Percentage Problem: A vs B

This question asks us to determine the percentage difference in earnings between two people, A and B. Specifically, we know that A earns more than B, and we need to find out by what percentage B earns less than A.

A Earns More Than B: The Details

We are given that A earns \(33\frac{1}{3}%\) more than B. Let's understand this statement:

• First, convert the mixed fraction percentage into a simple fraction. \(33\frac{1}{3}%\) is equivalent to \( \frac{100}{3}%\).
• To convert this percentage into a fraction of the whole, we divide by 100: \( \frac{100/3}{100} = \frac{100}{3 \times 100} = \frac{1}{3} \).
• This means A's earnings are \( \frac{1}{3} \) greater than B's earnings.

Earnings Calculation: Base and Increase

To calculate the actual percentage decrease for B, let's assume a base earning for B. A convenient way to handle percentages is to assume B earns 100 units.

Let B's earnings = \( 100 \text{ units} \).

Since A earns \( \frac{1}{3} \) more than B, the extra amount A earns is:

Extra amount = \( \frac{1}{3} \times \text{B's earnings} = \frac{1}{3} \times 100 = \frac{100}{3} \text{ units} \).

Now, we can find A's total earnings:

A's earnings = B's earnings + Extra amount

A's earnings = \( 100 + \frac{100}{3} = \frac{300}{3} + \frac{100}{3} = \frac{400}{3} \text{ units} \).

B Earns Less Than A: The Percentage

The question asks how much B earns less than A. To find this percentage, we first need the difference in their earnings, and then express this difference as a percentage of A's earnings (since A's earnings are the new reference point).

Difference in earnings = A's earnings - B's earnings

Difference = \( \frac{400}{3} - 100 = \frac{400}{3} - \frac{300}{3} = \frac{100}{3} \text{ units} \).

Now, we calculate the percentage decrease relative to A's earnings:

Percentage less = \( \left( \frac{\text{Difference}}{\text{A's earnings}} \right) \times 100% \)

Substituting the values:

Percentage less = \( \left( \frac{100/3}{400/3} \right) \times 100% \)

Simplify the fraction \( \frac{100/3}{400/3} \), which is \( \frac{100}{400} \) or \( \frac{1}{4} \).

Percentage less = \( \frac{1}{4} \times 100% \)

Percentage less = \( 25% \).

Percentage Formula Shortcut

A quicker way to solve this is using a standard formula. If one value increases by \( x% \) compared to another, the second value decreases by \( \frac{x}{100+x} \times 100% \) compared to the first.

Here, \( x = 33\frac{1}{3} = \frac{100}{3} \).

Using the formula:

Percentage decrease = \( \frac{100/3}{100 + 100/3} \times 100% \)

Calculate the denominator: \( 100 + \frac{100}{3} = \frac{300+100}{3} = \frac{400}{3} \).

Substitute back into the formula:

Percentage decrease = \( \frac{100/3}{400/3} \times 100% \)

Percentage decrease = \( \frac{100}{400} \times 100% \)

Percentage decrease = \( \frac{1}{4} \times 100% = 25% \).

Result: The Lower Percentage for B

Both methods confirm that B earns 25% less than A.