Fractions — Multiple Choice Questions

Mercy solved $\dfrac{2}{7}$ part of an exercise while John solved $\dfrac{4}{5}$ of it, who solved lesser part?

1. Mercy

2. John

3. none of these

Answer

Comparing $\dfrac{2}{7} \text{ and } \dfrac{4}{5}$ by cross-multiplication:

$2 \times 5 = 10 \text{ and } 7 \times 4 = 28$

Since $10 \lt 28$, we have $\dfrac{2}{7} \lt \dfrac{4}{5}$.

∴ Mercy solved the lesser part.

Hence, Option 1 is the correct option.

Fractions $\dfrac{7}{8}, \dfrac{17}{12} \text{ and } \dfrac{41}{48}$ in the form of their equivalent fractions are:

1. $\dfrac{315}{360}, \dfrac{510}{360} \text{ and } \dfrac{126}{360}$

2. $\dfrac{315}{180}, \dfrac{510}{180} \text{ and } \dfrac{64}{180}$

3. $\dfrac{7}{360}, \dfrac{17}{360} \text{ and } \dfrac{41}{360}$

4. none of these

Answer

LCM of the denominators 8, 12 and 48 = 48.

Converting each fraction to denominator 48:

$\dfrac{7}{8} = \dfrac{7 \times 6}{8 \times 6} = \dfrac{42}{48} \\[1em] \dfrac{17}{12} = \dfrac{17 \times 4}{12 \times 4} = \dfrac{68}{48} \\[1em] \dfrac{41}{48} = \dfrac{41}{48}$

The equivalent fractions are $\dfrac{42}{48}, \dfrac{68}{48} \text{ and } \dfrac{41}{48}$.

Since 48 does not divide 360 a whole number of times for $\dfrac{41}{48}$ (as $360 \div 48 = 7.5$), the fractions cannot all be written with denominator 360, so none of the given options is correct.

Hence, Option 4 is the correct option.

The difference between the fractions $\dfrac{3}{5} \text{ and } \dfrac{7}{10}$ is:

1. $-\dfrac{1}{10}$

2. $\dfrac{1}{10}$

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

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

Answer

The difference between two fractions is found by subtracting the smaller from the larger.

LCM of 5 and 10 = 10.

$\Rightarrow \dfrac{7}{10} - \dfrac{3}{5} \\[1em] = \dfrac{7}{10} - \dfrac{6}{10} \\[1em] = \dfrac{7 - 6}{10} = \dfrac{1}{10}$

Hence, Option 2 is the correct option.

$2\dfrac{1}{3} + 4\dfrac{1}{3} - 3\dfrac{1}{3}$ is equal to:

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

2. 3

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

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

Answer

$\Rightarrow 2\dfrac{1}{3} + 4\dfrac{1}{3} - 3\dfrac{1}{3}\\[1em] = \dfrac{7}{3} + \dfrac{13}{3} - \dfrac{10}{3} \\[1em] = \dfrac{7 + 13 - 10}{3} = \dfrac{10}{3} = 3\dfrac{1}{3}$

Hence, Option 1 is the correct option.

Which of the following is greater? $\dfrac{2}{7} \text{ of } \dfrac{3}{4} \text{ or } \dfrac{3}{5} \text{ of } \dfrac{5}{8}$

1. $\dfrac{2}{7} \text{ of } \dfrac{3}{4}$

2. $\dfrac{3}{5} \text{ of } \dfrac{5}{8}$

3. none of these

Answer

$\dfrac{2}{7} \text{ of } \dfrac{3}{4} = \dfrac{2}{7} \times \dfrac{3}{4} = \dfrac{6}{28} = \dfrac{3}{14} \\[1em] \dfrac{3}{5} \text{ of } \dfrac{5}{8} = \dfrac{3}{5} \times \dfrac{5}{8} = \dfrac{15}{40} = \dfrac{3}{8}$

Comparing $\dfrac{3}{14} \text{ and } \dfrac{3}{8}$ (equal numerators), the fraction with the smaller denominator is greater.

Since $8 \lt 14$, we have $\dfrac{3}{8} \gt \dfrac{3}{14}$.

∴ $\dfrac{3}{5} \text{ of } \dfrac{5}{8}$ is greater.

Hence, Option 2 is the correct option.

By what number should $7\dfrac{1}{3}$ be multiplied to get $5\dfrac{2}{3}$?

1. $\dfrac{22}{17}$

2. $\dfrac{17}{11}$

3. $\dfrac{17}{22}$

4. none of these

Answer

Required number $= 5\dfrac{2}{3} \div 7\dfrac{1}{3}$

$= \dfrac{17}{3} \div \dfrac{22}{3} = \dfrac{17}{3} \times \dfrac{3}{22} = \dfrac{17}{22}$

Hence, Option 3 is the correct option.

The cost of $3\dfrac{3}{4}$ kg apples is ₹ 600; the rate of apple per kg is:

1. ₹ 40

2. ₹ 80

3. ₹ 160

4. ₹ $\dfrac{1}{600}$

Answer

Rate per kg $= 600 \div 3\dfrac{3}{4}$

$= 600 \div \dfrac{15}{4} = 600 \times \dfrac{4}{15} = \dfrac{2400}{15} = 160$

Hence, Option 3 is the correct option.

$\dfrac{1}{3}$ part of a work is done in one hour. The part of work done in $2\dfrac{1}{5}$ hours is:

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

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

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

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

Answer

Work done in 1 hour $= \dfrac{1}{3}$

∴ Work done in $2\dfrac{1}{5} \text { hours }= \dfrac{1}{3} \times 2\dfrac{1}{5}$

Hence, Option 4 is the correct option.

