Understanding Compound Percent Problems

A compound percent problem involves applying two or more percent changes one after another. The most important rule to remember is that each new percent change is applied to the result of the previous step, not to the original starting amount.

Because of this, consecutive percent increases and decreases do not simply cancel each other out!

The Common Misconception

Imagine a price goes up by 20%, and then is discounted by 20%. You might guess the price goes back to normal. Let's see why this is false by using a starting value of $100$.

1. Step 1 (20% Increase): The price goes up by 20% of $100$. $100 \times 0.20 = 20$ New Price: $100 + 20 = 120$

2. Step 2 (20% Decrease): Now, the 20% discount is applied to the new price ($120$), not the original $100$. $120 \times 0.20 = 24$ Final Price: $120 - 24 = 96$

The final price is $96$, which is a 4% overall decrease from the original $100$.

Example: Investment Gains and Losses

Problem: You invest $1,000$. It gains 10% the first year and loses 10% the second year. How much do you have?

Step 1: Year 1 Gain

• Gain = $1000 \times 0.10 = 100$
• End of Year 1 = $1000 + 100 = 1100$

Step 2: Year 2 Loss

• Loss = $1100 \times 0.10 = 110$
• End of Year 2 = $1100 - 110 = 990$

You are left with $990$.

The Multiplier Method (Shortcut)

Instead of calculating the change and adding or subtracting it, you can use decimal multipliers to solve compound percent problems in one single equation.

• A 20% increase means you keep 100% and add 20% (120% total), making your multiplier 1.20.
• A 20% decrease means you keep 100% minus 20% (80% total), making your multiplier 0.80.

You can chain these multipliers together by multiplying them: $\text{Final Value} = \text{Original Value} \times \text{Multiplier}_1 \times \text{Multiplier}_2$

Using our first example with a starting price of $100: $\text{Final Price} = 100 \times 1.20 \times 0.80$ $\text{Final Price} = 100 \times 0.96 = 96$

This shortcut makes it incredibly easy to calculate any number of consecutive percent changes!