Exercise: 4-I
Multiple Choice Type
Q1: \(\frac{48}{72}\) in simplest form is
Step 1: Write the given fraction
\[
\frac{48}{72}
\]Step 2: Find the HCF (Highest Common Factor) of 48 and 72
Factors of 48 = 1, 2, 3, 4, 6, 8, 12, 16, 24, 48
Factors of 72 = 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
HCF = 24
Step 3: Divide numerator and denominator by their HCF (24)
\[
\frac{48 \div 24}{72 \div 24} = \frac{2}{3}
\]Step 4: Check all options and match with \(\frac{2}{3}\)
Answer: c. \(\frac{2}{3}\)
Q2: \(\frac{42}{54}\) in simplest form is
Step 1: Write the given fraction
\[
\frac{42}{54}
\]Step 2: Find the HCF (Highest Common Factor) of 42 and 54
Factors of 42 = 1, 2, 3, 6, 7, 14, 21, 42
Factors of 54 = 1, 2, 3, 6, 9, 18, 27, 54
HCF = 6
Step 3: Divide numerator and denominator by their HCF (6)
\[
\frac{42 \div 6}{54 \div 6} = \frac{7}{9}
\]Step 4: Check all options and match with \(\frac{7}{9}\)
Answer: d. \(\frac{7}{9}\)
Q3: \(\frac{91}{114}\) in simplest form is
Step 1: Write the given fraction
\[
\frac{91}{114}
\]Step 2: Find the HCF (Highest Common Factor) of 91 and 114
Factors of 91 = 1, 7, 13, 91
Factors of 114 = 1, 2, 3, 6, 19, 38, 57, 114
Common factors are only 1, so check prime factorization:
– 91 = \(7 \times 13\)
– 114 = \(2 \times 3 \times 19\)
No common prime factors other than 1.
Step 3: Since HCF = 1, the fraction \(\frac{91}{114}\) is already in simplest form.
Check options to find equivalent fraction.
Answer: None of the options
Q4: \(\frac{117}{143}\) in simplest form is
Step 1: Write the given fraction
\[
\frac{117}{143}
\]Step 2: Find the HCF (Highest Common Factor) of 117 and 143
Prime factorization:
– \(117 = 3 \times 3 \times 13 = 3^2 \times 13\)
– \(143 = 11 \times 13\)
Step 3: Identify the common factors
Common factor = 13
Step 4: Divide numerator and denominator by 13 to simplify:
\[
\frac{117 \div 13}{143 \div 13} = \frac{9}{11}
\]Step 5: Therefore, the fraction in simplest form is
\[
\frac{9}{11}
\]Answer: b. \(\frac{9}{11}\)
Q5: If \(\frac{3}{4}\) is equivalent to \(\frac{x}{20}\), then x = ?
Step 1: Given the equivalent fractions, set up the equation using cross multiplication:
\[
\frac{3}{4} = \frac{x}{20}
\]
Cross multiply:
\[
3 \times 20 = 4 \times x \\
60 = 4x
\]Step 2: Solve for \(x\):
\[
x = \frac{60}{4} = 15
\]Step 3: Thus, the value of \(x\) is 15.
Answer: b. 15
Q6: If \(\frac{3}{x}\) is equivalent to \(\frac{45}{60}\), then the value of x is
Step 1: Write the equation for equivalence of two fractions:
\[
\frac{3}{x} = \frac{45}{60}
\]Step 2: Cross multiply to solve for \(x\):
\[
3 \times 60 = 45 \times x \\
180 = 45x
\]Step 3: Divide both sides by 45 to isolate \(x\):
\[
x = \frac{180}{45} = 4
\]Answer: a. \(4\)
Q7: Which of the following are the like fractions?
Step 1: Recall that like fractions are fractions having the same denominator.
Step 2: Check denominators for each option:
a. Denominators are 3, 5, 7, 9 — different denominators, so not like fractions.
b. Denominators are 3, 4, 5, 6 — different denominators, so not like fractions.
c. Denominators are 5, 5, 5, 5 — all denominators are the same, so these are like fractions.
Answer: c. \(\frac{1}{5}, \frac{2}{5}, \frac{3}{5}, \frac{4}{5}\)
Q8: Which of the following is an improper fraction?
Step 1: Recall the definition:
An improper fraction is a fraction where the numerator is equal to or greater than the denominator.
Step 2: Check each option:
a. \(\frac{5}{7}\): numerator (5) < denominator (7), so it is a proper fraction.
b. \(\frac{4}{7}\): numerator (4) < denominator (7), so it is a proper fraction.
c. \(\frac{7}{7}\): numerator (7) = denominator (7), so it is an improper fraction.
d. \(\frac{2}{3}\): numerator (2) < denominator (3), so it is a proper fraction.
Answer: c. \(\frac{7}{7}\)
Q9: Which of the following is a proper fraction?
Step 1: Recall the definition:
A proper fraction is a fraction where the numerator is less than the denominator, i.e., numerator < denominator.
Step 2: Analyze each option:
a. \(\frac{5}{5}\) → numerator = denominator ⇒ not a proper fraction.
b. \(\frac{6}{5}\) → numerator > denominator ⇒ improper fraction.
c. \(\frac{4}{3}\) → numerator > denominator ⇒ improper fraction.
d. \(\frac{3}{4}\) → numerator < denominator ⇒ proper fraction.
