Mathematics of Percentages: Beginner to Advanced

Part I: Foundations

What Is a Percentage?

A percentage expresses a quantity as a fraction of 100. The symbol $%$ is shorthand for $\div 100$.

\[\text{percentage} = \frac{\text{part}}{\text{whole}} \times 100\]

Think of it as "out of 100." If 3 out of every 10 apples are red, that is 30 out of every 100 — or $30\%$.

Table 1. Core equivalences
Fraction	Decimal	Percentage
$\frac{1}{2}$	$0.5$	$50\%$
$\frac{1}{4}$	$0.25$	$25\%$
$\frac{3}{4}$	$0.75$	$75\%$
$\frac{1}{5}$	$0.2$	$20\%$
$\frac{1}{8}$	$0.125$	$12.5\%$
$\frac{1}{3}$	$0.\overline{3}$	$33.\overline{3}\%$
$\frac{2}{3}$	$0.\overline{6}$	$66.\overline{6}\%$
$\frac{1}{10}$	$0.1$	$10\%$
$\frac{3}{8}$	$0.375$	$37.5\%$

Converting Between Forms

Fraction to percentage

Divide numerator by denominator, multiply by 100:

\[\frac{a}{b} \times 100 = \text{percentage}\]

Percentage to decimal

Divide by 100 (move the decimal point two places left):

\[p\% = \frac{p}{100}\]

Decimal to fraction

Write the decimal over the appropriate power of 10 and simplify:

\[0.375 = \frac{375}{1000} = \frac{3}{8}\]

The Three Basic Percentage Questions

Every percentage problem is one of three types:

Type 1: Find the part

What is $p\%$ of $n$?

\[\text{part} = \frac{p}{100} \times n\]

Type 2: Find the percentage

$x$ is what percent of $n$?

\[p = \frac{x}{n} \times 100\]

Type 3: Find the whole

$x$ is $p\%$ of what number?

\[n = \frac{x \times 100}{p}\]

Memorize this triangle. Cover the unknown, and the remaining two show the operation:

\[\text{Part} = \text{Whole} \times \frac{\text{Percent}}{100}\]

Foundations — Problems

Set A: Conversions (10 problems)

A1. Convert $\frac{7}{20}$ to a percentage.

\[\frac{7}{20} = \frac{7 \times 5}{20 \times 5} = \frac{35}{100} = 35\%\]

A2. Convert $0.045$ to a percentage.

\[0.045 \times 100 = 4.5\%\]

A3. Convert $72\%$ to a fraction in lowest terms.

\[\frac{72}{100} = \frac{18}{25}\]

A4. Convert $\frac{5}{6}$ to a percentage. Round to two decimal places.

\[\frac{5}{6} = 0.8333\ldots = 83.33\%\]

A5. Convert $0.008$ to a percentage.

\[0.008 \times 100 = 0.8\%\]

A6. Convert $125\%$ to a decimal.

\[\frac{125}{100} = 1.25\]

A percentage greater than $100\%$ means more than the whole.

A7. Convert $\frac{11}{40}$ to a percentage.

\[\frac{11}{40} = 0.275 = 27.5\%\]

A8. Convert $0.3\%$ to a decimal.

\[\frac{0.3}{100} = 0.003\]

Watch the trap — $0.3\%$ is not $0.3$. It is $0.003$.

A9. Convert $\frac{7}{8}$ to a percentage.

\[\frac{7}{8} = 0.875 = 87.5\%\]

A10. Which is larger: $\frac{3}{7}$ or $42\%$?

\[\frac{3}{7} = 0.42857\ldots = 42.857\%\]

$\frac{3}{7} > 42\%$ by $0.857$ percentage points.

Set B: Finding the Part (10 problems)

B1. What is $15\%$ of $240$?

\[\frac{15}{100} \times 240 = 0.15 \times 240 = 36\]

B2. What is $6\%$ of $1{,}500$?

\[0.06 \times 1500 = 90\]

B3. What is $0.5\%$ of $10{,}000$?

\[0.005 \times 10000 = 50\]

B4. What is $33\frac{1}{3}\%$ of $270$?

\[\frac{1}{3} \times 270 = 90\]

B5. What is $120\%$ of $85$?

\[1.20 \times 85 = 102\]

B6. A recipe calls for $250$g of flour. You want to make $75\%$ of the recipe. How much flour?

\[0.75 \times 250 = 187.5\text{g}\]

B7. A city has $340{,}000$ residents. $62\%$ are registered voters. How many registered voters?

\[0.62 \times 340000 = 210{,}800\]

B8. A hard drive is $2{,}000$GB. $37.5\%$ is used. How many GB are free?

\[\text{used} = 0.375 \times 2000 = 750\text{GB}\]

\[\text{free} = 2000 - 750 = 1{,}250\text{GB}\]

B9. What is $8.25\%$ of $\$1{,}250$?

\[0.0825 \times 1250 = \$103.13\]

B10. An alloy is $14\%$ nickel. A bar weighs $3.5$kg. How many grams of nickel?

