Fractions — Multiple Choice Questions

$\dfrac{48}{72}$ in simplest form is

1. $\dfrac{8}{9}$

2. $\dfrac{3}{4}$

3. $\dfrac{2}{3}$

4. none of these

Answer

By prime factorization,

$\Rightarrow \dfrac{48}{72} \\[1em] \Rightarrow \dfrac{2 \times 2 \times 2 \times 2 \times 3}{2 \times 2 \times 2 \times 3 \times 3} \\[1em] \Rightarrow \dfrac{2}{3}.$

Hence, option 3 is the correct option.

$\dfrac{42}{54}$ in simplest form is

1. $\dfrac{3}{4}$

2. $\dfrac{2}{3}$

3. $\dfrac{3}{7}$

4. $\dfrac{7}{9}$

Answer

By prime factorization,

$\Rightarrow \dfrac{42}{54} \\[1em] \Rightarrow \dfrac{2 \times 3 \times 7}{2 \times 3 \times 3 \times 3} \\[1em] \Rightarrow \dfrac{7}{9}.$

Hence, option 4 is the correct option.

$\dfrac{91}{114}$ in simplest form is

1. $\dfrac{13}{8}$

2. $\dfrac{7}{8}$

3. $\dfrac{91}{114}$

4. $\dfrac{9}{13}$

Answer

HCF of 91 and 114 = 1.

Thus,

$\dfrac{91}{114}$ in simplest form is $\dfrac{91}{114}$.

Hence, option 3 is the correct option.

$\dfrac{117}{143}$ in simplest form is

1. $\dfrac{117}{143}$

2. $\dfrac{9}{11}$

3. $\dfrac{13}{11}$

4. $\dfrac{9}{13}$

Answer

By prime factorization,

$\Rightarrow \dfrac{117}{143} \\[1em] \Rightarrow \dfrac{13 \times 9}{13 \times 11} \\[1em] \Rightarrow \dfrac{9}{11}.$

Hence, option 2 is the correct option.

If $\dfrac{3}{4}$ is equivalent to $\dfrac{x}{20}$, then x = ?

1. 12

2. 15

3. 18

4. 16

Answer

Given, $\dfrac{3}{4}$ is equivalent to $\dfrac{x}{20}$

$\dfrac{3}{4} = \dfrac{3 \times 5}{4 \times 5} = \dfrac{15}{20}$

Comparing,

$\dfrac{x}{20}$ and $\dfrac{15}{20}$

Thus, x = 15.

Hence, option 2 is the correct option.

If $\dfrac{3}{x}$ is equivalent to $\dfrac{45}{60}$, then the value of x is

1. 4

2. 5

3. 6

4. 20

Answer

Given, $\dfrac{3}{x}$ is equivalent to $\dfrac{45}{60}$

$\Rightarrow \dfrac{3}{x} = \dfrac{45}{60} \\[1em] \Rightarrow x = \dfrac{3 \times 60}{45} \\[1em] \Rightarrow x = \dfrac{60}{15} = 4.$

Hence, option 1 is the correct option.

Which of the following are the like fractions?

1. $\dfrac{2}{3}, \dfrac{2}{5}, \dfrac{2}{7}, \dfrac{2}{9}$

2. $\dfrac{2}{3}, \dfrac{3}{4}, \dfrac{4}{5}, \dfrac{5}{6}$

3. $\dfrac{1}{5}, \dfrac{2}{5}, \dfrac{3}{5}, \dfrac{4}{5}$

4. none of these

Answer

Like fractions are fractions that have the same denominator.

In option 3,

$\dfrac{1}{5}, \dfrac{2}{5}, \dfrac{3}{5}, \dfrac{4}{5}$

Denominators = 5, 5, 5 and 5

All denominators are same.

So, they are like fractions.

Hence, option 3 is the correct option.

Which of the following is an improper fraction?

1. $\dfrac{5}{7}$

2. $\dfrac{4}{7}$

3. $\dfrac{7}{7}$

4. $\dfrac{2}{3}$

Answer

An improper fraction is a fraction in which the numerator is greater than or equal to the denominator.