Answer: d. \(\frac{3}{4}\)
Q10: Which of the following statements is correct:
Step 1: Convert both fractions to a common denominator for comparison.
\[
\frac{2}{3} = \frac{2 \times 5}{3 \times 5} = \frac{10}{15}, \quad \frac{3}{5} = \frac{3 \times 3}{5 \times 3} = \frac{9}{15}
\]Step 2: Compare the new numerators:
\[
\frac{10}{15} \gt \frac{9}{15} \Rightarrow \frac{2}{3} \gt \frac{3}{5}
\]Answer: a. \(\frac{2}{3} \gt \frac{3}{5}\)
Q11: The smallest of the fractions \(\frac{3}{5},\frac{5}{6},\frac{7}{10},\frac{2}{3}\) is
Step 1: First, convert all fractions to a common denominator to compare them.
The LCM of the denominators 5, 6, 10, and 3 is 30.
Step 2: Convert each fraction:
\[
\frac{3}{5} = \frac{3 \times 6}{5 \times 6} = \frac{18}{30} \\
\frac{5}{6} = \frac{5 \times 5}{6 \times 5} = \frac{25}{30} \\
\frac{7}{10} = \frac{7 \times 3}{10 \times 3} = \frac{21}{30} \\
\frac{2}{3} = \frac{2 \times 10}{3 \times 10} = \frac{20}{30}
\]Step 3: Now compare:
\[
\frac{18}{30} < \frac{20}{30} < \frac{21}{30} < \frac{25}{30}
\]Answer: b. \(\frac{3}{5}\)
Q12: The largest of the fractions \(\frac{5}{6},\frac{5}{7},\frac{5}{9},\frac{5}{11}\) is
Step 1: All fractions have the same numerator \(5\).
So, the fraction with the **smallest denominator** will be the **largest**.
Step 2: Compare the denominators:
\[
6,\ 7,\ 9,\ 11 \Rightarrow \text{smallest is } 6
\]Step 3: So, the largest fraction is:
\[
\frac{5}{6}
\]Answer: d. \(\frac{5}{6}\)
Q13: The smallest of the fractions \(\frac{8}{13},\frac{9}{13},\frac{10}{13},\frac{11}{13}\) is
Step 1: All fractions have the **same denominator** \(13\).
So, the fraction with the **smallest numerator** will be the **smallest**.
Step 2: Compare the numerators:
\[
8,\ 9,\ 10,\ 11 \Rightarrow \text{smallest is } 8
\]Step 3: So, the smallest fraction is:
\[
\frac{8}{13}
\]Answer: c. \(\frac{8}{13}\)
Q14: The smallest of the fractions \(\frac{3}{4},\frac{5}{6},\frac{7}{12},\frac{2}{3}\) is
Step 1: Find the LCM of all denominators:
Denominators = 4, 6, 12, 3
LCM(4, 6, 12, 3) = 12
Step 2: Convert all fractions to like denominators (denominator = 12)
\[
\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12} \\
\frac{5}{6} = \frac{5 \times 2}{6 \times 2} = \frac{10}{12} \\
\frac{7}{12} = \frac{7}{12} \\
\frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12}
\]Step 3: Now compare:
\[
\frac{9}{12},\ \frac{10}{12},\ \frac{7}{12},\ \frac{8}{12} \\
\Rightarrow \text{Smallest is } \frac{7}{12}
\]Answer: d. \(\frac{7}{12}\)
Q15: \(\frac{37}{5}=\ ?\)
Step 1: Divide numerator by denominator
\[
\frac{37}{5} = 7\ \text{quotient},\ 2\ \text{remainder}
\]Step 2: Write in mixed fraction form
\[
\frac{37}{5} = 7\frac{2}{5}
\]Answer: b. \(7\frac{2}{5}\)
Q16: The reciprocal of \(1\frac{5}{7}\) is
Step 1: Convert the mixed number to an improper fraction
\[
1\frac{5}{7} = \frac{7 \times 1 + 5}{7} = \frac{12}{7}
\]Step 2: Find the reciprocal
\[
\text{Reciprocal of } \frac{12}{7} = \frac{7}{12}
\]Answer: c. \(\frac{7}{12}\)
Q17: \(1\div\frac{5}{7}=\ ?\)
Step 1: Recall the rule:
\[
\text{Dividing by a fraction is the same as multiplying by its reciprocal}
\]
So,
\[
1 \div \frac{5}{7} = 1 \times \frac{7}{5}
\]Step 2: Multiply
\[
1 \times \frac{7}{5} = \frac{7}{5}
\]Answer: d. none of these
Q18: \(\frac{4}{5}\) of a number is 64. Half of that number is
Step 1: Let the number be \(x\). Given:
\[
\frac{4}{5} \times x = 64
\]Step 2: Solve for \(x\):
\[
x = 64 \times \frac{5}{4} \\
x = 64 \times \frac{5}{4} = 64 \times 1.25 = 80
\]Step 3: Find half of \(x\):
\[
\frac{1}{2} \times 80 = 40
\]Answer: b. 40