\[0.14 \times 3500 = 490\text{g}\]

Set C: Finding the Percentage (10 problems)

C1. $18$ is what percent of $72$?

\[\frac{18}{72} \times 100 = 25\%\]

C2. You scored $42$ out of $60$ on a test. What is your percentage?

\[\frac{42}{60} \times 100 = 70\%\]

C3. A team won $19$ out of $25$ games. What is their win percentage?

\[\frac{19}{25} \times 100 = 76\%\]

C4. $7$ out of $200$ products were defective. What is the defect rate?

\[\frac{7}{200} \times 100 = 3.5\%\]

C5. You read $135$ pages of a $450$-page book. What percentage have you read?

\[\frac{135}{450} \times 100 = 30\%\]

C6. A student answered $47$ out of $52$ questions correctly. What percentage? Round to one decimal.

\[\frac{47}{52} \times 100 = 90.4\%\]

C7. Out of $8{,}400$ emails sent, $378$ bounced. What is the bounce rate?

\[\frac{378}{8400} \times 100 = 4.5\%\]

C8. A company earned $\$2.4M$ revenue and $\$360K$ profit. What is the profit margin?

\[\frac{360{,}000}{2{,}400{,}000} \times 100 = 15\%\]

C9. An election candidate received $12{,}870$ votes out of $39{,}000$ total. What percentage? Round to one decimal.

\[\frac{12870}{39000} \times 100 = 33.0\%\]

C10. A server was down $43$ minutes in a $30$-day month. What is the uptime percentage? (Round to four decimals.)

\[\text{total minutes} = 30 \times 24 \times 60 = 43{,}200\]

\[\text{uptime} = \frac{43200 - 43}{43200} \times 100 = \frac{43157}{43200} \times 100 = 99.9005\%\]

Set D: Finding the Whole (10 problems)

D1. $45$ is $30\%$ of what number?

\[n = \frac{45 \times 100}{30} = \frac{4500}{30} = 150\]

D2. $\$63$ is $9\%$ sales tax on what purchase price?

\[\text{price} = \frac{63}{0.09} = \$700\]

D3. A student got $36$ questions right, which was $80\%$ of the test. How many total questions?

\[n = \frac{36}{0.80} = 45 \text{ questions}\]

D4. $15\%$ of a company’s employees are engineers. There are $72$ engineers. How many total employees?

\[n = \frac{72}{0.15} = 480\]

D5. After a $20\%$ discount, a shirt costs $\$44$. What was the original price?

\[44 = n \times 0.80 \quad \Rightarrow \quad n = \frac{44}{0.80} = \$55\]

D6. $0.5\%$ of a city’s population has a rare condition. That is $1{,}230$ people. What is the population?

\[n = \frac{1230}{0.005} = 246{,}000\]

D7. A tip of $\$13.50$ represents $18\%$ of the bill. What was the bill?

\[\text{bill} = \frac{13.50}{0.18} = \$75\]

D8. After a $5\%$ raise, an employee earns $\$63{,}000$. What was the original salary?

\[63000 = n \times 1.05 \quad \Rightarrow \quad n = \frac{63000}{1.05} = \$60{,}000\]

D9. A battery is at $35\%$ and shows $2{,}100$mAh remaining. What is the full capacity?

\[n = \frac{2100}{0.35} = 6{,}000\text{mAh}\]

D10. An investment lost $12\%$ and is now worth $\$22{,}000$. What was the original value?

\[22000 = n \times 0.88 \quad \Rightarrow \quad n = \frac{22000}{0.88} = \$25{,}000\]

Part II: Intermediate Applications

Percentage Increase and Decrease

Percentage increase

\[\%\text{ increase} = \frac{\text{new} - \text{original}}{\text{original}} \times 100\]

Percentage decrease

\[\%\text{ decrease} = \frac{\text{original} - \text{new}}{\text{original}} \times 100\]

The asymmetry trap: A $p\%$ increase followed by a $p\%$ decrease does not return to the original.

\[100 \xrightarrow{+25\%} 125 \xrightarrow{-25\%} 93.75\]

General formula for the net effect:

\[\text{net change} = -\left(\frac{p}{100}\right)^2 \times 100\%\]

This is always a loss. The larger $p$, the bigger the loss.

Discounts, Tax, and Tips

Discount formula

\[\text{sale price} = \text{original} \times \left(1 - \frac{d}{100}\right)\]

Tax formula

\[\text{total} = \text{price} \times \left(1 + \frac{t}{100}\right)\]

Discount then tax (order matters)

\[\text{total} = \text{original} \times \left(1 - \frac{d}{100}\right) \times \left(1 + \frac{t}{100}\right)\]

Mental math shortcut for tips

10% tip: move decimal one place left.
15% tip: find 10%, add half of it.
20% tip: find 10%, double it.
25% tip: find half, then halve again (or divide by 4).

