(33\) and \(48\).
Since \(33 < 48\), we conclude that: \(\frac{3}{8} < \frac{6}{11}\).
Answer: \(\frac{3}{8} < \frac{6}{11}\).
iii. \(\frac{5}{18}\) and \(\frac{11}{21}\)
Step 1: To compare \(\frac{5}{18}\) and \(\frac{11}{21}\) using cross multiplication, we multiply the numerator of the first fraction by the denominator of the second fraction, and vice versa.
Cross multiplication: \(5 \times 21 = 105\) and \(18 \times 11 = 198\).
Step 2: Now, compare the results: \(105\) and \(198\).
Since \(105 < 198\), we conclude that: \(\frac{5}{18} < \frac{11}{21}\).
Answer: \(\frac{5}{18} < \frac{11}{21}\).
Q7: Arrange the given fractions in ascending order by making the denominators equal:
i. \(\frac{1}{3}, \frac{2}{5}, \frac{3}{4}\) and \(\frac{1}{6}\)
Step 1: We need to find the Least Common Denominator (LCD) for \(\frac{1}{3}\), \(\frac{2}{5}\), \(\frac{3}{4}\), and \(\frac{1}{6}\). The LCM of 3, 5, 4, and 6 is 60.
Step 2: Convert each fraction to have the denominator 60:
\(\frac{1}{3} = \frac{1 \times 20}{3 \times 20} = \frac{20}{60}\)
\(\frac{2}{5} = \frac{2 \times 12}{5 \times 12} = \frac{24}{60}\)
\(\frac{3}{4} = \frac{3 \times 15}{4 \times 15} = \frac{45}{60}\)
\(\frac{1}{6} = \frac{1 \times 10}{6 \times 10} = \frac{10}{60}\)
Step 3: Arrange the fractions in ascending order by comparing the numerators: \(\frac{10}{60}, \frac{20}{60}, \frac{24}{60}, \frac{45}{60}\)
Thus, the ascending order is: \(\frac{1}{6} < \frac{1}{3} < \frac{2}{5} < \frac{3}{4}\).
Answer: \(\frac{1}{6} < \frac{1}{3} < \frac{2}{5} < \frac{3}{4}\).
ii. \(\frac{5}{6}, \frac{7}{8}, \frac{11}{12}\) and \(\frac{3}{10}\)
Step 1: We need to find the Least Common Denominator (LCD) for \(\frac{5}{6}\), \(\frac{7}{8}\), \(\frac{11}{12}\), and \(\frac{3}{10}\). The LCM of 6, 8, 12, and 10 is 120.
Step 2: Convert each fraction to have the denominator 120:
\(\frac{5}{6} = \frac{5 \times 20}{6 \times 20} = \frac{100}{120}\)
\(\frac{7}{8} = \frac{7 \times 15}{8 \times 15} = \frac{105}{120}\)
\(\frac{11}{12} = \frac{11 \times 10}{12 \times 10} = \frac{110}{120}\)
\(\frac{3}{10} = \frac{3 \times 12}{10 \times 12} = \frac{36}{120}\)
Step 3: Arrange the fractions in ascending order by comparing the numerators: \(\frac{36}{120}, \frac{100}{120}, \frac{105}{120}, \frac{110}{120}\)
Thus, the ascending order is: \(\frac{3}{10} < \frac{5}{6} < \frac{7}{8} < \frac{11}{12}\).
Answer: \(\frac{3}{10} < \frac{5}{6} < \frac{7}{8} < \frac{11}{12}\).
iii. \(\frac{5}{7}, \frac{3}{8}, \frac{9}{14}\) and \(\frac{20}{21}\)
Step 1: We need to find the Least Common Denominator (LCD) for \(\frac{5}{7}\), \(\frac{3}{8}\), \(\frac{9}{14}\), and \(\frac{20}{21}\). The LCM of 7, 8, 14, and 21 is 168.
Step 2: Convert each fraction to have the denominator 168:
\(\frac{5}{7} = \frac{5 \times 24}{7 \times 24} = \frac{120}{168}\)
\(\frac{3}{8} = \frac{3 \times 21}{8 \times 21} = \frac{63}{168}\)
\(\frac{9}{14} = \frac{9 \times 12}{14 \times 12} = \frac{108}{168}\)
\(\frac{20}{21} = \frac{20 \times 8}{21 \times 8} = \frac{160}{168}\)
Step 3: Arrange the fractions in ascending order by comparing the numerators: \(\frac{63}{168}, \frac{108}{168}, \frac{120}{168}, \frac{160}{168}\)
Thus, the ascending order is: \(\frac{3}{8} < \frac{9}{14} < \frac{5}{7} < \frac{20}{21}\).
