Fractions — Multiple Choice Questions

In the given number line, identify the values of A and B.

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

2. $A = \dfrac{2}{3}, B = 1\dfrac{3}{4}$

3. $A = 3 \text{ and } B = 7$

4. $A = 2 \text{ and } B = 6$

Answer

On the number line, point A lies between 0 and 1. Therefore, its value is greater than 0 and less than 1.

The interval between 0 and 1 is divided into 3 equal parts, and point A lies on the second division mark.

$\therefore A = \dfrac{2}{3}$

Point B lies between 1 and 2. Therefore, its value is greater than 1 and less than 2.

The interval between 1 and 2 is divided into 4 equal parts, and point B lies on the third division mark after 1.

$\therefore B = 1 + \dfrac{3}{4} = 1\dfrac{3}{4}$

Hence, option 2 is the correct option.

In $\dfrac{3}{5}, \dfrac{2}{5}, \dfrac{3}{4}, \dfrac{2}{7} \text{ and } \dfrac{4}{5}$, like fractions are:

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

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

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

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

Answer

Like fractions have the same denominator. The fractions with denominator 5 are $\dfrac{3}{5}, \dfrac{2}{5} \text{ and } \dfrac{4}{5}$.

Hence, option 4 is the correct option.

Out of $\dfrac{15}{7}, \dfrac{15}{8}, \dfrac{15}{11} \text{ and } \dfrac{15}{13}$, the largest fraction is:

1. $\dfrac{15}{7}$

2. $\dfrac{15}{8}$

3. $\dfrac{15}{11}$

4. $\dfrac{15}{13}$

Answer

These fractions have the same numerator 15. Among fractions with the same numerator, the one with the smallest denominator is the largest. The smallest denominator is 7.

Hence, option 1 is the correct option.

Out of $\dfrac{7}{15}, \dfrac{8}{15}, \dfrac{11}{15} \text{ and } \dfrac{13}{15}$, the least fraction is:

1. $\dfrac{7}{15}$

2. $\dfrac{8}{15}$

3. $\dfrac{11}{15}$

4. $\dfrac{13}{15}$

Answer

These are like fractions (same denominator 15). Among like fractions, the one with the smallest numerator is the least. The smallest numerator is 7.

Hence, option 1 is the correct option.

$\dfrac{7}{16} + \dfrac{13}{16} - \dfrac{5}{16}$ is equal to:

1. $\dfrac{15}{16}$

2. $\dfrac{25}{16}$

3. 0

4. 1

Answer

$\Rightarrow \dfrac{7}{16} + \dfrac{13}{16} - \dfrac{5}{16} = \dfrac{7 + 13 - 5}{16} \\[1em] = \dfrac{15}{16}.$

Hence, option 1 is the correct option.

$31\dfrac{2}{3} - 16\dfrac{2}{3} - 15\dfrac{2}{3}$ is equal to:

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

2. 0

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

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

Answer

$\Rightarrow 31\dfrac{2}{3} - 16\dfrac{2}{3} - 15\dfrac{2}{3} \\[1em] \text{Changing mixed fraction into improper fraction}\\[1em] = \dfrac{95}{3} - \dfrac{50}{3} - \dfrac{47}{3} \\[1em] = \dfrac{95 - 50 - 47}{3} \\[1em] = \dfrac{-2}{3}.$

Hence, option 4 is the correct option.

12 ÷ 12 = 1, 0 ÷ 12 = 0; then 12 ÷ 0 is:

1. 1

2. 0

3. not defined

Answer

Division by zero is not defined.

Hence, option 3 is the correct option.

$\dfrac{1}{4} \times \dfrac{1}{4} \div \dfrac{1}{4} \text{ of } \dfrac{1}{4}$ is equal to:

1. 1

2. 16

3. $\dfrac{1}{16}$

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

Answer

Solving,

$\Rightarrow \dfrac{1}{4} \times \dfrac{1}{4} \div \dfrac{1}{4} \text{ of } \dfrac{1}{4} \\[1em] \Rightarrow \dfrac{1}{4} \times \dfrac{1}{4} \div \dfrac{1}{16} \\[1em] \Rightarrow \dfrac{1}{4} \times \dfrac{1}{4} \times 16 \\[1em] \Rightarrow \dfrac{16}{16} \\[1em] \Rightarrow 1.$

Hence, option 1 is the correct option.

$\dfrac{1}{4} \text{ of } \dfrac{1}{4} \div \dfrac{1}{4} \times \dfrac{1}{4}$ is equal to:

1. 1

2. 16

3. $\dfrac{1}{16}$

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

Answer

Solving 'of' first,

$\Rightarrow \dfrac{1}{4} \text{ of } \dfrac{1}{4} = \dfrac{1}{4} \times \dfrac{1}{4} = \dfrac{1}{16}.\\[1em] \Rightarrow \dfrac{1}{16} \div \dfrac{1}{4} \times \dfrac{1}{4} = \dfrac{1}{16} \times \dfrac{4}{1} \times \dfrac{1}{4} \\[1em] = \dfrac{1}{16}.$

Hence, option 3 is the correct option.

If $\dfrac{x}{3} = 15$, then $\dfrac{x}{5}$ is:

1. 45

2. 9

3. 75

4. none of these

Answer

$\Rightarrow \dfrac{x}{3} = 15 \\[1em] \Rightarrow x = 15 \times 3 = 45.\\[1em] \Rightarrow \dfrac{x}{5} = \dfrac{45}{5} = 9.$

Hence, option 2 is the correct option.

Assertion (A): 7 ÷ 7 x 7 - 7 = 0

Reason (R): $7 \times \dfrac{1}{7} \times 7 - 7 = 7 - 7 = 0$

1. A is true, R is false.

2. A is false, R is true.

3. Both A and R are true.

4. Both A and R are false.

Answer

According to BODMAS,

Checking A: 7 ÷ 7 x 7 - 7 = 1 x 7 - 7 = 7 - 7 = 0. So A is true.

Checking R: $7 \times \dfrac{1}{7} \times 7 - 7 = 7 - 7 = 0$. So R is true and correctly explains A.

Hence, option 3 is the correct option.

Assertion (A): $\dfrac{1}{2}$ of a fraction is 15, then $\dfrac{1}{5}$ of the same fraction is 6.

Reason (R): The fraction is 30.

1. A is true, R is false.

2. A is false, R is true.

3. Both A and R are true.

4. Both A and R are false.

Answer

If $\dfrac{1}{2}$ of a fraction is 15, then the fraction = 15 × 2 = 30, so R is true.

Now, $\dfrac{1}{5} \text{ of } 30 = \dfrac{1}{5} \times 30$ = 6, so A is true.

Both A and R are true, and R correctly explains A.

Hence, option 3 is the correct option.

Statement 1: If $x = 1\dfrac{1}{2} \div 3\dfrac{1}{2} \times 2\dfrac{1}{3}$, then the value of $x$ is 1.

Statement 2: $1\dfrac{1}{2} \div 3\dfrac{1}{2} \times 2\dfrac{1}{3} = \dfrac{3}{2} \times \dfrac{2}{7} \times \dfrac{7}{3} = 1$

1. Both the statements are true.

2. Both the statements are false.

3. Statement 1 is true, and statement 2 is false.

4. Statement 1 is false, and statement 2 is true.

Answer

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

So, the value of $x$ is 1, and both statements are true with statement 2 showing the correct working.

Hence, option 1 is the correct option.