The commission of a broker decreases from 10% to 8%. Find what percent should he increase his business so as to retain his original income?

Questions : The commission of a broker decreases from 10% to 8%. Find what percent should he increase his business so as to retain his original income?

Answer:  

Step 1: Consider the original and new commission rates

• The broker's original commission was 10% of the business.
• The new commission is 8% of the business.
• Let the original business amount be 100 units (assuming for easy calculation).
• The original income from commission was: $$\frac{10}{100} \times 100 = 10 \text{ units}$$
• After the commission is reduced to 8%, let the new business amount be X units to retain the same commission of 10 units.

Step 2: Form the equation

$$\frac{8}{100} \times X = 10$$ $$X = \frac{10 \times 100}{8} = 125$$

Step 3: Find the percentage increase

$$\text{Increase} = \frac{125 - 100}{100} \times 100 = \frac{25}{100} \times 100 = 25\%$$

Thus, the broker must increase his business by 25% to maintain his original income.

Alternative 

1. Focus on the commission rates: The commission decreased from 10% to 8%. This means the new commission is 8/10 = 4/5 of the original commission.

2. Inverse Relationship: Since the income needs to stay the same, the business volume needs to increase by the inverse of that fraction. The inverse of 4/5 is 5/4.

3. Calculate the Increase: 5/4 represents the new business volume compared to the old. This means the new business is 1 1/4 or 1.25 times the old business. The increase is therefore 1.25 - 1 = 0.25

4. Convert to Percentage: 0.25 represents a 25% increase.

Therefore, the broker needs to increase his business by 25%.

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