Percent Problems

💯 Percent Problems

Part 1 of 5 — What a Percent Really Is

Topics in This Part

🔑 Key Concept: A percent is just a fraction with a hidden denominator of $100$. The word percent literally means "per hundred" — cent is the Latin root for hundred (think century, centimeter). Master that one idea and every percent problem becomes a fraction problem you already know how to do.

Percent Means "Out of 100"

When you see $37\%$, read it as "37 out of 100":

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

Imagine a $10 \times 10$ grid with $100$ little squares. Shading $37$ of them shows you exactly what $37\%$ looks like.

💡 Watch the edges: $100\%$ is the whole thing. More than $100\%$ (like $150\%$) means more than the whole, and less than $1\%$ means a tiny sliver.

Concept Check 🎯

Percent → Decimal: Move the Point Left

To turn a percent into a decimal, divide by 100 — which just slides the decimal point two places to the left and drops the $\%$ sign.

$48\% = 48.0 \to 0.48 \qquad 7\% = 07.0 \to 0.07 \qquad 250\% = 250.0 \to 2.50$

⚠️ Don't forget the leading zero. $7\%$ becomes $0.07$, not $0.7$. You must fill the empty place with a zero: $0\underline{0}7 \to 0.07$.

Decimal → Percent: Move the Point Right

Going the other way, multiply by 100 — slide the point two places to the right and add a $\%$ sign.

$0.92 \to 92\% \qquad 0.05 \to 5\% \qquad 1.3 \to 130\%$

Convert Percents and Decimals 🧮

Fill in each blank.

1) $84\% = \,?$ (as a decimal) 2) $3\% = \,?$ (as a decimal) 3) $0.6 = \,?$ (as a percent — enter just the number, e.g. type 60 for $60\%$)

Percent → Fraction (and Simplify)

Write the percent over $100$, then reduce:

$40\% = \frac{40}{100} = \frac{2}{5} \qquad 75\% = \frac{75}{100} = \frac{3}{4}$

The "Benchmark" Percents Worth Memorizing

These come up constantly — knowing them by heart makes mental math fast.

🔑 Tip: $\frac{1}{4}$ of something is the same as $25\%$ of it. Spotting these matches lets you skip the calculator.

Match the Forms 🔽

Pick the equivalent value for each.

💯 Percent Problems

Part 2 of 5 — Finding the Percent OF a Number

🔑 The Big Idea: In a percent problem, the word "of" means multiply. To find a percent of a number, change the percent to a decimal and multiply.

"Of" Means Multiply

To find a percent of a number, follow two steps:

1. Convert the percent to a decimal (move the point two places left).
2. Multiply by the number.

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

Worked Example: What is $30\%$ of $80$?

$30\% \text{ of } 80 = 0.30 \times 80 = 24$

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

Worked Example: What is $5\%$ of $200$?

$5\% \text{ of } 200 = 0.05 \times 200 = 10$

⚠️ Sanity check your answer. Since $30\%$ is less than half, $30\%$ of $80$ should be less than $40$ — and $24$ is. If you ever get a part bigger than the whole (with a percent under $100\%$), you slipped a decimal point.

The 10% Trick (Fast Mental Math)

$10\%$ of any number is found by moving its decimal point one place left — just divide by $10$.

$10\% \text{ of } 60 = 6 \qquad 10\% \text{ of } 45 = 4.5$

Once you know $10\%$, you can build other percents:

Example: $20\%$ of $50$ (mental)

$10\%$ of $50$ is $5$, so $20\%$ is $5 \times 2 = \mathbf{10}$.

💡 This is exactly how people figure out a tip in their head — find $10\%$, then scale it.

Concept Check 🎯

The Two-Step Recap

Whenever you need a percent of a number, run the same two moves:

$\underbrace{\text{percent}}_{\text{move point 2 left}} \;\longrightarrow\; \text{decimal} \;\times\; \text{whole} \;=\; \text{part}$

💡 Keep the table's pattern in mind as you work the drill below — decimal first, then multiply.

Find the Part 🧮

Compute each. (Convert to a decimal, then multiply.)

1) $25\%$ of $60 = \,?$ 2) $10\%$ of $130 = \,?$ 3) $150\%$ of $40 = \,?$

Estimate First, Compute Second

Good math students predict the size of an answer before grinding the numbers. Anchor every percent to these landmarks:

💡 Estimating first turns a calculation into a quick check. If $50\%$ of $88$ "should be about half," then $44$ feels right and $176$ obviously doesn't.

Reason Before You Compute 🔽

Without a calculator, decide what's true.

💯 Percent Problems

Part 3 of 5 — The Three Kinds of Percent Questions

🔑 One Setup, Three Questions: Every percent problem fits the relationship below. Whichever piece is missing, that's what you solve for.

$\frac{\text{part}}{\text{whole}} = \frac{\text{percent}}{100}$

The Percent Proportion

Line up the numbers in this proportion and cross-multiply to solve for the unknown:

$\frac{\text{part}}{\text{whole}} = \frac{\%}{100}$

• The whole comes right after the word "of."
• The part is usually right next to the word "is."
• The percent is the number with the $\%$ sign.

The Three Questions

💡 Part 2 already covered "find the part." Now we tackle the other two.

Finding the Percent: "_ is what percent of _?"

Question: $12$ is what percent of $48$?

Use $\dfrac{\text{part}}{\text{whole}}$, then convert to a percent:

$\frac{12}{48} = 0.25 = 25\%$

So $12$ is $\mathbf{25\%}$ of $48$.

Worked Example: $9$ is what percent of $20$?

