Percents, Discounts and Interest

🏷️ Percents, Discounts and Interest

Part 1 of 7 — Percents on the AR Subtest

Topic Overview

What these questions look like

The official scope of Arithmetic Reasoning (AR) names "interest and percentage" outright, and one of the official sample topics is percent depreciation. So percent problems are a regular part of AR. They show up as everyday money stories:

• a jacket that is 25% off,
• sales tax or a tip added to a bill,
• a paycheck with base pay plus commission,
• a truck that loses 10% of its value each year,
• a loan or a savings account that earns simple interest.

The math is almost always one of three moves: find a percent of a number, find what percent one number is of another, or find the whole when you know a part. What trips people up is the story around the math: which number is the starting value, whether the question wants the discount or the price you actually pay, and whether "18 months" has been turned into years.

Percents on test day

Some questions may be unscored tryout items, and you can't tell which ones, so treat every question as scored. If you take a paper version, the pace is much faster, so the shortcuts in Part 2 matter even more.

Why it matters

AR is one of the four subtests in your AFQT score (AR + MK + 2 × VE, using standard scores, turned into a 1–99 percentile), and the AFQT decides whether you can enlist. AR also feeds job line scores such as the Army GT and Air Force G scores (AR + VE) and the Marine Corps GT score (AR + MC + VE). Percent problems are some of the most learnable points on AR: the same handful of setups appears again and again.

The AR habits, applied to percents

1. Read the last sentence first. Does it ask for the discount or the sale price? The interest or the total owed? The commission or the total pay? Circle it on your scratch paper before you start.
2. Translate words to an equation. "Is" means equals and "of" means times: "What is 15% of 60?" becomes $\text{answer} = 0.15 \times 60$.
3. Track units. Interest rates are per year, so time must be in years. Write "dollars" or "%" next to each number so you don't hand in a percent when the question wants dollars.
4. Estimate first. 10% of anything is easy (move the decimal point one place left). Use it to know roughly where the answer sits.
5. Use shortcuts instead of long multiplication. 25% is one quarter, 20% is one fifth, 15% is 10% plus half of that.
6. Check reasonableness. A sale price must be less than the original. A price with tax must be more. Interest on a small loan for one year is a small number.
7. Use the answer choices. On "what was the original price?" questions, you can test each choice: take the discount off and see which one lands on the sale price.

Key Fact: On percent problems, most wrong answers come from answering a different question than the one asked, not from bad arithmetic. Read the last sentence first.

Worked Example: a sale price

A jacket regularly priced at $80 is on sale for 15% off. What is the sale price?

$12 · $65 · $68 · $92

Step 1 — Last sentence. We want the sale price: what you actually pay, in dollars.

Step 2 — Estimate. 10% of $80 is $8, so 15% off is a bit more than $8 off. The sale price should be a little less than $72.

Step 3 — Compute with shortcuts.

• 10% of 80: move the decimal point, $80 \to 8$
• 5% of 80: half of that, $8 \div 2 = 4$
• 15% of 80: $8 + 4 = 12$, so the discount is $12
• Sale price: $80 - 12 = 68$, so $68

Step 4 — Check against the story and the choices. $68 is less than $80 and just under our estimate of $72. Here is where the other choices come from:

Key Fact: Every percent problem has a base, the "whole" the percent is taken of. Here the base is the original price, $80.

Warm-up. Two quick percent problems. Read the last sentence first. 🔥

Warm-up, continued. ✍️

A pair of work boots costs $60 and is on sale for 20% off.

1) What is the sale price, in dollars? (number only)

Key Takeaways — Part 1

• Percent problems on AR are money stories: discounts, tax and tips, commission, depreciation and simple interest
• Every problem has a base (the whole the percent is taken of); find it first
• Read the last sentence: discount or sale price? interest or total?
• No calculator: lean on 10%, 5%, 25% and 50% shortcuts, and estimate before you compute
• A sale price is below the original; a price with tax is above it

Up Next: Part 2 — percent of a number, the three question types, and the mental-math shortcuts that replace a calculator.

