Percent Applications

💯 Percent Applications

Part 1 of 5 — The Three Pieces of Every Percent Problem

Topics in This Part

🔑 Key Concept: Almost every percent problem — tips, tax, discounts, interest — is the same equation in disguise: $\text{part} = \text{percent} \times \text{whole}$. Master that one relationship in Part 1 and the rest of the lesson is just plugging in.

What "Percent" Really Means

Percent means "per hundred." So $37\%$ literally means $37$ out of $100$:

$37\% = \frac{37}{100} = 0.37$

To use a percent in arithmetic, convert it to a decimal by moving the decimal point two places left (or dividing by $100$):

💡 Heads up: $100\% = 1$ (the whole thing), and percents above $100\%$ are perfectly normal — they just mean "more than the whole." $150\%$ of a number is one-and-a-half times that number.

Concept Check 🎯

The Percent Equation

Every basic percent problem has three pieces, and the equation that links them is:

$\text{part} = \text{percent (as a decimal)} \times \text{whole}$

• The whole is the total you start with (it usually follows the word "of").
• The percent is the rate, written as a decimal.
• The part is the amount you end up with.

Worked Example: Find $30\%$ of $80$

Here the whole is $80$, the percent is $0.30$, and we want the part:

$\text{part} = 0.30 \times 80 = 24$

So $30\%$ of $80$ is $\mathbf{24}$.

🔑 Translation tip: The word "of" means multiply, and "is" means equals. "What is $30\%$ of $80$?" becomes $\text{part} = 0.30 \times 80$.

Find the Part 🧮

Use $\text{part} = \text{percent} \times \text{whole}$. Convert each percent to a decimal first.

1) What is $15\%$ of $60$? 2) What is $25\%$ of $200$? 3) What is $8\%$ of $50$?

Spotting the Pieces in Words

Before you can plug into the equation, you have to label the pieces hiding in a sentence. Two signal words do most of the work:

• The number after "of" is the whole.
• The phrase "what is" marks the part you're solving for.

For "What is $20\%$ of $150$?" the whole is $150$, the percent is $0.20$, and the part is the unknown. Try labeling the next one yourself.

Name the Pieces 🔽

In the question "What is $20\%$ of $150$?", identify each piece.

You've Got the Engine

Every percent application — sales tax, a $20\%$-off sale, a $15\%$ tip, the interest on a savings account — runs on the same engine:

$\text{part} = \text{percent} \times \text{whole}$

So far we've solved for the part. But sometimes you're handed the part and asked for the percent or the whole. That's Part 2.

💯 Percent Applications

Part 2 of 5 — Solving for the Percent or the Whole

🔑 The Idea: The same equation $\text{part} = \text{percent} \times \text{whole}$ can be rearranged to find any missing piece. If you know two of the three, you can always find the third by dividing.

Rearranging the Percent Equation

Start from $\text{part} = \text{percent} \times \text{whole}$ and divide to isolate what you need:

Worked Example: "$18$ is what percent of $72$?"

Here the part is $18$ and the whole is $72$. Use $\text{percent} = \dfrac{\text{part}}{\text{whole}}$:

$\text{percent} = \frac{18}{72} = 0.25 = 25\%$

💡 Don't forget the last step! Dividing gives a decimal ($0.25$). Move the point two places right to report it as a percent ($25\%$).

Choose the Right Move 🔽

Match each question to the operation that solves it.

Worked Example: "$45$ is $30\%$ of what number?"

This time the part is $45$ and the percent is $0.30$; the whole is unknown. Use $\text{whole} = \dfrac{\text{part}}{\text{percent}}$:

$\text{whole} = \frac{45}{0.30} = 150$

✅ Check: $30\%$ of $150$ is $0.30 \times 150 = 45$ ✓

⚠️ Common mistake: When the percent is missing you divide part by whole; when the whole is missing you divide part by percent. Mixing these up is the #1 percent error. Always ask: "Which two pieces do I already have?"

