Percent Increase and Decrease — Solve percent increase or decrease problems. - SHSAT Math

SECTION 1

Where Did Percents Come From?

People have compared amounts for thousands of years. Ancient traders needed a fair way to describe how prices changed. The word percent comes from the Latin phrase per centum, which means "out of one hundred." Using 100 as a base made it easy for merchants to compare deals.

Over time, percent calculations became essential in everyday life. Today you see them in store discounts, test scores, sports stats, and even the SHSAT! Understanding percent increase and percent decrease helps you describe how things change.

~3000 BCE • Ancient Tax Collectors

Egyptians and Babylonians used fractions to figure taxes and trade amounts, often comparing parts out of fixed totals.

~100 CE • Roman "Per Centum"

Roman Emperor Augustus taxed goods at rates like 1/100 of the sale price. The phrase per centum (per hundred) became common.

1400s • The % Symbol Appears

Italian merchants shortened "per cento" into a symbol that eventually became the % sign we use today.

Modern Day • Percents Everywhere

Sale discounts, interest rates, test scores, and standardized exams like the SHSAT all rely on percent calculations.

So here is the big question this lesson answers: if you know the original value and the new value, how do you find the percent change? And if you know the percent change, how do you find the new value? Let's find out.

SECTION 2

Core Ideas Behind Percent Change

Before jumping into formulas, let's nail down a few key ideas. These four principles are the building blocks for every percent increase or decrease problem you will see on the SHSAT.

Original (Starting) Value

This is the value you start with before the change happens. It is the number in the denominator of the percent-change fraction.

Amount of Change

Subtract the smaller value from the larger value. This tells you how much something went up or down. It goes in the numerator.

Direction Matters

If the new value is bigger, it is a percent increase. If the new value is smaller, it is a percent decrease.

Always Divide by the Original

The percent change is always relative to the original value, not the new value. This is the most common mistake students make!

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KEY TAKEAWAY

Think of percent change like comparing a slice of pizza to the whole original pizza. The "slice" is how much changed. The "whole pizza" is always the starting amount. You figure out what fraction that slice is, then express it out of 100.

SECTION 3

Seeing Percent Change

A picture can make percent change click instantly. The bar diagram below shows what happens when a value of 200 increases by 25% and when it decreases by 25%. Notice that the amount of change (50) is the same in both cases, but the direction is different.

The green section shows the 25% increase (adding 50 to 200). The red dashed section shows the 25% decrease (removing 50 from 200). In both cases, the change of 50 is divided by the original value of 200.

Look at the diagram above carefully. The original bar (200) is the same for both. The green bar extends to the right for an increase. The dashed red bar shows the portion that was removed for a decrease. The key lesson: the original is your reference point every single time.

SECTION 4

The Formulas You Need

There are two main tasks on the SHSAT. First, you might need to find the percent change between two values. Second, you might need to find the new value after a given percent change. Here are the formulas for each.

FINDING PERCENT CHANGE

Percent Change = (Amount of Change ÷ Original) × 100

Amount of Change = |New Value − Original Value|. The vertical bars mean "take the positive version." Original = the starting value (before the change). Multiply by 100 to turn the decimal into a percent.

FINDING THE NEW VALUE (INCREASE)

New Value = Original × (1 + Percent ÷ 100)

Example: a 20% increase means you multiply the original by (1 + 0.20) = 1.20. The "1" keeps the original, and the "0.20" adds the increase.

FINDING THE NEW VALUE (DECREASE)

New Value = Original × (1 − Percent ÷ 100)

Example: a 30% decrease means you multiply the original by (1 − 0.30) = 0.70. You keep 70% of the original.

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SHSAT Shortcut

Instead of finding the amount of change first, you can multiply the original by a single decimal. For a 15% increase, multiply by 1.15. For a 15% decrease, multiply by 0.85. This one-step method saves time on the test!

SECTION 5

Step-by-Step Breakdown

Let's walk through the decision process. The flowchart below shows you exactly what to do when you see a percent-change problem. Follow the arrows!

This flowchart shows the two main paths. On the left path, you find the percent change between two known values. On the right path, you find a new value after a given percent change.

Problem Type	What You Know	What You Find	Key Formula
Find the % change	Original and New values	The percent	(Change ÷ Original) × 100
Find the new value (increase)	Original and percent increase	The new value	Original × (1 + r)
Find the new value (decrease)	Original and percent decrease	The new value	Original × (1 − r)
Find the original	New value and percent change	The original	New Value ÷ (1 ± r)

Summary of percent-change problem types (r = percent ÷ 100)

SECTION 6

Worked Examples

Let's work through two complete examples, one for percent increase and one for percent decrease.

Example 1: A store raised the price of a jacket from $80 to $100. What is the percent increase?

1
Step 1 — Identify the Original and New values

The original price is $80 (before the change). The new price is $100 (after the change).
2
Step 2 — Find the Amount of Change

Amount of Change = New − Original = $100 − $80 = $20. Since the new value is bigger, this is an increase.

Amount of Change = $20
3
Step 3 — Divide by the Original

$20 ÷ $80 = 0.25. This is the decimal form of the percent change.

0.25
4
Step 4 — Multiply by 100

0.25 × 100 = 25. The price increased by 25%.

25% increase

Example 2: A town's population was 6,000. After a 15% decrease, what is the new population?

1
Step 1 — Identify the Original and the Percent

The original population is 6,000. The percent decrease is 15%.
2
Step 2 — Convert the Percent to a Decimal

15% ÷ 100 = 0.15.
3
Step 3 — Subtract from 1 (because it's a decrease)

1 − 0.15 = 0.85. This means the town keeps 85% of its population.