$\$73.40 \rightarrow 10\% = \$7.34 \rightarrow 20\% = \$14.68$

Markup vs. Margin

These are not interchangeable:

Metric	Formula	Base
Markup	\(\frac{\text{profit}}{\text{cost}} \times 100\)	Cost
Margin	\(\frac{\text{profit}}{\text{revenue}} \times 100\)	Revenue

Metric

Formula

Base

Markup

$\frac{\text{profit}}{\text{cost}} \times 100$

Cost

Margin

$\frac{\text{profit}}{\text{revenue}} \times 100$

Revenue

Conversion between them

\[\text{margin} = \frac{\text{markup}}{100 + \text{markup}} \times 100 \qquad \text{markup} = \frac{\text{margin}}{100 - \text{margin}} \times 100\]

Intermediate — Problems

Set E: Increase and Decrease (10 problems)

E1. A shirt was $40 and now costs $52. What is the percentage increase?

\[\frac{52 - 40}{40} \times 100 = \frac{12}{40} \times 100 = 30\%\]

E2. A stock went from $84 to $63. What is the percentage decrease?

\[\frac{84 - 63}{84} \times 100 = \frac{21}{84} \times 100 = 25\%\]

E3. A town’s population grew from 15,000 to 19,500. What is the percentage increase?

\[\frac{19500 - 15000}{15000} \times 100 = \frac{4500}{15000} \times 100 = 30\%\]

E4. Rent increased from $1,800 to $2,070. What is the percentage increase?

\[\frac{2070 - 1800}{1800} \times 100 = \frac{270}{1800} \times 100 = 15\%\]

E5. After a 40% increase, a gym membership costs $84/month. What was the original price?

\[84 = n \times 1.40 \quad \Rightarrow \quad n = \frac{84}{1.40} = \$60\]

E6. A cryptocurrency dropped 35% on Monday, then rose 35% on Tuesday. Is the investor back to even? What is the net change?

\[\text{Let original} = 100\]

\[100 \times 0.65 = 65 \quad \rightarrow \quad 65 \times 1.35 = 87.75\]

Net change: $-12.25\%$. The investor is not back to even.

By formula: $-\left(\frac{35}{100}\right)^2 \times 100 = -12.25\%$

E7. Last year a company had 240 employees. This year it has 312. What is the percentage growth?

\[\frac{312 - 240}{240} \times 100 = \frac{72}{240} \times 100 = 30\%\]

E8. A car’s value dropped from $32,000 to $24,320 in one year. What was the depreciation rate?

\[\frac{32000 - 24320}{32000} \times 100 = \frac{7680}{32000} \times 100 = 24\%\]

E9. A product price increased by 10%, then increased again by 10%. Is the total increase 20%?

\[\text{Let original} = 100\]

\[100 \times 1.10 \times 1.10 = 100 \times 1.21 = 121\]

Total increase: $21\%$, not $20\%$. The extra $1\%$ comes from the second increase acting on the already-increased value.

E10. After a 15% pay cut, an employee earns $5,950/month. What was the original salary? How much of a raise (%) would restore it?

Original:

\[5950 = n \times 0.85 \quad \Rightarrow \quad n = \frac{5950}{0.85} = \$7{,}000\]

Raise needed to restore:

\[\frac{7000 - 5950}{5950} \times 100 = \frac{1050}{5950} \times 100 \approx 17.65\%\]

A $15\%$ cut requires a $17.65\%$ raise to recover — the asymmetry trap in action.

Set F: Discounts, Tax, and Shopping (10 problems)

F1. A $160 jacket is 25% off. What is the sale price?

\[160 \times 0.75 = \$120\]

F2. A $50 meal has 8.25% tax. What is the total?

\[50 \times 1.0825 = \$54.13\]

F3. A $75 item is 20% off, then you pay 9% sales tax. What is the final price?

\[75 \times 0.80 \times 1.09 = 60 \times 1.09 = \$65.40\]

F4. You paid $89.25 for an item after 15% off. What was the original price?

\[n = \frac{89.25}{0.85} = \$105\]

F5. Store A sells a TV for $800 with 30% off. Store B sells it for $600 with 10% off. Which is cheaper?

Store A: $800 \times 0.70 = \$560$

Store B: $600 \times 0.90 = \$540$

Store B is cheaper by $\$20$.

F6. Your dinner bill is $86.40. You want to tip 20% on the pre-tax amount. Tax was 8%. What was the pre-tax amount and what tip do you leave?

\[\text{pre-tax} = \frac{86.40}{1.08} = \$80\]

\[\text{tip} = 80 \times 0.20 = \$16\]

\[\text{total paid} = 86.40 + 16 = \$102.40\]

F7. An item is marked "Buy 2, get 1 free." What is the effective discount percentage?

You pay for 2 items but receive 3.

\[\text{discount} = \frac{1}{3} \times 100 = 33.\overline{3}\%\]

F8. A store marks up goods by 60% then offers a 25% "sale." What is the net markup over cost?

\[\text{Let cost} = 100\]

\[100 \times 1.60 = 160 \quad \rightarrow \quad 160 \times 0.75 = 120\]

Net markup: $20\%$ over cost. The "sale" is partially manufactured margin.

