Understanding Percentages

Arithmetic is the math of counting, sharing, and comparing amounts. It covers the four operations plus fractions, decimals, percents, and ratios.

Understanding Percentages: the central idea

A percentage is a ratio per hundred. Every percent problem depends on identifying the part, the rate, and the correct whole or original value.

Words you need

Percent

A percent is a count of parts out of 100, so 45 percent means 45 out of every 100.

Base

The base, also called the whole, is the amount the percent is taken of, and it is usually the number right after the word of.

Percent change

Percent change is the difference between the new and the original amount, divided by the original amount, then written as a percent.

Percentage point

A percentage point is the plain gap between two percents, so a rate moving from 4 percent to 6 percent rises 2 percentage points, which is a 50 percent increase in the rate itself.

Discount

A discount is the amount taken off an original price, and paying after a 25 percent discount is the same as paying 75 percent of the price.

Simple interest

Simple interest is interest found with $I=P\times r\times t$, so 500 dollars at 4 percent for 3 years earns $500\times0.04\times3=60$ dollars.

What to know before this lesson

Know decimal place value, fractions out of one hundred, multiplication, and the meaning of a reference whole.

If one of those prerequisites is uncertain, use the Arithmetic subject guide to locate the earlier concept before memorizing a procedure.

Percentages: a worked example

Every step, with the arithmetic

1. Step 1 - Turn the discount percent into a decimal$25\%\div100=0.25$
2. Step 2 - Find the discount on a 60-dollar jacket$0.25\times60=15$ dollars off
3. Step 3 - Subtract to get the sale price$60-15=45$ dollars
4. Step 4 - Confirm with the one-step versionPaying after 25 percent off means paying 75 percent: $0.75\times60=45$
5. Step 5 - Apply the 8 percent tax to the sale price, not the old price$0.08\times45=3.60$ dollars of tax
6. Step 6 - Add the tax on$45+3.60=48.60$ dollars
7. Step 7 - Do the whole thing with multipliers$60\times0.75\times1.08=48.60$ dollars

$18\%$ of $250$ is $0.18(250)=45$. The word “of” signals multiplication and $250$ is the reference whole.

Percent, decimal, fraction, and the multiplier that does the work

The step-by-step method for percentages

1. Translate the percent to a decimal by dividing by $100$.
2. Name which quantity is the whole, then use part = rate × whole or rearrange that relationship.
3. For percent change, divide the change by the original value and interpret increase or decrease.

How to check your answer

Estimate $20\%$ of $250$ as $50$, so an exact result of $45$ has the expected size.

Use rounding, inverse operations, and benchmark fractions such as one-half or one-quarter to verify the sign and size of the answer.

A mistake that changes the mathematics

A $20\%$ increase followed by a $20\%$ decrease does not return to the start because the second percent uses a different base.

Where you will actually use this

Leaving a tip

An 18 percent tip on a 42-dollar meal is $0.18\times42=7.56$ dollars, so the total is 49.56 dollars. A fast mental check: 10 percent is 4.20 and half of that is 2.10, and $4.20+2.10+1.26$ lands in the same place.

Turning a test score into a grade

Getting 43 questions right out of 50 is $\tfrac{43}{50}=0.86$, which is 86 percent. The denominator is the whole, so the same 43 correct answers out of 60 questions would only be $43\div60\approx71.7$ percent.

Earning interest on savings

Putting 2,000 dollars in an account paying 5 percent simple interest for 2 years earns $2000\times0.05\times2=200$ dollars, leaving a balance of 2,200 dollars.

How percentages connects to the rest of arithmetic

• Fractions — Every percent is a fraction over 100, so knowing that $\tfrac14=25$ percent lets you swap to whichever form makes the arithmetic easier.
• Ratios and proportions — Percent problems can be set up as the proportion part over whole equals percent over 100, and cross multiplying solves for whichever piece is missing.
• Exponentials and logarithms — Repeating a percent increase is exponential growth, since applying a multiplier of 1.05 for $n$ years is the same as multiplying by $1.05^n$.

Try a transfer problem

An item rises from $80$ to $92$ and later falls to $84$. Calculate each percent change using its own original value.

Show the worked answer

First change: the item goes from 80 to 92, an increase of 12. Divide by the original 80: $12\div80=0.15$, a 15 percent increase. Second change: the item goes from 92 to 84, a drop of 8. This time the original value is 92, so $8\div92\approx0.087$, about an 8.7 percent decrease. Overall, the item moved from 80 to 84, an increase of $4\div80=0.05$, or 5 percent. The two percents do not simply subtract to 6.3 percent because each one is measured against a different base, 80 for the rise and 92 for the fall. With multipliers: $80\times1.15\times0.913\approx84$.

Once you have an answer, check it with the percent calculator. Work the problem yourself first: the calculator confirms the arithmetic, but choosing the method is the part that transfers to the next question.

Work without copying the example. When finished, use the relevant focused calculator or formula reference to check the setup and result, then correct the first line where your reasoning changed. When the method feels reliable, move to arithmetic practice questions.

Questions about percentages

Why does a 20 percent increase followed by a 20 percent decrease not return to the start?

Because the second percent is taken from a bigger number. Starting at 100, a 20 percent increase gives 120, and 20 percent of 120 is 24, so the drop lands at 96, not 100. In multipliers, $1.20\times0.80=0.96$, a 4 percent net loss no matter what the starting amount was.

How do I find the original price when I only know the sale price?

Divide instead of multiply. If a jacket costs 45 dollars after 25 percent off, then 45 is 75 percent of the original, so the original is $45\div0.75=60$ dollars. Taking 25 percent of 45 and adding it back gives 56.25, which is wrong, because the discount was never based on 45.

What is the difference between percent and percentage points?

Percentage points measure the plain gap between two percents; percent measures relative change. An interest rate moving from 4 percent to 6 percent rises 2 percentage points, but that is a 50 percent increase in the rate because $2\div4=0.5$. News stories mix these up often.

Can a percent be larger than 100?

Yes, whenever the part is bigger than the whole it is compared to. A price that goes from 20 dollars to 50 dollars rises by 30 dollars, and $30\div20=1.5$, a 150 percent increase. Likewise 150 percent of 40 is $1.5\times40=60$.