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 $100$. Master that one idea and every percent problem — tips, taxes, discounts, interest — becomes the same problem in a costume.

Percent = "Out of 100"

The word percent comes from per centum — "per hundred." So:

$x\% = \frac{x}{100}$

That means $37\%$ literally means $\dfrac{37}{100}$, or $37$ out of every $100$.

Three Ways to Write the Same Value

💡 Notice: $100\%$ of something is the whole thing, and $200\%$ is twice as much. Percents above $100$ are completely normal.

Converting Between Forms

Two moves do almost everything:

Percent → Decimal: divide by $100$ (move the decimal point two places left). $64\% = 0.64 \qquad 8\% = 0.08 \qquad 150\% = 1.5$

Decimal → Percent: multiply by $100$ (move the decimal point two places right). $0.3 = 30\% \qquad 0.07 = 7\% \qquad 1.25 = 125\%$

⚠️ Watch the zeros. $8\%$ is $0.08$, not $0.8$. Forgetting the leading zero is the #1 percent error.

Concept Check 🎯

Conversion Drill 🧮

Convert each value. Enter decimals as decimals (e.g. 0.42).

1) $42\%$ as a decimal $= \,?$ 2) $3\%$ as a decimal $= \,?$ 3) $1.4$ as a percent $= \,?$ (enter just the number, e.g. 140)

Finding the Percent OF a Number

To find a percent of a number, turn the percent into a decimal and multiply. The word "of" means times.

$x\% \text{ of } N = \frac{x}{100} \cdot N$

Worked Example: $20\%$ of $80$

$20\% \text{ of } 80 = 0.20 \cdot 80 = 16$

Worked Example: $15\%$ of $60$

$15\% \text{ of } 60 = 0.15 \cdot 60 = 9$

💡 Mental-math shortcut: $10\%$ of any number just moves the decimal one place left. $10\%$ of $250$ is $25$. Then $5\%$ is half of that ($12.5$), and $20\%$ is double it ($50$).

Find the Part 🧮

Compute each. Multiply the decimal form by the number.

1) $25\%$ of $40 = \,?$ 2) $10\%$ of $90 = \,?$ 3) $30\%$ of $200 = \,?$

💯 Percent Problems

Part 2 of 5 — The Percent Proportion & The Three Questions

🔑 One tool, three questions. Every basic percent problem is asking for one of three things: the part, the whole, or the percent. A single proportion handles all three.

The Percent Proportion

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

The whole is the amount after the word "of." The part is usually the amount after "is."

💡 A useful phrase: "is over of equals percent over 100." $\frac{\text{is}}{\text{of}} = \frac{\%}{100}$

The Three Questions

Fill in everything you know, then cross-multiply and solve.

Question 1 — Finding the Part

Worked Example: What is $35\%$ of $140$?

Set up the proportion with the part unknown:

$\frac{p}{140} = \frac{35}{100}$

Cross-multiply:

$100p = 35 \cdot 140 = 4900 \;\Rightarrow\; p = \frac{4900}{100} = 49$

✅ Check with multiplication: $0.35 \cdot 140 = 49$ ✓ — same answer as Part 1's method, because they're the same idea.

Question 2 — Finding the Percent

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

Now the percent is unknown:

$\frac{18}{72} = \frac{x}{100}$

Cross-multiply:

$72x = 18 \cdot 100 = 1800 \;\Rightarrow\; x = \frac{1800}{72} = 25$

So $18$ is $\mathbf{25\%}$ of $72$.

⚠️ Order matters. "$18$ is what percent of $72$" puts $72$ (the "of" number) on the bottom. Flipping them gives the wrong answer.

Set Up the Proportion 🔽

Read each question and choose the correct setup using $\dfrac{\text{is}}{\text{of}} = \dfrac{\%}{100}$.

Concept Check 🎯

Part & Percent Drill 🧮

Use the proportion (or multiply).

1) What is $45\%$ of $80$? $= \,?$ 2) $30$ is what percent of $120$? $= \,?$ (enter just the number) 3) What is $8\%$ of $250$? $= \,?$

💯 Percent Problems

Part 3 of 5 — Finding the Whole (the Reverse Problem)

🔑 The trickiest of the three. When the whole is missing, students often divide by the wrong number. The proportion keeps you safe — but there's also a slick one-step method.

Finding the Whole

Worked Example: $24$ is $30\%$ of what number?