🏷️ Percents, Discounts and Interest

Part 2 of 7 — Percent of a Number and Mental-Math Shortcuts

Percent means "per hundred"

$37\%$ means 37 out of every 100, or $\frac{37}{100} = 0.37$. To use a percent in arithmetic, turn it into a decimal or a fraction:

• Percent → decimal: move the decimal point two places left. $8\% = 0.08$, $45\% = 0.45$, $150\% = 1.5$
• Decimal → percent: move it two places right. $0.6 = 60\%$, $0.035 = 3.5\%$

Percents worth knowing as fractions

Key Fact: "Of" means multiply. 25% of 60 is $\frac{1}{4} \times 60 = 15$.

The three question types

Every basic percent question fits one sentence: part = percent × whole. "Is" means equals and "of" means times, so translate the words directly. The question tells you two of the three pieces; you find the third.

Worked Example 1 — find the part

A company of 160 soldiers is asked to volunteer for a blood drive, and 15% sign up. How many sign up?

Estimate: 10% of 160 is 16, so the answer is a bit more than 16.

$10\% \text{ of } 160 = 16, \qquad 5\% \text{ of } 160 = 8, \qquad 15\% \text{ of } 160 = 16 + 8 = 24$

24 soldiers sign up. That is a bit more than 16, as estimated.

Worked Example 2 — find the percent

A mechanic has finished 12 of the 48 work orders on the board. What percent has she finished?

The whole is the 48 orders; the part is the 12 finished.

$\frac{12}{48} = \frac{1}{4} = 25\%$

Reasonable? 12 is clearly less than half of 48, and 25% is less than half. ✓

Worked Example 3 — find the whole

A recruit has saved $30, which is 20% of the price of a new phone. How much does the phone cost?

Set up "30 is 20% of what": $30 = 0.20 \times w$, so $w = 30 \div 0.20 = 150$. Or think in fractions: 20% is one fifth, so the whole is $5 \times 30 = 150$. The phone costs $150.

Check: 20% of 150 is $\frac{150}{5} = 30$. ✓ And the whole must be bigger than the part when the percent is under 100%.

Mental-math shortcuts (your calculator replacement)

Start from 10% and build everything else from it.

The swap trick

$a\%$ of $b$ is the same as $b\%$ of $a$, because both are $\frac{a \times b}{100}$. Use whichever is easier:

$8\% \text{ of } 50 = 50\% \text{ of } 8 = 4 \qquad 4\% \text{ of } 75 = 75\% \text{ of } 4 = 3$

Worked Example 4 — a 35% problem without a calculator

A unit's 60-hour training block spends 35% of its hours on land navigation. How many hours is that?

$25\% \text{ of } 60 = 15, \qquad 10\% \text{ of } 60 = 6, \qquad 35\% \text{ of } 60 = 15 + 6 = 21$

21 hours. Estimate check: 35% is a bit more than a third, and a third of 60 is 20. ✓

Check: the three question types. Set up part = percent × whole each time. 🎯

Use the shortcuts. No calculator. ✍️

1) What is 5% of 360? (number only)

2) Use the swap trick: what is 8% of 25? (number only)

3) 45 is what percent of 180? (number only, in percent)

Translate and convert. 🔽

Key Takeaways — Part 2

• Percent means per hundred: move the decimal point two places to convert
• Part = percent × whole. "Is" means equals, "of" means times
• Find the part: multiply. Find the percent: part ÷ whole. Find the whole: part ÷ percent
• Build percents from 10% (move the point): 5% is half, 20% is double, 15% = 10% + 5%
• 25% = divide by 4, 20% = divide by 5, 50% = divide by 2
• Swap trick: 8% of 50 = 50% of 8

Up Next: Part 3 — percent change, discounts, markups and depreciation, and why a 20% discount followed by a 10% discount is not 30% off.

🏷️ Percents, Discounts and Interest

Part 3 of 7 — Percent Change, Discounts and Depreciation

Percent change

When a value goes up or down, the percent change compares the change to the original value, the one you started with:

$\text{percent change} = \frac{\text{amount of change}}{\text{original value}}$

Worked Example 1 — a percent increase

An apartment's monthly rent goes from $800 to $880. By what percent did the rent increase?

Change: $880 - 800 = 80$. Original: 800.

