Lesson 67: Percent of Change

New Concept

Percents can describe a change, like an increase or decrease. We use a special three-row ratio table (Original, Change, New) to solve these problems.

What’s next

You're just getting started! Next, you’ll tackle worked examples using this ratio table to find sale prices, original costs, and the exact amount of a change.

Property

To find the original price after a discount, use a proportion comparing the original percent (100%) to the new percent (100% - discount %).

Examples

• A jacket bought at a 40% off sale cost 48 dollars. What was the original price?
  $$\frac{100}{60} = \frac{x}{48} \rightarrow 60x = 4800 \rightarrow x = 80$$
  . The original price was 80 dollars.
• A video game is on sale for 25% off and costs 45 dollars. Find its original price.
  $$\frac{100}{75} = \frac{x}{45} \rightarrow 75x = 4500 \rightarrow x = 60$$
  . The original price was 60 dollars.

Explanation

When an item is discounted, its sale price is a new, smaller percentage of the original. The original is always 100%. If you get 30% off, you pay 70%. We use this simple logic to create a proportion that pits the original percent against the new percent to find the starting price, our mystery number!

Property

To find the specific amount of change from an increase, use a proportion comparing the percent increase to the new total percent (100% + increase %).

Examples

• A collectible's value rose 10% in one year to 550 dollars. By how many dollars did its value increase?
  $$\frac{10}{110} = \frac{c}{550} \rightarrow 110c = 5500 \rightarrow c = 50$$
  . The value increased by 50 dollars.
• A town's population grew by 30% to 13,000 people. How many new people moved to the town?
  $$\frac{30}{130} = \frac{c}{13000} \rightarrow 130c = 390000 \rightarrow c = 3000$$
  . The population increased by 3000 people.

Explanation

When a value goes up, the new amount is more than 100% of the original. If a price rises by 20%, the new price is 120% of the old one. We can then set up a proportion comparing the percent increase to the total new percent, which allows us to find the exact dollar amount of that increase.

Property

A ratio table with three rows helps organize percent problems: one for the Original (100%), one for the Change (Increase + or Decrease –), and one for the New total.

Examples

• For a 20% discount on a 50 dollar item, the New % is $100\% - 20\% = 80\%$. To find the sale price:
  $$\frac{80}{100} = \frac{x}{50} \rightarrow 100x = 4000 \rightarrow x=40$$
  . The sale price is 40 dollars.
• A store marks up a 30 dollar item by 60%. The New % is $100\% + 60\% = 160\%$. To find the store price:
  $$\frac{160}{100} = \frac{x}{30} \rightarrow 100x = 4800 \rightarrow x=48$$
  . The store price is 48 dollars.

Explanation

This awesome table keeps our numbers organized! The 'Original' is always your 100% baseline. Just add the percent change for an increase or subtract it for a decrease to find the 'New' percentage. This structure lets you create a simple proportion to solve for any missing piece, whether it’s the old price or the change amount.