F9. You have a 10% off coupon and the store is running a 30% off sale. If they stack (applied sequentially), what is the total discount?

\[(1 - 0.30)(1 - 0.10) = 0.70 \times 0.90 = 0.63\]

\[\text{total discount} = 1 - 0.63 = 37\%\]

Not $40\%$ — sequential discounts always yield less than the sum.

F10. A wholesaler offers trade discounts of 20/10/5 (applied successively) on a list price of $1,000. What is the net price?

\[1000 \times 0.80 \times 0.90 \times 0.95 = 1000 \times 0.684 = \$684\]

Equivalent single discount:

\[1 - 0.684 = 31.6\%\]

Set G: Profit, Loss, and Business (10 problems)

G1. A shop buys a widget for $25 and sells it for $40. What is the markup percentage? What is the margin?

Markup (based on cost):

\[\frac{40 - 25}{25} \times 100 = 60\%\]

Margin (based on revenue):

\[\frac{40 - 25}{40} \times 100 = 37.5\%\]

G2. A product costs $80 to make. The company wants a 40% margin. What should the selling price be?

\[\text{margin} = \frac{\text{price} - 80}{\text{price}} = 0.40\]

\[\text{price} - 80 = 0.40 \times \text{price} \quad \Rightarrow \quad 0.60 \times \text{price} = 80\]

\[\text{price} = \frac{80}{0.60} = \$133.33\]

G3. Revenue increased from $1.2M to $1.5M. Expenses increased from $900K to $1.2M. Did profitability improve or worsen?

Before:

\[\text{margin} = \frac{1.2M - 0.9M}{1.2M} \times 100 = 25\%\]

After:

\[\text{margin} = \frac{1.5M - 1.2M}{1.5M} \times 100 = 20\%\]

Profitability worsened by 5 percentage points despite higher absolute profit.

G4. A retailer bought 500 units at $12 each. 80% sold at $20. The rest sold at 50% off. What was the overall profit margin?

Cost: $500 \times 12 = \$6{,}000$

Revenue from full-price: $400 \times 20 = \$8{,}000$

Revenue from clearance: $100 \times 10 = \$1{,}000$

Total revenue: $\$9{,}000$

\[\text{margin} = \frac{9000 - 6000}{9000} \times 100 = 33.\overline{3}\%\]

G5. An investor bought stock at $45, it rose to $54, then fell to $48. What is the percentage gain from the original purchase?

\[\frac{48 - 45}{45} \times 100 = \frac{3}{45} \times 100 = 6.\overline{6}\%\]

G6. A company’s profit margin is 12%. Revenue is $850,000. What is the profit in dollars?

\[0.12 \times 850000 = \$102{,}000\]

G7. A product’s cost increased by 8% but the selling price stayed at $50. The old margin was 25%. What is the new margin?

Old cost: $50 \times 0.75 = \$37.50$

New cost: $37.50 \times 1.08 = \$40.50$

\[\text{new margin} = \frac{50 - 40.50}{50} \times 100 = 19\%\]

Margin dropped 6 percentage points.

G8. A freelancer charges $75/hr. Their effective tax rate is 28%. What is their after-tax hourly rate? How much must they charge to net $75/hr after tax?

After-tax: $75 \times 0.72 = \$54/\text{hr}$

To net $75:

\[\frac{75}{0.72} = \$104.17/\text{hr}\]

G9. A SaaS company has 2,000 customers. Monthly churn is 3%. How many customers remain after 6 months (no new acquisitions)?

\[2000 \times (1 - 0.03)^6 = 2000 \times (0.97)^6 = 2000 \times 0.8330 = 1{,}666\]

G10. Cost of goods sold is $340,000. Operating expenses are $85,000. Revenue is $500,000. Calculate the gross margin and net margin.

Gross margin:

\[\frac{500000 - 340000}{500000} \times 100 = 32\%\]

Net margin:

\[\frac{500000 - 340000 - 85000}{500000} \times 100 = \frac{75000}{500000} \times 100 = 15\%\]

Part III: Advanced Concepts

Compound Interest

\[A = P\left(1 + \frac{r}{n}\right)^{nt}\]

Where:

$A$ = final amount
$P$ = principal
$r$ = annual rate (decimal)
$n$ = compounding periods per year
$t$ = years

The Rule of 72

A quick approximation for how long it takes money to double:

\[t_{\text{double}} \approx \frac{72}{r\%}\]

At $6\%$: $\frac{72}{6} = 12$ years. (Exact: $11.90$ years.)

Successive Percentage Changes

When applying $n$ successive changes $p_1, p_2, \ldots, p_n$:

\[\text{final} = \text{original} \times \prod_{i=1}^{n} \left(1 + \frac{p_i}{100}\right)\]

Where $p_i$ is negative for decreases.