$\frac{80}{800} = \frac{1}{10} = 10\%$

Worked Example 2 — a percent decrease

A pair of headphones drops from $60 to $45. By what percent did the price decrease?

Change: $60 - 45 = 15$. Original: 60 (the price before the drop).

$\frac{15}{60} = \frac{1}{4} = 25\%$

If you divide by the new price instead, you get $\frac{15}{45} = 33\frac{1}{3}\%$, a wrong answer that is often waiting among the choices.

Key Fact: Percent change always divides by the original (starting) value, whether the value went up or down.

Discounts and markups in one step

You can find a discount and subtract it (two steps), or multiply by what is left (one step):

Worked Example 3 — a discount

A $56 jacket is 25% off. What is the sale price?

One step: 75% is three quarters, so $56 \div 4 = 14$ and $14 \times 3 = 42$. The sale price is $42.

Two-step check: the discount is 25% of 56, which is $14, and $56 - 14 = 42$. ✓

Worked Example 4 — a markup

A store pays $40 for a backpack and marks the price up 30%. What is the store's selling price?

Markup: 30% of 40 is $3 \times 4 = 12$. Selling price: $40 + 12 = 52$, so $52. (One step: $40 \times 1.3 = 52$.)

A markup is a percent increase on the store's cost; a discount is a percent decrease on the original price.

Depreciation: losing value over time

Depreciation means something (a vehicle, a piece of equipment) loses value as it gets older. On AR it is a percent decrease, sometimes for one year and sometimes for two years in a row.

Worked Example 5 — one year

A truck worth $24,000 loses 15% of its value in its first year. What is it worth after one year?

$10\% \text{ of } 24{,}000 = 2{,}400, \qquad 5\% = 1{,}200, \qquad 15\% = 3{,}600$

Value after one year: $24{,}000 - 3{,}600 = 20{,}400$, so $20,400.

Worked Example 6 — two years in a row

A generator worth $18,000 loses 10% of its value each year. What is it worth after two years?

Work one year at a time, and take each year's percent of the value at the start of that year:

After two years it is worth $14,580. Taking 20% off the original in one shot gives $14,400, which is wrong: the second year's 10% is taken from the smaller value, $16,200, so the second loss is smaller.

Key Fact: Percents applied one after another are not added. Each new percent is taken of the new value.

Successive discounts and reverse problems

Two discounts in a row

A $100 drill is 20% off, and a coupon takes another 10% off the sale price. What do you pay?

• After 20% off: $100 \times 0.8 = 80$
• After 10% off the $80: $80 \times 0.9 = 72$

You pay $72. That is $28 off in total, a 28% discount, not 30%.

Working backward to the original price

After a 10% discount, a tool bag costs $63. What was the original price?

The sale price is 90% of the original: $\text{original} \times 0.9 = 63$. Undo the multiplication by dividing:

$63 \div 0.9 = 630 \div 9 = 70$

The original price was $70. Check: 10% of 70 is 7, and $70 - 7 = 63$. ✓

The tempting mistake is to add 10% of $63 back on: $63 + 6.30 = 69.30$. That is wrong because the 10% discount was taken of the original price, not of $63.

Key Fact: To find an original price, divide the new amount by the multiplier (0.9 for 10% off, 0.8 for 20% off). If you are unsure, test each answer choice: take the discount off and see which one lands on the sale price.

Check: change, depreciation and reverse problems. 🎯

Discounts, decreases and markups. ✍️

1) A $70 coat is 30% off. What is the sale price? (number only, in dollars)

2) A price falls from $80 to $68. What is the percent decrease? (number only, in percent)

3) A shop pays $25 for a hat and marks it up 40%. What is the selling price? (number only, in dollars)

Key Takeaways — Part 3

• Percent change = change ÷ original value, for increases and decreases alike
• Discount in one step: multiply by what is left (25% off → $\times 0.75$); markup: multiply by more than 1 (30% up → $\times 1.3$)
• Depreciation over two years: work year by year, each percent taken of that year's starting value
• Successive percents are not added: 20% off then 10% off is 28% off, not 30%
• Reverse problems: divide by the multiplier ($63 after 10% off → $63 \div 0.9 = 70$); never just add the percent back

Up Next: Part 4 — sales tax, tips, discount-then-tax, and commission on a paycheck.