In option 3, $\dfrac{7}{7}$; numerator = denominator = 7

Hence, option 3 is the correct option.

Which of the following is a proper fraction ?

1. $\dfrac{5}{5}$

2. $\dfrac{6}{5}$

3. $\dfrac{4}{3}$

4. $\dfrac{3}{4}$

Answer

A proper fraction is a fraction in which the numerator is less than the denominator.

In option 4, $\dfrac{3}{4}$; numerator < denominator

Hence, option 4 is the correct option.

Which of the following statements is correct ?

1. $\dfrac{2}{3} \gt \dfrac{3}{5}$

2. $\dfrac{2}{3} \lt \dfrac{3}{5}$

3. $\dfrac{2}{3} = \dfrac{3}{5}$

4. $\dfrac{2}{3}$ and $\dfrac{3}{5}$ cannot be compared

Answer

Comparing both fractions; $\dfrac{2}{3}$ and $\dfrac{3}{5}$

Cross multiply: 2 x 5 and 3 x 3

⇒ 10 and 9

Now, comparing both numbers;

⇒ 10 > 9

⇒ $\dfrac{2}{3} \gt \dfrac{3}{5}$

Hence, option 1 is the correct option.

The smallest of the fractions $\dfrac{3}{5}, \dfrac{5}{6}, \dfrac{7}{10}, \dfrac{2}{3}$ is

1. $\dfrac{2}{3}$

2. $\dfrac{3}{5}$

3. $\dfrac{5}{6}$

4. $\dfrac{7}{10}$

Answer

Given fractions: $\dfrac{3}{5}, \dfrac{5}{6}, \dfrac{7}{10}, \dfrac{2}{3}$

LCM of the denominators (5, 6, 10, 3) = 30

$\Rightarrow \dfrac{3 \times 6}{5 \times 6}, \dfrac{5 \times 5}{6 \times 5}, \dfrac{7 \times 3}{10 \times 3}, \dfrac{2 \times 10}{3 \times 10}\\[1em] \Rightarrow \dfrac{18}{30}, \dfrac{25}{30}, \dfrac{21}{30}, \dfrac{20}{30}$

Among fractions with the same denominator, the one with the smaller numerator is smaller.

So, $\dfrac{18}{30} \lt \dfrac{20}{30} \lt \dfrac{21}{30} \lt \dfrac{25}{30}$

$\dfrac{3}{5} \lt \dfrac{2}{3} \lt \dfrac{7}{10} \lt \dfrac{5}{6}$

Hence, option 2 is the correct option.

The largest of the fractions $\dfrac{5}{6}, \dfrac{5}{7}, \dfrac{5}{9}, \dfrac{5}{11}$ is

1. $\dfrac{5}{11}$

2. $\dfrac{5}{9}$

3. $\dfrac{5}{7}$

4. $\dfrac{5}{6}$

Answer

Given fractions: $\dfrac{5}{6}, \dfrac{5}{7}, \dfrac{5}{9}, \dfrac{5}{11}$

LCM of the denominators (6, 7, 9, 11) = 1,386

$\Rightarrow \dfrac{5 \times 231}{6 \times 231}, \dfrac{5 \times 198}{7 \times 198}, \dfrac{5 \times 154}{9 \times 154}, \dfrac{5 \times 126}{11 \times 126}\\[1em] \Rightarrow \dfrac{1155}{1386}, \dfrac{990}{1386}, \dfrac{770}{1386}, \dfrac{630}{1386}$

Among fractions with the same denominator, the one with the smaller numerator is smaller.

So, $\dfrac{630}{1386} \lt \dfrac{770}{1386} \lt \dfrac{990}{1386} \lt \dfrac{1155}{1386}$

$\dfrac{5}{11} \lt \dfrac{5}{9} \lt \dfrac{5}{7} \lt \dfrac{5}{6}$

Hence, option 4 is the correct option.