Set up with the whole unknown:

$\frac{24}{w} = \frac{30}{100}$

Cross-multiply:

$30w = 24 \cdot 100 = 2400 \;\Rightarrow\; w = \frac{2400}{30} = 80$

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

✅ Check: $30\%$ of $80 = 0.30 \cdot 80 = 24$ ✓

The One-Step Shortcut

Since "part $=$ percent (as a decimal) $\times$ whole," you can solve for the whole by dividing:

$\text{whole} = \frac{\text{part}}{\text{percent as a decimal}}$

Same Example, Faster

$w = \frac{24}{0.30} = 80$

Another: $45$ is $75\%$ of what?

$w = \frac{45}{0.75} = 60$

⚠️ Direction check: finding the whole means the answer should be bigger than the part (when the percent is under $100\%$). If you divide $24 \div 0.30$ and get something smaller than $24$, you divided the wrong way.

Which Operation? 🔽

Decide what to do, then read off the result.

Mixed Concept Check 🎯

These mix all three question types — read carefully before you set up.

Find the Whole 🧮

Use whole $=$ part $\div$ decimal percent.

1) $14$ is $50\%$ of what number? $= \,?$ 2) $9$ is $25\%$ of what number? $= \,?$ 3) $63$ is $90\%$ of what number? $= \,?$

💯 Percent Problems

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

🔑 Real money lives here. Markups, markdowns, tips, and sales tax are all percent of change problems. One formula and one shortcut cover them all.

Percent of Change

$\text{percent change} = \frac{\text{new} - \text{old}}{\text{old}} \times 100\%$

• A positive result is a percent increase.
• A negative result is a percent decrease.

⚠️ Always divide by the ORIGINAL (old) amount, not the new one. The "old" value is your $100\%$ baseline.

Worked Example: Price rises from $40 to $50

$\frac{50 - 40}{40} \times 100\% = \frac{10}{40} \times 100\% = 25\% \text{ increase}$

Worked Example: Price drops from $80 to $60

$\frac{60 - 80}{80} \times 100\% = \frac{-20}{80} \times 100\% = -25\% \;(\text{a } 25\% \text{ decrease})$

Concept Check 🎯

The Multiplier Shortcut (Tips, Tax, Discounts)

Instead of finding the change and then adding or subtracting, multiply once by a smart number.

🔑 Rule: To increase by $p\%$, multiply by $\left(1 + \dfrac{p}{100}\right)$. To decrease by $p\%$, multiply by $\left(1 - \dfrac{p}{100}\right)$.

Worked Example: $60 meal with a $20\%$ tip

$60 \times 1.20 = 72 \;\Rightarrow\; \$72 \text{ total}$

Worked Example: $80 shoes, $25\%$ off

$80 \times 0.75 = 60 \;\Rightarrow\; \$60 \text{ sale price}$

Pick the Multiplier 🔽

Money Math 🧮

Use the multiplier shortcut. Enter dollar amounts as numbers (e.g. 64.80).

1) A $120 bike is $40\%$ off. Sale price $= \,?$ 2) A $25 meal with $8\%$ tax. Total $= \,?$ 3) A population of $400$ grows by $15\%$. New population $= \,?$

💯 Percent Problems

Part 5 of 5 — Mixed Applications & Mastery Check

You can now (1) convert percents, (2) find the part, (3) find the percent and the whole, and (4) handle percent change, tips, tax, and discounts. Time to combine them — including a few classic traps.

Quick Reference

⚠️ Two famous traps:

1. A $20\%$ increase followed by a $20\%$ decrease does not return to the start. ($100 \to 120 \to 96$.)
2. Percent change is always measured against the original, never the new value.

A Classic Trap: Up Then Down

A $100 stock rises $20\%$, then falls $20\%$. Is it back to $100?

Step 1 — Up $20\%$: $100 \times 1.20 = 120$. Step 2 — Down $20\%$: $120 \times 0.80 = 96$.

It ends at $96 — a net $4\%$ loss! The second $20\%$ is taken from the larger $120, so it removes more than the first $20\%$ added.

💡 Why: $1.20 \times 0.80 = 0.96$, which is a $4\%$ decrease overall — no matter the starting price.

Trap Check 🎯

Mixed Mastery Drill 🧮

1) $36$ is what percent of $48$? $= \,?$ (enter just the number) 2) $21$ is $35\%$ of what number? $= \,?$ 3) A $90 coat is $30\%$ off, then $10\%$ sales tax is added. Final price $= \,?$

Exit Quiz ✅

Answer all three to finish the lesson.