Answer: \(\frac{3}{8} < \frac{9}{14} < \frac{5}{7} < \frac{20}{21}\).
Q8: Arrange the given fractions in descending order by making the numerators equal:
i. \(\frac{5}{6}, \frac{4}{15}, \frac{8}{9}\) and \(\frac{1}{3}\)
Step 1: We need to find the Least Common Numerator (LCN) for \(\frac{5}{6}\), \(\frac{4}{15}\), \(\frac{8}{9}\), and \(\frac{1}{3}\). The LCM of the numerators 5, 4, 8, and 1 is 40.
Step 2: Convert each fraction to have the numerator 40:
\(\frac{5}{6} = \frac{5 \times 8}{6 \times 8} = \frac{40}{48}\)
\(\frac{4}{15} = \frac{4 \times 10}{15 \times 10} = \frac{40}{150}\)
\(\frac{8}{9} = \frac{8 \times 5}{9 \times 5} = \frac{40}{45}\)
\(\frac{1}{3} = \frac{1 \times 40}{3 \times 40} = \frac{40}{120}\)
Step 3: Arrange the fractions in descending order by comparing the numerators: \(\frac{40}{45}, \frac{40}{48}, \frac{40}{120}, \frac{40}{150}\)
Thus, the descending order is: \(\frac{8}{9} > \frac{5}{6} > \frac{1}{3} > \frac{4}{15}\).
Answer: \(\frac{8}{9} > \frac{5}{6} > \frac{1}{3} > \frac{4}{15}\).
ii. \(\frac{3}{7}, \frac{4}{9}, \frac{5}{7}\) and \(\frac{8}{11}\)
Step 1: We need to find the Least Common Numerator (LCN) for \(\frac{3}{7}\), \(\frac{4}{9}\), \(\frac{5}{7}\), and \(\frac{8}{11}\). The LCM of the numerators 3, 4, 5, and 8 is 120.
Step 2: Convert each fraction to have the numerator 120:
\(\frac{3}{7} = \frac{3 \times 40}{7 \times 40} = \frac{120}{280}\)
\(\frac{4}{9} = \frac{4 \times 30}{9 \times 30} = \frac{120}{270}\)
\(\frac{5}{7} = \frac{5 \times 24}{7 \times 24} = \frac{120}{168}\)
\(\frac{8}{11} = \frac{8 \times 15}{11 \times 15} = \frac{120}{165}\)
Step 3: Arrange the fractions in descending order by comparing the numerators: \(\frac{120}{165}, \frac{120}{168}, \frac{120}{270}, \frac{120}{280}\)
Thus, the descending order is: \(\frac{8}{11} > \frac{5}{7} > \frac{4}{9} > \frac{3}{7}\).
Answer: \(\frac{8}{11} > \frac{5}{7} > \frac{4}{9} > \frac{3}{7}\).
iii. \(\frac{1}{10}, \frac{6}{11}, \frac{8}{11}\) and \(\frac{3}{5}\)
Step 1: We need to find the Least Common Numerator (LCN) for \(\frac{1}{10}\), \(\frac{6}{11}\), \(\frac{8}{11}\), and \(\frac{3}{5}\). The LCM of the numerators 1, 6, 8, and 3 is 24.
Step 2: Convert each fraction to have the numerator 120:
\(\frac{1}{10} = \frac{1 \times 24}{10 \times 24} = \frac{24}{240}\)
\(\frac{6}{11} = \frac{6 \times 4}{11 \times 4} = \frac{24}{44}\)
\(\frac{8}{11} = \frac{8 \times 3}{11 \times 3} = \frac{24}{33}\)
\(\frac{3}{5} = \frac{3 \times 8}{5 \times 8} = \frac{24}{40}\)
Step 3: Arrange the fractions in descending order by comparing the numerators: \(\frac{24}{33}, \frac{24}{40}, \frac{24}{44}, \frac{24}{240}\)
Thus, the descending order is: \(\frac{8}{11} > \frac{3}{5} > \frac{6}{11} > \frac{1}{10}\).
Answer: \(\frac{8}{11} > \frac{3}{5} > \frac{6}{11} > \frac{1}{10}\).
