🏷️ 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, 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 $\frac{1}{4} \times 60 = 15$.

🏷️ 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.

🏷️ 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.

🏷️ 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.

🏷️ 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.

🏷️ 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.