🏷️ Percents, Discounts and Interest

Part 4 of 7 — Sales Tax, Tips and Commission

Sales tax

Sales tax is a percent of the price added on. Two ways to get the total:

• Two steps: find the tax, then add it to the price.
• One step: multiply by $1$ plus the tax rate as a decimal. A 6% tax means multiply by $1.06$.

Worked Example 1 — sales tax

A shirt costs $40, and the sales tax is 6%. What is the total cost?

Tax: 6% of 40. Use the swap trick: 6% of 40 = 40% of 6 = 2.4, so the tax is $2.40.

Total: $40 + 2.40 = 42.40$, so $42.40. (One step: $40 \times 1.06 = 42.40$.)

Reasonable? The total is a little more than $40, because tax only adds a small amount. ✓

Tips

A tip is a percent of the bill, usually 15% or 20%. Use the 10% shortcut:

Worked Example 2 — a tip

A dinner bill is $46. What is a 15% tip, and what is the total with a 20% tip?

• 10% of 46 is 4.60, and 5% is half of that, 2.30
• 15% tip: $4.60 + 2.30 = 6.90$, so $6.90
• 20% tip: double 10%, $2 \times 4.60 = 9.20$, so $9.20, and the total is $46 + 9.20 = 55.20$, or $55.20

Key Fact: Read the last sentence. "How much is the tip?" and "How much did she pay in all?" have different answers, and both are usually among the choices.

Discount first, then tax

When an item is on sale and taxed, the store takes the discount off first and then charges tax on the sale price.

Worked Example 3 — discount then tax

Boots priced at $80 are 25% off, and the sales tax is 7%. What is the total cost?

Step 1 — Sale price. 25% off means pay 75%: $80 \div 4 = 20$, so $80 - 20 = 60$, or $60.

Step 2 — Tax on the sale price. 7% of 60 is $7 \times 0.6 = 4.2$, so the tax is $4.20.

Step 3 — Total. $60 + 4.20 = 64.20$, so $64.20.

A common wrong path: tax on the original price ($80 + 5.60 = 85.60$), then subtract the $20 discount, which gives $65.60. That charges tax on money you never paid.

Commission

A commission is a percent of what a salesperson sells. Many jobs pay base pay plus commission.

Worked Example 4 — straight commission

A car salesperson earns a 6% commission. This week she sold $3,500 worth of accessories. What is her commission?

1% of 3,500 is 35, so 6% is $6 \times 35 = 210$. Her commission is $210.

Worked Example 5 — base pay plus commission

A sales associate earns $400 a week plus 5% of his sales. He sold $2,600 this week. What is his total pay for the week?

• Commission: 10% of 2,600 is 260, so 5% is 130
• Total pay: $400 + 130 = 530$, so $530

Reasonable? The commission is a small slice of the sales, and the total is a bit more than the base pay. ✓

Worked Example 6 — finding the sales from the commission

An installer earned a $360 commission at a rate of 8%. How much did he sell?

This is a "find the whole" question: $360 = 0.08 \times \text{sales}$.

$\text{sales} = 360 \div 0.08 = 36{,}000 \div 8 = 4{,}500$

He sold $4,500. Check: 1% of 4,500 is 45, and $8 \times 45 = 360$. ✓ The sales must be much bigger than the commission.

Check: tips and paychecks. Read the last sentence before you choose. 🎯

Tax, discount-then-tax, and commission. ✍️

1) An item costs $50, and the sales tax is 8%. What is the total cost? (number only, in dollars)

2) A $200 bike is 30% off, and then a 5% sales tax is charged on the sale price. What is the total cost? (number only, in dollars)

3) A salesperson earns a 5% commission on $3,600 of sales. What is the commission? (number only, in dollars)

Key Takeaways — Part 4

• Sales tax: total = price × (1 + rate). A 6% tax means multiply by 1.06
• Tips: 10% (move the point), then halve for 5% or double for 20%
• Discount first, then tax on the sale price
• Commission = rate × sales; total pay = base + commission
• Sales from a commission: commission ÷ rate (a "find the whole" problem)
• Know whether the question wants the tax, tip or commission or the final total

Up Next: Part 5 — simple interest: $I = P \times r \times t$, with time in years.

