Question 44·Medium·Percentages
A store increases the regular price of a laptop by . A week later, the store advertises a discount on the new, higher price. Relative to the laptop’s original regular price, what is the overall percent change after both adjustments?
For successive percent changes, avoid adding or subtracting the percentages directly. Instead, translate each change into a multiplier: increasing by means multiply by , and decreasing by means multiply by . Multiply these factors in order to get a single overall multiplier, then compare that number to 1 to see the total percent increase or decrease relative to the original. Using an easy starting value like 100 can also make the arithmetic and interpretation quick on test day.
Hints
Use a variable or an easy number
Let the original price be a simple value, like dollars (or even dollars), and track what happens to it after each percentage change.
Turn each percent change into a multiplier
A 15% increase means multiply by . A 20% discount means you pay 80% of the price, so multiply by . Apply these multipliers in order to your original price.
Compare final price to original price
Once you have an expression for the final price in terms of the original (like some number times ), compare that number to . How far above or below is it, and what percent is that?
Desmos Guide
Compute the overall multiplier
In Desmos, type 1.15*0.8 and look at the numerical result. This value tells you what fraction of the original price the laptop costs after both changes.
Interpret the multiplier as a percent change
Compare the Desmos result to 1. Determine how much less or more than 1 it is, and convert that difference into a percentage to decide whether the overall change is an increase or a decrease and by how many percent.
Step-by-step Explanation
Represent the original price with a variable
Let the laptop’s original regular price be dollars.
A 15% increase means you add 15% of to , which is the same as multiplying by . So, after the 15% increase, the new price is .
Apply the 20% discount to the increased price
A 20% discount means you pay 80% of the price, which is the same as multiplying by .
So, starting from the increased price , the price after the discount is:
Combine the multipliers
Now multiply the two factors:
So the final price is:
This means the laptop now costs times its original price .
Compare the final price to the original price
The original price is , and the final price is .
The change in price is:
This change, , is 8% of the original price .
Because the final price is lower than the original by 8%, the overall percent change is an 8% decrease, which corresponds to choice D.