# Lesson 8: Percent Increase and Decrease with Equations

Let’s use equations to represent increases and decreases.

## 8.1: From 100 to 106

How do you get from one number to the next using multiplication or division?

• From 100 to 106
• From 100 to 90
• From 90 to 100
• From 106 to 100

## 8.2: Interest and Depreciation

1. Money in a particular savings account increases by about 6% after a year. How much money will be in the account after one year if the initial amount is \$100? \$50? \$200? \$125? $x$ dollars? If you get stuck, consider using diagrams or a table to organize your work.
2. The value of a new car decreases by about 15% in the first year. How much will a car be worth after one year if its initial value was \$1,000? \$5,000? \$5,020?$x$dollars? If you get stuck, consider using diagrams or a table to organize your work. ## 8.3: Matching Equations Match an equation to each of these situations. Be prepared to share your reasoning. 1. The water level in a reservoir is now 52 meters. If this was a 23% increase, what was the initial depth? 2. The snow is now 52 inches deep. If this was a 77% decrease, what was the initial depth?$0.23x = 520.77x = 521.23x = 521.77x = 52$## 8.4: Representing Percent Increase and Decrease: Equations 1. The gas tank in dad’s car holds 12 gallons. The gas tank in mom’s truck holds 50% more than that. How much gas does the truck’s tank hold? Explain why this situation can be represented by the equation$(1.5) \boldcdot 12 = t$. Make sure that you explain what$t$represents. 2. Write an equation to represent each of the following situations. 1. A movie theater decreased the size of its popcorn bags by 20%. If the old bags held 15 cups of popcorn, how much do the new bags hold? 2. After a 25% discount, the price of a T-shirt was \$12. What was the price before the discount?
3. Compared to last year, the population of Boom Town has increased by 25%.The population is now 6,600. What was the population last year?

## Summary

We can use equations to express percent increase and percent decrease. For example, if $y$ is 15% more than $x$,

we can represent this using any of these equations:

$y = x + 0.15x$

$y = (1 + 0.15)x$

$y = 1.15x$

So if someone makes an investment of $x$ dollars, and its value increases by 15% to \$1250, then we can write and solve the equation$1.15x =1250$to find the value of the initial investment. Here is another example: if$a$is 7% less than$b$, we can represent this using any of these equations:$a = b - 0.07ba = (1-0.07)ba = 0.93b$So if the amount of water in a tank decreased 7% from its starting value of$b$to its ending value of 348 gallons, then you can write$0.93b = 348\$.

Often, an equation is the most efficient way to solve a problem involving percent increase or percent decrease.