🏷️ Percents, Discounts and Interest

Part 5 of 7 — Simple Interest

The formula

Interest is money paid for the use of money. A bank pays you interest on savings; you pay a lender interest on a loan. AR uses simple interest, which is figured on the original amount only:

$I = P \times r \times t$

The total in a savings account, or the total owed on a loan, is the principal plus the interest: $P + I$.

Worked Example 1 — a loan over several years

Marcus borrows $2,000 at 6% simple interest for 3 years. How much interest does he pay, and how much does he owe in all?

One year of interest: 6% of 2,000 is $6 \times 20 = 120$. Three years: $120 \times 3 = 360$.

$I = 2{,}000 \times 0.06 \times 3 = 360$

He pays $360 in interest and owes $2{,}000 + 360 = 2{,}360$, or $2,360 in all.

Key Fact: Find one year of interest first (rate × principal), then multiply by the number of years. It is the same as $P \times r \times t$, and it is easier to do in your head.

Time in months: convert to years

The rate is per year, so time has to be in years. Divide the months by 12:

Worked Example 2 — six months

$1,500 is saved at 4% simple interest for 6 months. How much interest does it earn?

One year: 4% of 1,500 is $4 \times 15 = 60$. Six months is half a year: $60 \times \frac{1}{2} = 30$. It earns $30.

Using 6 as if it were years gives $60 \times 6 = 360$, twelve times too much. Reasonableness check: half a year of interest must be less than one year's $60.

Worked Example 3 — eighteen months

$2,200 is borrowed at 5% simple interest for 18 months. How much interest is owed?

One year: 5% of 2,200 is half of 220, which is 110. Eighteen months is $1\frac{1}{2}$ years: $110 + 55 = 165$. The interest is $165.

Finding the rate or the time

If the problem gives the interest and asks for the rate or the time, use the same formula and divide.

Worked Example 4 — find the rate

A deposit of $800 earns $96 in simple interest over 2 years. What is the yearly interest rate?

Interest per year: $96 \div 2 = 48$. As a percent of the principal: $\frac{48}{800} = \frac{6}{100} = 6\%$.

Check: 6% of 800 is 48, and $48 \times 2 = 96$. ✓

Worked Example 5 — find the time

$1,200 is invested at 5% simple interest. How many years will it take to earn $180 in interest?

One year earns 5% of 1,200, which is 60. Years: $180 \div 60 = 3$. It takes 3 years.

Savings or loan, the math is the same

Worked Example 6 — equal monthly payments

Tasha borrows $1,800 for a used motorcycle at 5% simple interest for 2 years. She repays the loan plus interest in equal monthly payments. How much is each payment?

• Interest: 5% of 1,800 is 90 per year, so $90 \times 2 = 180$
• Total to repay: $1{,}800 + 180 = 1{,}980$
• Number of payments: 2 years × 12 = 24 months
• Each payment: $1{,}980 \div 24 = 82.50$, so $82.50

Estimate: $1{,}800 \div 24$ is 75, so each payment should be a bit more than $75. ✓

Check: simple interest. Rate as a decimal, time in years. 🎯

Interest, totals and payments. ✍️

1) $4,000 is saved at 3% simple interest for 5 years. How much interest does it earn? (number only, in dollars)

2) Ray borrows $1,200 at 8% simple interest for 18 months. How much does he owe in all, principal plus interest? (number only, in dollars)

3) A $1,000 loan at 8% simple interest for 1 year is repaid, with interest, in 12 equal monthly payments. How much is each payment? (number only, in dollars)

Key Takeaways — Part 5

• Simple interest: $I = P \times r \times t$, with the rate as a decimal and the time in years
• Shortcut: one year of interest (rate × principal), then multiply by the years
• Months → years: divide by 12 (6 months = $\frac{1}{2}$ year, 18 months = 1.5 years)
• Total owed or saved = principal + interest
• Find the rate: (interest per year) ÷ principal. Find the time: interest ÷ (one year of interest)
• Equal monthly payments: (principal + interest) ÷ number of months

Up Next: Part 6 — the traps the test sets on percent problems, and how to dodge each one.