A rectangular sheet of paper is $11\dfrac{1}{2}$ cm long and $8\dfrac{1}{2}$ cm wide. Its perimeter is:

1. 20 cm

2. 40 cm

3. $\dfrac{391}{4}$ cm

4. 38 cm

Answer

Perimeter $= 2 \times (\text{length} + \text{breadth})$

$= 2 \times \left(11\dfrac{1}{2} + 8\dfrac{1}{2}\right) \\[1em] = 2 \times \left(\dfrac{23}{2} + \dfrac{17}{2}\right) \\[1em] = 2 \times \dfrac{40}{2}\\[1em] = 2 \times 20 = 40 \text{ cm}$

Hence, Option 2 is the correct option.

Peter can read a novel in 17 days. If he devotes $2\dfrac{1}{7}$ hours per day, in how many hours (in all) will he read the whole novel:

1. $17 \div 2\dfrac{1}{7}$ hours

2. $2\dfrac{1}{7} \div 17$ hours

3. $\left(2\dfrac{1}{7} \times 17\right)$ hours

4. none of these

Answer

Total hours = hours per day x number of days $= \left(2\dfrac{1}{7} \times 17\right)$ hours

Hence, Option 3 is the correct option.

$\dfrac{1}{3} - \left[\dfrac{1}{3} - \Bigg\lbrace \dfrac{1}{3} - \left(\dfrac{1}{3} - \overline{\dfrac{1}{3} - \dfrac{1}{3}}\right)\Bigg\rbrace\right]$ is equal to:

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

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

3. 0

4. 1

Answer

Removing brackets in order — vinculum first, then parentheses, curly brackets and square brackets:

$\dfrac{1}{3} - \left[\dfrac{1}{3} - \Bigg\lbrace \dfrac{1}{3} - \left(\dfrac{1}{3} - \overline{\dfrac{1}{3} - \dfrac{1}{3}}\right)\Bigg\rbrace\right] \\[1em] = \dfrac{1}{3} - \left[\dfrac{1}{3} - \Bigg\lbrace \dfrac{1}{3} - \left(\dfrac{1}{3} - 0\right)\Bigg\rbrace\right] \\[1em] = \dfrac{1}{3} - \left[\dfrac{1}{3} - \Bigg\lbrace \dfrac{1}{3} - \dfrac{1}{3}\Bigg\rbrace\right] \\[1em] = \dfrac{1}{3} - \left[\dfrac{1}{3} - 0\right] \\[1em] = \dfrac{1}{3} - \dfrac{1}{3} = 0$

Hence, Option 3 is the correct option.

$\dfrac{1}{5} \div \dfrac{3}{7} \times \dfrac{5}{4}$ is equal to:

1. $\dfrac{1}{5} \times \dfrac{7 \times 4}{3 \times 5}$

2. $\dfrac{1}{5} \times \dfrac{7}{3} \times \dfrac{5}{4}$

3. $5 \times \dfrac{3 \times 5}{7 \times 4}$

4. none of these

Answer

To divide by $\dfrac{3}{7}$, we multiply by its reciprocal $\dfrac{7}{3}$:

$\dfrac{1}{5} \div \dfrac{3}{7} \times \dfrac{5}{4} = \dfrac{1}{5} \times \dfrac{7}{3} \times \dfrac{5}{4}$

Hence, Option 2 is the correct option.

$\Bigg\lbrace \dfrac{3}{5} \div \left(\dfrac{3}{5} + \dfrac{2}{5}\right)\Bigg\rbrace \div \dfrac{3}{5}$ is equal to:

1. $\dfrac{9}{25}$

2. $\dfrac{25}{9}$

3. 0

4. 1

Answer

$\Rightarrow \Bigg\lbrace \dfrac{3}{5} \div \left(\dfrac{3}{5} + \dfrac{2}{5}\right)\Bigg\rbrace \div \dfrac{3}{5} \\[1em] = \Bigg\lbrace \dfrac{3}{5} \div \dfrac{5}{5}\Bigg\rbrace \div \dfrac{3}{5} \\[1em] = \Bigg\lbrace \dfrac{3}{5} \div 1\Bigg\rbrace \div \dfrac{3}{5} = \dfrac{3}{5} \div \dfrac{3}{5} \\[1em] = \dfrac{3}{5} \times \dfrac{5}{3} = 1$

Hence, Option 4 is the correct option.

What fraction of an hour is 40 minutes?

1. $\dfrac{1}{40}$

2. $\dfrac{40}{60}$

3. $\dfrac{60}{40}$

4. 2400

Answer

1 hour = 60 minutes.

∴ Required fraction $= \dfrac{40}{60}$

Hence, Option 2 is the correct option.

$\dfrac{3}{11}$ of Shyam's salary is ₹ 7,260, then his salary is:

1. ₹ 13300

2. ₹ 26600

3. ₹ 13320

4. ₹ 26620

Answer

Let Shyam's salary be ₹ $x$.

$\dfrac{3}{11} \times x = 7260 \\[1em] \Rightarrow x = 7260 \times \dfrac{11}{3} = 2420 \times 11 = 26620$

Hence, Option 4 is the correct option.

The number that should divide $2\dfrac{1}{3}$ to get 1 is:

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

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

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

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

Answer

Let the required number be $x$.

$\Rightarrow 2\dfrac{1}{3} \div x = 1 \\[1em] \Rightarrow \dfrac{7}{3} \div x = 1 \\[1em] \Rightarrow x = \dfrac{7}{3} = 2\dfrac{1}{3}$

Hence, Option 3 is the correct option.