Depreciation

Straight-line

\[\text{annual depreciation} = \frac{\text{cost} - \text{salvage value}}{\text{useful life}}\]

Declining balance (exponential)

\[V(t) = V_0 \left(1 - \frac{d}{100}\right)^t\]

Weighted Averages

\[\bar{x}_w = \frac{\sum_{i=1}^{n} w_i \cdot x_i}{\sum_{i=1}^{n} w_i}\]

Elasticity (Economics)

Price elasticity of demand:

\[E_d = \frac{\%\text{ change in quantity demanded}}{\%\text{ change in price}} = \frac{\Delta Q / Q}{\Delta P / P}\]

$|E_d| > 1$ — elastic (sensitive to price)
$|E_d| < 1$ — inelastic (insensitive to price)
$|E_d| = 1$ — unit elastic

Percentile Rank

\[\text{percentile rank} = \frac{\text{number of values below } x}{\text{total number of values}} \times 100\]

Percentage Points vs. Percentages

These are different concepts:

Percentage points — absolute difference between two percentages
Percent change — relative change

If interest goes from $4\%$ to $5\%$:

The increase is 1 percentage point
The increase is $\frac{5-4}{4} \times 100 = 25\%$ (a 25% increase)

Confusing these leads to major misinterpretation of data.

Advanced — Problems

Set H: Compound Interest and Growth (10 problems)

H1. $5,000 at 6% compounded monthly for 10 years. What is the final amount?

\[A = 5000\left(1 + \frac{0.06}{12}\right)^{120} = 5000(1.005)^{120} \approx \$9{,}096.98\]

H2. Same as H1 but compounded continuously. Compare the two.

\[A = 5000 \cdot e^{0.06 \times 10} = 5000 \cdot e^{0.6} \approx \$9{,}110.59\]

Continuous compounding yields $\$13.61$ more.

H3. How long to double $1,000 at 8% compounded annually? Use both Rule of 72 and exact calculation.

Rule of 72: $\frac{72}{8} = 9$ years

Exact:

\[2 = (1.08)^t \quad \Rightarrow \quad t = \frac{\ln 2}{\ln 1.08} = \frac{0.6931}{0.07696} \approx 9.01 \text{ years}\]

Rule of 72 is remarkably accurate here.

H4. You want $50,000 in 15 years. The account pays 5% compounded quarterly. How much must you deposit today?

\[P = \frac{50000}{\left(1 + \frac{0.05}{4}\right)^{60}} = \frac{50000}{(1.0125)^{60}} = \frac{50000}{2.1072} \approx \$23{,}728.47\]

H5. A bacteria colony doubles every 4 hours. Starting from 500 cells, how many after 24 hours? Express the growth as a percentage.

Doublings: $\frac{24}{4} = 6$

\[500 \times 2^6 = 500 \times 64 = 32{,}000\]

Percentage growth:

\[\frac{32000 - 500}{500} \times 100 = 6{,}300\%\]

H6. Inflation averages 3.2% per year. What is the purchasing power of $100 after 20 years?

\[100 \times (1 - 0.032)^{20} = 100 \times (0.968)^{20} \approx \$52.14\]

Your $\$100$ buys less than half as much after 20 years.

H7. Account A: 4.8% compounded daily. Account B: 4.9% compounded annually. Which has the higher effective annual rate?

Account A effective annual rate:

\[\left(1 + \frac{0.048}{365}\right)^{365} - 1 = (1.00013151)^{365} - 1 \approx 0.04917 = 4.917\%\]

Account B: $4.9\%$ (already annual)

Account B has a higher effective rate by $0.017$ percentage points. Barely — the frequent compounding nearly closes the gap.

H8. A credit card charges 22% APR compounded daily. What is the effective annual rate?

\[\left(1 + \frac{0.22}{365}\right)^{365} - 1 = (1.0006027)^{365} - 1 \approx 0.2461 = 24.61\%\]

You’re actually paying $24.61\%$, not $22\%$. This is why credit card debt is devastating.

H9. An investment returns +15%, -8%, +12%, -3%, +20% over 5 years. What is the overall return and the compound annual growth rate (CAGR)?

Overall multiplier:

\[1.15 \times 0.92 \times 1.12 \times 0.97 \times 1.20 = 1.3808\]

Overall return: $38.08\%$

CAGR:

\[\text{CAGR} = (1.3808)^{1/5} - 1 = 1.0667 - 1 = 6.67\%\]

H10. You deposit $200/month into an account earning 6% compounded monthly. What is the balance after 30 years? (Future value of annuity.)

\[FV = PMT \times \frac{(1 + r)^n - 1}{r}\]

where $r = \frac{0.06}{12} = 0.005$ and $n = 360$:

\[FV = 200 \times \frac{(1.005)^{360} - 1}{0.005} = 200 \times \frac{6.0226 - 1}{0.005} = 200 \times 1004.52 = \$200{,}903.01\]

Total deposited: $200 \times 360 = \$72{,}000$. Interest earned: $\$128{,}903$.