🏷️ Percents, Discounts and Interest

Part 6 of 7 — Test Strategy and Common Traps

The traps, one by one

The wrong answer choices on percent problems are not random. Each one is the result of a specific, predictable slip. Learn the slips and you will recognize the traps before you fall into them.

Key Fact: The trap answers are usually sitting right there among the four choices. Finding a choice that matches your number does not prove it is right; check it against the question and the story.

Timing, estimating and back-solving

Timing

On the CAT-ASVAB you have about 3.5 minutes per AR question on average. A one-step percent question ("What is 20% of 85?") should take under a minute with the shortcuts. That banked time pays for the multi-step items: discount then tax, two years of depreciation, interest with monthly payments. Since you can't go back, spend the time to finish each problem carefully rather than rushing and moving on.

Estimate to eliminate

Before you compute, decide what the answer must look like:

• A sale price is less than the original. Any choice above the original is out.
• A price with tax or a tip is a little more than the bill. A choice that doubles it is out.
• An original price (from a sale price) is more than the sale price.
• Interest for a year or two is usually much smaller than the principal; a choice bigger than the principal probably used the rate as a whole number.

Back-solving a reverse problem

After a 25% discount, a radio costs $36. What was the original price?

$27 · $45 · $48 · $61

Instead of setting up an equation, test the choices. 25% off means you pay three quarters:

• $48: $48 \div 4 = 12$, and $48 - 12 = 36$ ✓

The original price was $48. The trap choice $45 comes from adding 25% of $36 back on; $27 takes another 25% off; $61 adds $25.

The reasonableness checklist

1. Did I answer the question in the last sentence (final price, or the discount/tax/interest)?
2. Did I divide by the original value for a percent change?
3. Did I apply back-to-back percents one at a time?
4. Is my time in years and my rate a decimal?
5. Does the answer make sense in the story (sale price below original, total above the bill)?

Spot the trap. Each question has a trap answer waiting. Find the real answer. 🕵️

Key Takeaways — Part 6

• Wrong base: percent change divides by the original value
• Successive percents are applied one at a time, never added
• Reverse problems: divide by the multiplier, or back-solve from the choices
• Answer the question asked: final amount vs the discount, tax, tip or interest
• Interest: time in years (months ÷ 12), rate as a decimal
• Estimate to eliminate: a sale price is below the original; a total with tax is above the price

Up Next: Part 7 — a mixed mastery set covering every skill in this topic.

🏷️ Percents, Discounts and Interest

Part 7 of 7 — Mixed Mastery Set

This set mixes every skill in the topic, in roughly the order of difficulty you would meet on the CAT-ASVAB: a few quick one-step items, then multi-step items, then the kind that comes after a run of right answers.

For each problem:

1. Read the last sentence first: final amount, or the discount, tax, commission or interest?
2. Find the base: what is the percent taken of?
3. Estimate, then compute with the shortcuts (10%, 5%, 25%, one year of interest).
4. Check against the story and the trap list from Part 6.

No calculator. Use scratch paper, just like at MEPS.

Mastery set, part A. One- and two-step problems. 🎯

Mastery set, part B. Multi-step problems. Watch for the traps. 🧠

Mastery set, part C. ✍️

1) A practice test has 85 questions. Lee answered 40% of them in the first 20 minutes. How many questions did he answer in that time? (number only)

2) A $200 tent is 25% off, and a coupon takes an extra 10% off the sale price. What is the final price? (number only, in dollars)

Percents, Discounts and Interest — Summary 🎓

The one-line checklist

Last sentence → base → estimate → compute → check the story.

What to do next

1. Take the exit quiz for this topic to lock in the skills.
2. Review the flashcards: the one-step multipliers (×0.75 for 25% off, ×1.06 for 6% tax) and $I = P \times r \times t$ are worth memorizing.
3. Move on to the next topic in the AR unit. Percents show up again inside ratio, rate and average problems, so the shortcuts you built here keep paying off.