Multiplier = 0.85
4
Step 4 — Multiply the Original by the Multiplier

6,000 × 0.85 = 5,100. The new population is 5,100.

New population = 5,100

SECTION 7

Common Mistakes & How to Avoid Them

Percent-change problems are not hard once you know the steps. But there are a few traps that catch students on the SHSAT every year. Here's what to watch out for.

Common Mistake	Why It's Wrong	Correct Approach
Dividing by the NEW value instead of the original	Percent change is always relative to where you started, not where you ended up.	Always put the original (starting) value in the denominator.
Forgetting to multiply by 100	You get 0.25 instead of 25%. The answer choices on the SHSAT will be in percent form.	After dividing, multiply by 100 to convert the decimal to a percent.
Confusing percent increase with percent decrease	Getting the direction wrong changes your answer completely.	Check: Is the new value bigger (increase) or smaller (decrease)?
Thinking a 50% increase then a 50% decrease returns to the original	A 50% increase on 100 gives 150. A 50% decrease on 150 gives 75, not 100!	The original changes after the first step, so the second percent applies to a different base.

Avoid these traps on the SHSAT

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KEY TAKEAWAY

Imagine you score 80 on a test, then improve to 100. Your friend says you improved "by 20 points." That's the amount of change. But percent change puts it in perspective: 20 out of 80 is a 25% increase. If you went from 200 to 220, that's also 20 points — but only a 10% increase. The starting value determines how big the percent change is.

SECTION 8

Connecting to Harder Problems

Once you master basic percent increase and decrease, the SHSAT can test you with trickier versions. Here is a quick look at how the concept connects to more advanced ideas you might see.

Basic Version	Advanced Version
One percent change on one value	Successive (back-to-back) percent changes — you must apply each step to the NEW value, not the original
Finding the percent change given both values	Finding the original value when you know the new value and the percent change (work backwards)
Whole-number percents like 25% or 40%	Fractional or decimal percents like 12.5% or 33⅓%
Simple percent of a number	Percent change combined with ratios or proportions in multi-step word problems

How today's concept scales to harder SHSAT questions

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Successive Percent Changes — Quick Example

A $200 item is marked up 10%, then discounted 10%. Is the final price $200? No! After a 10% increase: $200 × 1.10 = $220. After a 10% decrease: $220 × 0.90 = $198. The final price is $2 less than the original because the decrease acts on the larger amount ($220), not the original ($200).

For now, focus on the basic one-step problems. Once those feel easy, try the successive percent change problems — they show up on harder SHSAT questions and in high school algebra.

SECTION 9

Practice Problems

Try these five problems on your own. Work through each step before checking the answer. The problems get harder as you go!

Question 1: CONCEPTUAL

A shirt's price goes from $40 to $50. Is this a percent increase or a percent decrease? What number goes in the denominator when you calculate the percent change?

Show answer

This is a percent increase because the new price ($50) is greater than the original price ($40). The denominator is always the original value, which is $40. You would calculate ($10 ÷ $40) × 100 to find the percent increase. The answer is 25%.

Question 2: BASIC CALCULATION

A video game was originally $60 and is now on sale for $45. What is the percent decrease?

Show answer

First, find the amount of change: $60 − $45 = $15. Next, divide by the original: $15 ÷ $60 = 0.25. Finally, multiply by 100: 0.25 × 100 = 25%. The price decreased by 25%. Using the formula: Percent Change = (Amount of Change ÷ Original) × 100 = (15 ÷ 60) × 100 = 25%.

Question 3: INTERMEDIATE

A school had 480 students last year. This year enrollment increased by 15%. How many students does the school have now?

Show answer

Convert the percent to a decimal: 15% ÷ 100 = 0.15. Since it is an increase, add to 1: 1 + 0.15 = 1.15. Multiply: 480 × 1.15 = 552 students. You could also find 15% of 480 first: 480 × 0.15 = 72. Then add: 480 + 72 = 552. Both methods give the same answer.

Question 4: APPLIED

Maria bought a bicycle for $250. After one year, she sold it for $180. What was the percent decrease in the bicycle's value? Round to the nearest whole percent.

Show answer

Amount of change = $250 − $180 = $70. Divide by the original price: $70 ÷ $250 = 0.28. Multiply by 100: 0.28 × 100 = 28%. The bicycle lost 28% of its value over the year. Remember to divide by the original ($250), not the selling price ($180). If you divided by 180, you would incorrectly get about 38.9%.

Question 5: CRITICAL THINKING

A store increases the price of a $120 item by 25%, then later decreases the new price by 20%. What is the final price? Is it higher or lower than the original $120? By what percent did the price change overall?

Show answer

First increase: $120 × 1.25 = $150. Then decrease: $150 × 0.80 = $120. In this case, the final price is exactly the same as the original! The overall percent change is 0%. This is a special case — it does NOT always work out this way. A 25% increase followed by a 25% decrease would give $120 × 1.25 × 0.75 = $112.50, which is LESS than the original. The key insight is that the percent decrease applies to the new (larger) amount, not the original.

SUMMARY

Lesson Summary

Summary

Percent change measures how much a value has grown or shrunk compared to its original (starting) amount. To find the percent change, calculate the amount of change (|New − Original|), divide by the original, and multiply by 100. If the new value is bigger, it is an increase; if smaller, it is a decrease.

To find a new value after a percent change, use the multiplier shortcut: multiply the original by (1 + r) for an increase or (1 − r) for a decrease, where r is the percent as a decimal. Always remember that the original value goes in the denominator — never divide by the new value. Watch out for successive percent changes, where each step uses a new base. With these tools, you are ready to tackle any percent-change problem on the SHSAT!