Two numbers are respectively 30% and 40% less than a third number. What percent is the second of the first ?

Question

Accepted Answer

To solve the problem, we need to determine what percent the second number is of 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 40% less than the third number:
   $$
   \text{Second number} = x - 0.4x = 0.6x
   $$

Now, we need to find what percent the second number is of the first number:
$$
\text{Percentage} = \left(\frac{\text{Second number}}{\text{First number}}\right) \times 100 = \left(\frac{0.6x}{0.7x}\right) \times 100
$$

Simplifying the expression:
$$
= \left(\frac{0.6}{0.7}\right) \times 100 = \left(\frac{6}{7}\right) \times 100
$$

Calculating the fraction:
$$
= \frac{600}{7} \approx 85.7142857
$$

This value is equivalent to $ 85\frac{5}{7} \% $, which matches Option 1.

**Explanation for why Option 1 is correct:**
- The calculation shows that the second number is approximately $ 85.7142857\% $ of the first number, which is precisely $ 85\frac{5}{7} \% $. This confirms that Option 1 is the correct answer.

**Explanation for why other options are incorrect:**
- **Option 2: $ 86\frac{2}{5} \% $**: This percentage is higher than the calculated value of $ 85\frac{5}{7} \% $. Therefore, it does not match the correct calculation.
- **Option 3: $ 58\frac{5}{7} \% $**: This percentage is significantly lower than the calculated value. It does not reflect the correct relationship between the second and first numbers.
- **Option 4: $ 59\frac{3}{4} \% $**: Similar to Option 3, this percentage is also lower than the calculated value and does not accurately represent the percentage relationship.

Thus, Option 1 is the only correct choice based on the given calculations.