Decimal Fractions — Assertion-Reason Type Questions

Assertion (A): Representation of 6.25 as a vulgar fraction is $6\dfrac{1}{4}$.

Reason (R): A fraction is said to be a vulgar fraction if the denominator of its fractional part is a natural number but not of the form 10ⁿ, n∈N (natural number).

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

For the Assertion:

$6.25 = 6 + 0.25 = 6 + \dfrac{25}{100} = 6 + \dfrac{1}{4} = 6\dfrac{1}{4}$

In the fraction $6\dfrac{1}{4}$, the denominator (4) is a natural number and cannot be expressed as 10ⁿ, n∈N. So the Assertion (A) is true.

The Reason correctly states the condition for a fraction to be a vulgar fraction, so the Reason (R) is true.

∴ Both A and R are true.

Hence, Option 3 is the correct option.

Assertion (A): If the product of two decimal numbers is 17.55 and one of them is 6.5, then the other one is 2.7.

Reason (R): In division of decimal numbers, the dividend is not always exactly divisible.

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

For the Assertion, the other number $= 17.55 \div 6.5 = \dfrac{175.5}{65} = 2.7$. So the Assertion (A) is true.

The Reason states a true general fact that in division of decimal numbers, the dividend is not always exactly divisible. So the Reason (R) is true.

∴ Both A and R are true.

Hence, Option 3 is the correct option.

Assertion (A): 3.10 ÷ (0.1 × 0.1) = 3.1

Reason (R): In division of a decimal number by 10ⁿ, n∈N, shift the decimal point to the left by n places (digits).

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

For the Assertion:

$3.10 \div (0.1 \times 0.1) = 3.10 \div 0.01 = \dfrac{310}{1} = 310$

Since the result is 310 and not 3.1, the Assertion (A) is false.

The Reason correctly states that in division of a decimal number by 10ⁿ, the decimal point is shifted to the left by n places, so the Reason (R) is true.

∴ A is false, R is true.

Hence, Option 2 is the correct option.

Assertion (A): 9, 9.56, 9.2, 9.005 are all unlike decimals, hence addition operations can't be performed.

Reason (R): A whole number can also be expressed as a decimal number by putting a decimal after its unit's digit and after it as many zeroes as required.

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

For the Assertion, 9, 9.56, 9.2 and 9.005 are indeed unlike decimals (they have different numbers of decimal places). However, addition can still be performed by converting them into like decimals (9.000, 9.560, 9.200, 9.005). So the claim that "addition can't be performed" is wrong, and the Assertion (A) is false.

The Reason correctly states that a whole number can be expressed as a decimal number by placing a decimal point after the units digit and adding as many zeros to its right as required. So, the Reason (R) is true.

∴ A is false, R is true.

Hence, Option 2 is the correct option.