Set I: Depreciation, Decay, and Loss (8 problems)

I1. A $25,000 car depreciates at 15% per year. What is it worth after 5 years?

\[25000 \times (0.85)^5 = 25000 \times 0.4437 \approx \$11{,}092.63\]

I2. A machine costs $120,000 with a salvage value of $20,000 over 10 years. What is the annual straight-line depreciation? What percentage of original value is depreciated each year?

\[\text{annual} = \frac{120000 - 20000}{10} = \$10{,}000/\text{year}\]

\[\frac{10000}{120000} \times 100 = 8.\overline{3}\%\text{ per year}\]

I3. A radioactive substance decays at 4.5% per hour. Starting with 800g, how much remains after 12 hours?

\[800 \times (1 - 0.045)^{12} = 800 \times (0.955)^{12} = 800 \times 0.5735 \approx 458.8\text{g}\]

I4. A laptop loses 30% of its value in year 1, then 20% per year after. Bought for $2,000, what is it worth after 4 years?

\[2000 \times 0.70 \times (0.80)^3 = 2000 \times 0.70 \times 0.512 = \$716.80\]

I5. A car loses 18% per year. After how many full years is it worth less than half its original value?

Solve $(0.82)^t < 0.5$:

\[t > \frac{\ln(0.5)}{\ln(0.82)} = \frac{-0.6931}{-0.1985} \approx 3.49\]

After 4 full years. Verification: $(0.82)^4 = 0.4521 < 0.5 \checkmark$

I6. Equipment depreciates from $50,000 to $18,000 in 6 years (declining balance). What is the annual depreciation rate?

\[18000 = 50000 \times (1 - d)^6\]

\[(1 - d)^6 = 0.36 \quad \Rightarrow \quad 1 - d = 0.36^{1/6} = 0.8434\]

\[d = 1 - 0.8434 = 0.1566 = 15.66\%\]

I7. A drug’s concentration in the bloodstream decreases by 12% per hour. Starting at 400mg, when does it drop below 50mg?

\[50 = 400 \times (0.88)^t \quad \Rightarrow \quad (0.88)^t = 0.125\]

\[t = \frac{\ln(0.125)}{\ln(0.88)} = \frac{-2.0794}{-0.1278} \approx 16.3 \text{ hours}\]

I8. Two assets: Asset A costs $10,000 and loses 25% per year. Asset B costs $10,000 and loses $2,000 per year (straight-line). After how many years does Asset A become worth more than Asset B?

Asset A: $10000 \times (0.75)^t$

Asset B: $10000 - 2000t$

Test values:

Year	Asset A	Asset B
1	$7,500	$8,000
2	$5,625	$6,000
3	$4,219	$4,000
4	$3,164	$2,000

Asset A surpasses Asset B after year 2 (between years 2 and 3). The declining balance slows down while straight-line stays constant — a key difference for tax strategy.

Set J: Weighted Averages and Statistics (8 problems)

J1. A portfolio: 40% stocks (up 12%), 35% bonds (up 3%), 25% cash (up 0.5%). What is the overall return?

\[(0.40 \times 12) + (0.35 \times 3) + (0.25 \times 0.5) = 4.8 + 1.05 + 0.125 = 5.975\%\]

J2. A student’s grade is based on: Tests 50%, Homework 30%, Participation 20%. Scores: Tests 78, Homework 92, Participation 85. What is the final grade?

\[(0.50 \times 78) + (0.30 \times 92) + (0.20 \times 85) = 39 + 27.6 + 17 = 83.6\]

J3. A survey: 45% of respondents rated a service "excellent" (score 5), 30% "good" (4), 15% "average" (3), 7% "poor" (2), 3% "terrible" (1). What is the weighted average rating?

\[(0.45 \times 5) + (0.30 \times 4) + (0.15 \times 3) + (0.07 \times 2) + (0.03 \times 1)\]

\[= 2.25 + 1.20 + 0.45 + 0.14 + 0.03 = 4.07\]

J4. Three factories produce bolts. Factory A makes 50% of output (2% defect rate), Factory B makes 30% (3% defect rate), Factory C makes 20% (5% defect rate). What is the overall defect rate?

\[(0.50 \times 2) + (0.30 \times 3) + (0.20 \times 5) = 1.0 + 0.9 + 1.0 = 2.9\%\]

J5. In a class of 30 students, 10 scored an average of 90%, 12 scored an average of 75%, and 8 scored an average of 60%. What is the class average?

\[\frac{(10 \times 90) + (12 \times 75) + (8 \times 60)}{30} = \frac{900 + 900 + 480}{30} = \frac{2280}{30} = 76\%\]

J6. A student scored in the 85th percentile on a test with 200 students. How many students scored lower?

\[0.85 \times 200 = 170 \text{ students scored lower}\]

J7. In a dataset of 50 values, a score of 72 is greater than 38 of them. What is the percentile rank of 72?

\[\frac{38}{50} \times 100 = 76\text{th percentile}\]

J8. A mutual fund had these annual returns: Year 1: +22%, Year 2: -15%, Year 3: +18%, Year 4: +8%, Year 5: -5%. What is the arithmetic mean return? What is the geometric mean (CAGR)? Why do they differ?

