Two numbers are respectively 30% and 44% less than a third number. By what percent is the second number less than the first number?

Question

Accepted Answer

To solve the problem, we need to determine by what percentage the second number is less than the first number, given that both numbers are percentages less than a third number.

Let's denote the third number as $ x $.

1. The first number is 30% less than the third number:
   $$
   \text{First number} = x - 0.3x = 0.7x
   $$

2. The second number is 44% less than the third number:
   $$
   \text{Second number} = x - 0.44x = 0.56x
   $$

Now, we need to find by what percentage the second number is less than the first number. This can be calculated using the formula for percentage decrease:

$$
\text{Percentage decrease} = \left( \frac{\text{First number} - \text{Second number}}{\text{First number}} \right) \times 100
$$

Substituting the values we have:

$$
\text{Percentage decrease} = \left( \frac{0.7x - 0.56x}{0.7x} \right) \times 100
$$

$$
= \left( \frac{0.14x}{0.7x} \right) \times 100
$$

$$
= \left( \frac{0.14}{0.7} \right) \times 100
$$

$$
= 0.2 \times 100
$$

$$
= 20\%
$$

Thus, the second number is 20% less than the first number. This corresponds to Option 2, which is 0.2 when expressed as a decimal.

**Explanation for why each option is incorrect:**

1. **Option 1 (0.1):** This would imply a 10% decrease, which is incorrect because our calculation shows a 20% decrease.

2. **Option 3 (0.3):** This would imply a 30% decrease, which is incorrect because our calculation shows a 20% decrease.

3. **Option 4 (0.4):** This would imply a 40% decrease, which is incorrect because our calculation shows a 20% decrease.

Therefore, Option 2 is the correct answer, as it accurately represents the percentage by which the second number is less than the first number.