Applications of Percents: Percent Increase and Decrease

Percents show up everywhere outside the math classroom: store discounts, sales tax on a receipt, a tip at a restaurant, interest on savings, or how much a town's population has grown. All of these situations rely on the same core skill: finding the percent of a number and then either adding it to or subtracting it from an original amount. This lesson focuses on the two most common patterns you will see in real-world percent problems: percent increase and percent decrease.

A percent increase happens when a quantity grows, such as a price going up or a population expanding. A percent decrease happens when a quantity shrinks, such as a discounted price or a decrease in attendance. Both follow the same two-step process:

1. Find the amount of change: \( \)change\( = \)original amount\( \times \)percent rate\( \).
2. Add the change for an increase, or subtract it for a decrease, to get the new amount.

A jacket originally costs \(\$80\) and is on sale for \(25\%\) off. What is the sale price?

Step 1: Find the amount of change. \(80 \times 0.25 = 20\). The discount is \(\$20\).

Step 2: Subtract the change from the original price, since this is a decrease. \(80 - 20 = 60\).

The sale price is \(\$60\).

A phone case costs \(\$15\) before tax, and the sales tax rate is \(8\%\). What is the total cost?

Step 1: Find the tax amount. \(15 \times 0.08 = 1.20\).

Step 2: Add the tax to the original price, since tax increases the total. \(15 + 1.20 = 16.20\).

The total cost is \(\$16.20\). This is exactly the same two-step logic used to apply percentage in various real-life contexts, whether it is tax, a tip, or a markup.

A town has \(4{,}500\) residents. The population increases by \(6\%\) in one year. How many residents does the town have now?

Step 1: Find the increase. \(4500 \times 0.06 = 270\).

Step 2: Add the increase to the original population. \(4500 + 270 = 4770\).

The town now has \(4{,}770\) residents.

Sometimes a problem gives you the new amount and the percent change, and asks for the original amount. For example, a shirt's price dropped by \(20\%\) to a sale price of \(\$32\). Since the sale price is \(80\%\) of the original price, you can write \(0.80 \times x = 32\), so \(x = 32 \div 0.80 = 40\). The original price was \(\$40\).

Before solving problems like this, it helps to feel comfortable converting between representations using percents, fractions, and decimals, and reviewing how to multiply decimals, since every percent calculation involves multiplying by a decimal form of the rate.

• Read carefully to decide if the quantity is going up (increase) or going down (decrease).
• Identify the original amount and the given percent rate before calculating anything.
• Convert the percent to a decimal by dividing by \(100\) before multiplying.
• Check that your final answer makes sense: an increase should give a larger number, and a decrease should give a smaller one.

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