Arithmetic mean:

\[\frac{22 + (-15) + 18 + 8 + (-5)}{5} = \frac{28}{5} = 5.6\%\]

Geometric mean (CAGR):

\[(1.22 \times 0.85 \times 1.18 \times 1.08 \times 0.95)^{1/5} - 1\]

\[= (1.2583)^{0.2} - 1 = 1.0471 - 1 = 4.71\%\]

The geometric mean is always lower when returns vary. The arithmetic mean overstates actual growth because it ignores compounding and volatility. Always use CAGR for investment performance.

Set K: Mixed Advanced (10 problems)

K1. A drug is 95% effective. If 10,000 people take it, approximately how many are NOT protected?

\[10000 \times 0.05 = 500 \text{ people}\]

K2. The interest rate rose from 3.5% to 4.2%. A news headline says "interest rates rose 20%." Another says "rates rose 0.7 percentage points." Which is correct?

Both are correct — they measure different things.

Percentage point change: $4.2 - 3.5 = 0.7$ pp

Percent change: $\frac{0.7}{3.5} \times 100 = 20\%$

The "20% increase" sounds dramatic but is only 0.7 pp. Context matters.

K3. You need to score 85% overall. Four tests worth 25% each. You scored 78, 82, 90 on the first three. What minimum score do you need on test 4?

\[0.25(78 + 82 + 90 + x) = 85\]

\[78 + 82 + 90 + x = 340 \quad \Rightarrow \quad x = 90\]

K4. A population grows at 2.1% per year. The current population is 8.1 billion. What will it be in 25 years? When will it reach 10 billion?

In 25 years:

\[8.1 \times (1.021)^{25} = 8.1 \times 1.6796 \approx 13.6 \text{ billion}\]

When it reaches 10 billion:

\[10 = 8.1 \times (1.021)^t \quad \Rightarrow \quad (1.021)^t = 1.2346\]

\[t = \frac{\ln(1.2346)}{\ln(1.021)} = \frac{0.2107}{0.02078} \approx 10.1 \text{ years}\]

K5. A store raises prices by 8% then offers "8% off everything." Customers think they’re getting the old prices back. What is the actual net price change?

\[1.08 \times 0.92 = 0.9936\]

Net change: $-0.64\%$. Prices are actually slightly lower than original.

By formula: $-\left(\frac{8}{100}\right)^2 \times 100 = -0.64\%$

K6. A water tank is draining. It loses 8% of its current volume each hour. Starting full at 10,000 liters, how many hours until it has less than 2,000 liters?

\[2000 = 10000 \times (0.92)^t \quad \Rightarrow \quad (0.92)^t = 0.2\]

\[t = \frac{\ln(0.2)}{\ln(0.92)} = \frac{-1.6094}{-0.08338} \approx 19.3 \text{ hours}\]

After 20 full hours, the tank has less than 2,000 liters.

K7. A company offers two salary structures: Option A: $70,000 base with 5% annual raises. Option B: $65,000 base with 8% annual raises. After how many years does Option B overtake Option A? What is the total earnings difference over 10 years?

When B overtakes A:

\[65000 \times (1.08)^t = 70000 \times (1.05)^t\]

\[\left(\frac{1.08}{1.05}\right)^t = \frac{70000}{65000} = 1.07692\]

\[(1.02857)^t = 1.07692 \quad \Rightarrow \quad t = \frac{\ln(1.07692)}{\ln(1.02857)} = \frac{0.07411}{0.02817} \approx 2.63\]

Option B overtakes in year 3.

Total over 10 years:

Option A: $70000 \times \frac{(1.05)^{10} - 1}{0.05} = 70000 \times 12.578 = \$880{,}452$

Option B: $65000 \times \frac{(1.08)^{10} - 1}{0.08} = 65000 \times 14.487 = \$941{,}633$

Option B earns $\$61{,}181$ more over 10 years despite the lower start.

K8. A diagnostic test has 98% sensitivity and 95% specificity. A disease affects 1% of the population. If you test positive, what is the probability you actually have the disease? (Bayes' theorem with percentages.)

Using Bayes' theorem:

\[P(\text{disease} | +) = \frac{P(+ | \text{disease}) \times P(\text{disease})}{P(+)}\]

\[P(+) = P(+ | \text{disease}) \times P(\text{disease}) + P(+ | \text{no disease}) \times P(\text{no disease})\]

\[P(+) = (0.98)(0.01) + (0.05)(0.99) = 0.0098 + 0.0495 = 0.0593\]

\[P(\text{disease} | +) = \frac{0.0098}{0.0593} = 0.1653 = 16.53\%\]

Despite a 98% accurate test, a positive result only means a $16.53\%$ chance of disease. This is the base rate fallacy — when the disease is rare, false positives dominate.

K9. A mortgage of $350,000 at 6.5% for 30 years (compounded monthly). What is the monthly payment? What percentage of the first payment goes to interest vs. principal?

