The sum of two numbers is 520. If the bigger number is decreased by 4% and the smaller number is increased by 12%, then the obtained numbers are equal, then smaller number is :

Question

Accepted Answer

To solve the problem, we need to determine the smaller number when the sum of two numbers is 520, and after certain percentage changes, the numbers become equal.

Let's denote the smaller number as $ x $ and the bigger number as $ y $. According to the problem, we have:

$$ x + y = 520 $$

The problem states that if the bigger number $ y $ is decreased by 4%, and the smaller number $ x $ is increased by 12%, the two numbers become equal. Mathematically, this can be expressed as:

$$ y - 0.04y = x + 0.12x $$

Simplifying both sides, we get:

$$ 0.96y = 1.12x $$

Now, we have two equations:

1. $ x + y = 520 $
2. $ 0.96y = 1.12x $

From the second equation, we can express $ y $ in terms of $ x $:

$$ y = \frac{1.12x}{0.96} $$

$$ y = \frac{28x}{24} $$

$$ y = \frac{7x}{6} $$

Substitute this expression for $ y $ into the first equation:

$$ x + \frac{7x}{6} = 520 $$

Multiply through by 6 to eliminate the fraction:

$$ 6x + 7x = 3120 $$

$$ 13x = 3120 $$

$$ x = \frac{3120}{13} $$

$$ x = 240 $$

Thus, the smaller number is 240, which corresponds to Option 2.

**Explanation for why other options are incorrect:**

1. **Option 1 (210):** If the smaller number were 210, then the bigger number would be 310 (since 210 + 310 = 520). Applying the percentage changes, 310 decreased by 4% is 297.6, and 210 increased by 12% is 235.2. These numbers are not equal, so Option 1 is incorrect.

2. **Option 3 (270):** If the smaller number were 270, then the bigger number would be 250 (since 270 + 250 = 520). Applying the percentage changes, 250 decreased by 4% is 240, and 270 increased by 12% is 302.4. These numbers are not equal, so Option 3 is incorrect.

3. **Option 4 (300):** If the smaller number were 300, then the bigger number would be 220 (since 300 + 220 = 520). Applying the percentage changes, 220 decreased by 4% is 211.2, and 300 increased by 12% is 336. These numbers are not equal, so Option 4 is incorrect.

Therefore, the correct answer is indeed Option 2, where the smaller number is 240.