$\frac{9}{20} = 0.45 = 45\%$

⚠️ Order matters! It's $\frac{\text{part}}{\text{whole}}$, not whole over part. The "part" (the smaller piece you're measuring) goes on top. Flipping the fraction is the #1 mistake here.

Concept Check 🎯

Finding the Whole: "_ is _% of what?"

This is the trickiest one because the whole is missing. Use the proportion and solve.

Question: $18$ is $30\%$ of what number?

$\frac{18}{\text{whole}} = \frac{30}{100}$

Cross-multiply: $30 \times \text{whole} = 18 \times 100 = 1800$, so

$\text{whole} = \frac{1800}{30} = 60$

So $18$ is $30\%$ of $\mathbf{60}$.

💡 Shortcut: "part is p% of what" means $\text{whole} = \dfrac{\text{part}}{\text{decimal percent}}$. Here $\dfrac{18}{0.30} = 60$. Same answer, fewer steps.

⚠️ When the whole is missing, you divide the part by the decimal — you do not multiply. Multiplying would shrink your answer when it needs to grow.

Find the Missing Piece 🧮

1) $14$ is what percent of $56$? (enter just the number, e.g. 25 for $25\%$) 2) $21$ is $30\%$ of what number? 3) $6$ is $50\%$ of what number?

Read the Sentence to Pick the Method

Before computing anything, decide which of the three numbers is missing. The wording tells you:

🔑 Classify first, calculate second. Naming the question type stops you from multiplying when you should divide.

Which Question Is It? 🔽

For each sentence, choose what's being asked.

💯 Percent Problems

Part 4 of 5 — Percent Change: Tips, Tax, Discounts & Markups

🔑 Real-World Percents: Stores, restaurants, and banks all use percents to change a starting amount. The trick is knowing whether to add (tax, tip, markup) or subtract (sale, discount).

Discounts (Subtract) and Tax/Tip (Add)

Step 1: Find the percent of the original amount (just like Part 2). Step 2: Subtract it for a discount, or add it for tax or a tip.

Worked Example: A $40 shirt is $25\%$ off.

Discount amount: $0.25 \times 40 = 10$, i.e. $10. Sale price: $40 - 10 = 30$, so $30.

Worked Example: A $50 meal with a $20\%$ tip.

Tip amount: $0.20 \times 50 = 10$, i.e. $10. Total paid: $50 + 10 = 60$, so $60.

💡 One-step shortcut. A $25\%$ discount means you pay $75\%$: $0.75 \times 40 = 30$, giving $30. Adding a $20\%$ tip means you pay $120\%$: $1.20 \times 50 = 60$, giving $60. Same answers, one multiplication.

Concept Check 🎯

Percent Increase and Decrease

When an amount goes up or down, the percent change measures how big the change is compared to where it started:

$\text{percent change} = \frac{\text{amount of change}}{\text{original amount}} \times 100\%$

Worked Example: A price rises from $20 to $25.

Change $= 25 - 20 = 5$. Compare to the original ($20$):

$\frac{5}{20} = 0.25 = 25\% \text{ increase}$

Worked Example: A class shrinks from $30$ students to $24$.

Change $= 30 - 24 = 6$:

$\frac{6}{30} = 0.20 = 20\% \text{ decrease}$

⚠️ Always divide by the ORIGINAL amount, not the new one. The original is your starting point — that's what the change is being measured against.

Tips, Tax & Change 🧮

1) A $60 game is $20\%$ off. The sale price is $ $\,?$ 2) A $45 dinner gets a $20\%$ tip. The total is $ $\,?$ 3) A price goes from $50 to $60. The percent increase is $\,?$ (enter just the number, e.g. 20 for $20\%$)

The One Question to Ask: Up or Down?

Every real-world percent boils down to: does the price go up or down?

• Down (subtract): sales, discounts, markdowns → you pay less than $100\%$.
• Up (add): sales tax, tips, markups → you pay more than $100\%$.

The one-step multiplier captures both:

$\text{new amount} = (\,100\% \pm \text{rate}\,) \times \text{original}$

⚠️ A $40\%$ discount is not the same as paying $40\%$. A $40\%$ discount means you pay the remaining $60\%$. Always ask what fraction of the price is left.

Add or Subtract? 🔽

Decide how each situation changes the starting price.

💯 Percent Problems

Part 5 of 5 — Mixed Practice & Mastery Check

You can now (1) convert between percents, decimals, and fractions, (2) find a percent of a number, (3) solve all three percent questions, and (4) handle real-world discounts, tax, tips, and percent change. Let's put it all together.

Quick Reference

🔑 The whole engine is one relationship: $\dfrac{\text{part}}{\text{whole}} = \dfrac{\%}{100}$. Find the missing piece and you've solved the problem.

Mixed Practice 🎯

Your Three-Step Game Plan

For any percent problem on a test, run this checklist:

1. Classify — which is missing: part, percent, or whole?
2. Estimate — should the answer be small, about half, or near the whole?
3. Solve & check — compute, then confirm your answer matches the estimate.

🔑 The same engine handles plain percents and word problems: a discount is "find the part, then subtract," and a percent increase is "find the change, divide by the original." Different stories, one tool.

Put It Together 🧮

1) $18$ is what percent of $72$? (enter just the number) 2) A $90 bike is $30\%$ off. The sale price is $ $\,?$ 3) A population grows from $200$ to $250$. The percent increase is $\,?$ (just the number)

You're Ready

You've connected percents to decimals and fractions, found parts, percents, and wholes, and applied it all to tips, tax, discounts, and percent change. One last check confirms it all stuck.

💡 If a question ever feels stuck, fall back to the master relationship: $\dfrac{\text{part}}{\text{whole}} = \dfrac{\%}{100}$.

Exit Quiz ✅

Answer all three to finish the lesson.