Monthly payment:

\[M = P \times \frac{r(1+r)^n}{(1+r)^n - 1}\]

where $r = \frac{0.065}{12} = 0.005417$ and $n = 360$:

\[M = 350000 \times \frac{0.005417 \times (1.005417)^{360}}{(1.005417)^{360} - 1}\]

\[(1.005417)^{360} = 6.9916\]

\[M = 350000 \times \frac{0.005417 \times 6.9916}{6.9916 - 1} = 350000 \times \frac{0.03788}{5.9916} = 350000 \times 0.006322 = \$2{,}212.71\]

First payment breakdown:

Interest: $350000 \times 0.005417 = \$1{,}895.83 \quad (85.7\%)$

Principal: $2212.71 - 1895.83 = \$316.88 \quad (14.3\%)$

In the first payment, $85.7\%$ goes to the bank. This ratio reverses over 30 years.

K10. A country’s GDP grew at these real rates over 5 years: +3.2%, +2.8%, -1.5%, +4.1%, +3.6%. Inflation averaged 2.3% per year. What was the nominal CAGR? What was the total real growth?

Real growth multiplier:

\[1.032 \times 1.028 \times 0.985 \times 1.041 \times 1.036 = 1.1271\]

Total real growth: $12.71\%$

Real CAGR: $(1.1271)^{1/5} - 1 = 2.42\%$

Nominal CAGR (approximately, using Fisher equation):

\[(1 + r_{\text{nominal}}) \approx (1 + r_{\text{real}})(1 + r_{\text{inflation}})\]

\[r_{\text{nominal}} \approx (1.0242)(1.023) - 1 = 0.0478 = 4.78\%\]

Appendix: Formula Reference Card

Concept	Formula
Percentage of a number	\(\text{part} = \frac{p}{100} \times n\)
Percentage change	\(\frac{\text{new} - \text{old}}{\text{old}} \times 100\)
Reverse percentage (find original)	\(n = \frac{\text{final}}{1 \pm \frac{p}{100}}\)
Successive changes	\(\text{final} = \text{original} \times \prod(1 + \frac{p_i}{100})\)
Compound interest	\(A = P(1 + \frac{r}{n})^{nt}\)
Continuous compounding	\(A = Pe^{rt}\)
Rule of 72	\(t_{\text{double}} \approx \frac{72}{r\%}\)
Annuity (future value)	\(FV = PMT \times \frac{(1+r)^n - 1}{r}\)
Mortgage payment	\(M = P \times \frac{r(1+r)^n}{(1+r)^n - 1}\)
Declining balance depreciation	\(V(t) = V_0(1 - \frac{d}{100})^t\)
Weighted average	\(\bar{x}_w = \frac{\sum w_i x_i}{\sum w_i}\)
Markup	\(\frac{\text{profit}}{\text{cost}} \times 100\)
Margin	\(\frac{\text{profit}}{\text{revenue}} \times 100\)
CAGR	\((\frac{V_f}{V_i})^{1/t} - 1\)
Bayes' theorem	stem:[P(A
B) = \frac{P(B	A) \cdot P(A)}{P(B)}]

Concept

Formula

Percentage of a number

$\text{part} = \frac{p}{100} \times n$

Percentage change

$\frac{\text{new} - \text{old}}{\text{old}} \times 100$

Reverse percentage (find original)

$n = \frac{\text{final}}{1 \pm \frac{p}{100}}$

Successive changes

$\text{final} = \text{original} \times \prod(1 + \frac{p_i}{100})$

Compound interest

$A = P(1 + \frac{r}{n})^{nt}$

Continuous compounding

$A = Pe^{rt}$

Rule of 72

$t_{\text{double}} \approx \frac{72}{r\%}$

Annuity (future value)

$FV = PMT \times \frac{(1+r)^n - 1}{r}$

Mortgage payment

$M = P \times \frac{r(1+r)^n}{(1+r)^n - 1}$

Declining balance depreciation

$V(t) = V_0(1 - \frac{d}{100})^t$

Weighted average

$\bar{x}_w = \frac{\sum w_i x_i}{\sum w_i}$

Markup

$\frac{\text{profit}}{\text{cost}} \times 100$

Margin

$\frac{\text{profit}}{\text{revenue}} \times 100$

CAGR

$(\frac{V_f}{V_i})^{1/t} - 1$

Bayes' theorem

stem:[P(A

B) = \frac{P(B

A) \cdot P(A)}{P(B)}]

Fraction	Decimal	Percentage
\(\frac{1}{2}\)	\(0.5\)	\(50\%\)
\(\frac{1}{4}\)	\(0.25\)	\(25\%\)
\(\frac{3}{4}\)	\(0.75\)	\(75\%\)
\(\frac{1}{5}\)	\(0.2\)	\(20\%\)
\(\frac{1}{8}\)	\(0.125\)	\(12.5\%\)
\(\frac{1}{3}\)	\(0.\overline{3}\)	\(33.\overline{3}\%\)
\(\frac{2}{3}\)	\(0.\overline{6}\)	\(66.\overline{6}\%\)
\(\frac{1}{10}\)	\(0.1\)	\(10\%\)
\(\frac{3}{8}\)	\(0.375\)	\(37.5\%\)