The smallest of the fractions $\dfrac{8}{13}, \dfrac{9}{13}, \dfrac{10}{13}, \dfrac{11}{13}$ is

1. $\dfrac{11}{13}$

2. $\dfrac{10}{13}$

3. $\dfrac{8}{13}$

4. $\dfrac{9}{13}$

Answer

Given fractions: $\dfrac{8}{13}, \dfrac{9}{13}, \dfrac{10}{13}, \dfrac{11}{13}$

Among fractions with same denominator, the one with greater numerator is greater of the two.

So, $\dfrac{8}{13} \lt \dfrac{9}{13} \lt \dfrac{10}{13} \lt \dfrac{11}{13}$

Hence, option 3 is the correct option.

The smallest of the fractions $\dfrac{3}{4}, \dfrac{5}{6}, \dfrac{7}{12}, \dfrac{2}{3}$ is

1. $\dfrac{2}{3}$

2. $\dfrac{3}{4}$

3. $\dfrac{5}{6}$

4. $\dfrac{7}{12}$

Answer

Given fractions: $\dfrac{3}{4}, \dfrac{5}{6}, \dfrac{7}{12}, \dfrac{2}{3}$

LCM of the denominators (4, 6, 12, 3) = 12

$\Rightarrow \dfrac{3 \times 3}{4 \times 3}, \dfrac{5 \times 2}{6 \times 2}, \dfrac{7 \times 1}{12 \times 1}, \dfrac{2 \times 4}{3 \times 4}\\[1em] \Rightarrow \dfrac{9}{12}, \dfrac{10}{12}, \dfrac{7}{12}, \dfrac{8}{12}$

Among two fractions with same denominator, the one with greater numerator is greater of the two.

So, $\dfrac{7}{12} \lt \dfrac{8}{12} \lt \dfrac{9}{12} \lt \dfrac{10}{12}$

$\dfrac{7}{12} \lt \dfrac{2}{3} \lt \dfrac{3}{4} \lt \dfrac{5}{6}$

Hence, option 4 is the correct option.

$\dfrac{37}{5} = ?$

1. $7\dfrac{1}{5}$

2. $7\dfrac{2}{5}$

3. $7\dfrac{3}{5}$

4. $7\dfrac{4}{5}$

Answer

Given; $\dfrac{37}{5}$

Divide 37 by 5: 37 ÷ 5 = 7 with a remainder of 2.

The quotient is 7 and the remainder 2 becomes the numerator.

$\dfrac{37}{5} = 7\dfrac{2}{5}$

Hence, option 2 is the correct option.

The reciprocal of $1\dfrac{5}{7}$ is

1. $\dfrac{12}{7}$

2. $1\dfrac{7}{5}$

3. $\dfrac{7}{12}$

4. none of these

Answer

$1\dfrac{5}{7} = \dfrac{7 \times 1 + 5}{7} = \dfrac{12}{7}$.

The reciprocal of $\dfrac{12}{7}$ is $\dfrac{7}{12}$.

Hence, option 3 is the correct option.

1 ÷ $\dfrac{5}{7} = ?$

1. $\dfrac{2}{5}$

2. $\dfrac{5}{2}$

3. $\dfrac{7}{2}$

4. none of these

Answer

Given,

$\Rightarrow 1 ÷ \dfrac{5}{7}\\[1em] \Rightarrow 1 \times \dfrac{7}{5}\\[1em] \Rightarrow \dfrac{1 \times 7}{5}\\[1em] \Rightarrow \dfrac{7}{5}.$

Hence, option 4 is the correct option.

$\dfrac{4}{5}$ of a number is 64. Half of that number is

1. 32

2. 40

3. 80

4. 16

Answer

Given, $\dfrac{4}{5}$ of a number is 64.

Let the number be x.

$\Rightarrow \dfrac{4}{5} \times x = 64\\[1em] \Rightarrow x = 64 ÷ \dfrac{4}{5} \\[1em] \Rightarrow x = 64 \times \dfrac{5}{4} \\[1em] \Rightarrow x = \dfrac{64 \times 5}{4} \\[1em] \Rightarrow x = 16 \times 5 \\[1em] \Rightarrow x = 80.$

Half of the number = $\dfrac{1}{2} \times 80 = \dfrac{80}{2} = 40$.

Hence, option 2 is the correct option.