Concept Check 🎯

A Reliable Two-Question Strategy

For any percent problem, ask yourself two quick questions:

1. Which piece is missing — the part, the percent, or the whole?
2. Which formula matches that missing piece (multiply, or divide by the whole, or divide by the percent)?

That's it. The drill below mixes "find the percent" and "find the whole" questions on purpose, so you get practice telling them apart.

Find the Missing Piece 🧮

Decide whether the percent or the whole is missing, then solve. Enter percents as a number without the % sign (e.g. type 20 for $20\%$).

1) $30$ is what percent of $120$? (enter the percent) 2) $7$ is $35\%$ of what number? 3) $15$ is what percent of $25$? (enter the percent)

💯 Percent Applications

Part 3 of 5 — Tax, Tips & Discounts

🔑 Real-World Payoff: Sales tax, restaurant tips, and store discounts are all "percent of the price" problems. The trick is knowing whether you add the percent (tax, tip) or subtract it (discount).

Tax and Tips: Adding a Percent

A tax or tip is a part you add on top of the original price.

Worked Example: $50 dinner with an $18\%$ tip

Step 1 — Find the tip (the part): $0.18 \times 50 = 9$

Step 2 — Add it to the bill: $50 + 9 = 59$

The total is $59.

The One-Step Shortcut

Adding $18\%$ to a price is the same as taking $118\%$ of it:

$1.18 \times 50 = 59$

💡 Why $1.18$? You keep $100\%$ of the price plus the $18\%$ tip, and $100\% + 18\% = 118\% = 1.18$. The same logic gives $1.07$ for $7\%$ tax, $1.20$ for a $20\%$ tip, and so on.

Tax & Tip Practice 🧮

Find each total (price plus the added percent). Enter just the number of dollars.

1) A $40 shirt with $5\%$ sales tax. Total $= ?$ 2) A $30 meal with a $20\%$ tip. Total $= ?$ 3) A $200 purchase with $8\%$ tax. Total $= ?$

Discounts: Subtracting a Percent

A discount is a part you subtract from the original price.

Worked Example: $40 jacket, $25\%$ off

Step 1 — Find the discount (the part): $0.25 \times 40 = 10$

Step 2 — Subtract it from the price: $40 - 10 = 30$

The sale price is $30.

The One-Step Shortcut

If you take $25\%$ off, you pay the remaining $75\%$:

$0.75 \times 40 = 30$

🔑 The complement rule: "$25\%$ off" means "pay $75\%$," because $100\% - 25\% = 75\%$. Likewise $30\%$ off means pay $0.70$, and $10\%$ off means pay $0.90$.

Concept Check 🎯

Add or Subtract? Read the Word

The single most important step in these problems is deciding the direction:

The next drill deliberately mixes discounts and tips. Pause on each one and decide the direction before you multiply.

Add or Subtract? 🧮

Read carefully — decide whether the percent is added or subtracted, then find the final amount in dollars.

1) A $60 game, $20\%$ off. Sale price $= ?$ 2) A $25 lunch with a $16\%$ tip. Total $= ?$ 3) A $150 coat, $40\%$ off. Sale price $= ?$

💯 Percent Applications

Part 4 of 5 — Percent Change & Simple Interest

🔑 The Idea: "Percent increase" and "percent decrease" measure how much a value grew or shrank relative to where it started. The starting value is always the whole you divide by.

Percent Increase and Decrease

To find a percent change, compare the amount of change to the original value:

$\text{percent change} = \frac{\text{new} - \text{original}}{\text{original}}$

A positive result is an increase; a negative result is a decrease.

Worked Example: A price rises from $50 to $65

Step 1 — Find the change: $65 - 50 = 15$

Step 2 — Divide by the original: $\frac{15}{50} = 0.30 = 30\%$

That's a $\mathbf{30\%}$ increase.

⚠️ Always divide by the ORIGINAL, never the new value. Going $50 \to 65$ is a $30\%$ increase, but going $65 \to 50$ is a decrease of $\frac{15}{65} \approx 23\%$ — the same gap, a different percent, because the starting point changed.

Percent Change Practice 🧮

Use $\dfrac{\text{new} - \text{original}}{\text{original}}$. Enter the percent as a number (no % sign).

1) A plant grows from $80$ cm to $100$ cm. Percent increase $= ?$ 2) A price drops from $100$ to $80$. Percent decrease $= ?$ 3) A score rises from $40$ to $50$. Percent increase $= ?$

Simple Interest

When you save (or borrow) money, interest is a percent of the original amount, called the principal. Simple interest is paid on the principal each year:

$I = P \times r \times t$

where $P$ = principal, $r$ = annual rate (as a decimal), and $t$ = time in years.

Worked Example: $500 at $4\%$ for $3$ years

$I = 500 \times 0.04 \times 3 = 60$

You earn $60 in interest, so the account holds $500 + 60 = 560$, i.e. $560.

💡 The interest each year is $500 \times 0.04 = 20$, i.e. $20, and over $3$ years that's $3 \times 20 = 60$, i.e. $60. The formula just bundles those two multiplications together.

Concept Check 🎯

Interest vs. Total Balance

Watch the wording carefully — two different things get asked:

• Interest ($I$) is only the money earned: $I = Prt$.
• Total balance is the principal plus the interest: $P + I$.

For $500 at $4\%$ for $3$ years, the interest is $60 but the balance is $560. The next drill asks for one or the other, so read each question to the end.

Simple Interest Practice 🧮

Use $I = P \times r \times t$. Enter dollar amounts as a number.

1) $2,000 at $3\%$ for $4$ years. Interest $I = ?$ 2) $800 at $10\%$ for $2$ years. Interest $I = ?$ 3) $1,000 at $5\%$ for $1$ year — what is the total balance (principal $+$ interest)?

💯 Percent Applications

Part 5 of 5 — Reverse Percents & Mastery Check

You can now find a part, a percent, or a whole; apply tax, tips, and discounts; and compute percent change and interest. The last skill is the trickiest: working backward from a price that already includes a percent.

Reverse Percents: Finding the Original

When a price already has a discount baked in, you can't just add the percent back — you have to undo the multiplication.

Worked Example: After $20\%$ off, a coat costs $48. What was the original price?

"$20\%$ off" means the $48 is $80\%$ of the original. So:

$0.80 \times (\text{original}) = 48$

Divide both sides by $0.80$:

$\text{original} = \frac{48}{0.80} = 60$

The original price was $60.

⚠️ Why not just add $20\%$ to $48? That gives $48 \times 1.20 = 57.60$, i.e. $57.60 — wrong! Adding $20\%$ of $48 isn't the same as undoing $20\%$ of the (larger) original. Always divide by the complement, here $0.80$.

✅ Check: $20\%$ off $60 is $0.20 \times 60 = 12$, i.e. $12, and $60 - 12 = 48$, i.e. $48 ✓

Work Backward 🧮

Each price already includes the change. Find the original price in dollars by dividing.

1) After $25\%$ off, a bag costs $45. Original $= ?$ (divide by $0.75$) 2) After a $10\%$ tax, a tablet costs $220. Pre-tax price $= ?$ (divide by $1.10$) 3) After $40\%$ off, a chair costs $90. Original $= ?$

Quick Reference

🔑 Convert percents to decimals first, and when finding a percent, convert the decimal answer back to a percent at the end.

Mixed Practice 🎯

Ready for the Exit Quiz

You've covered the whole toolkit: finding any of the three pieces, applying tax and tips, taking discounts, measuring percent change, computing simple interest, and reversing a percent to recover the original.

The three questions below pull from across the lesson. Convert each percent to a decimal first, identify which piece is missing, and you'll be set. Good luck!

Exit Quiz ✅

Answer all three to finish the lesson.