Percents and Finance

TSI Math Study Guide

Percents and Finance

Percents are another way of expressing part of a whole. On the TSI they show up in real-world situations: discounts, markups, taxes, percent increase or decrease, simple and compound interest, and profit or loss.

The key skill: knowing which quantity is the whole, which is the part, and how the percent connects them.

Understanding percents

Percent — “per hundred.” 25% of a quantity means 0.25 times that quantity.

25% = 25100 = 0.25     35 = 0.6 = 60%     0.08 = 8%

The three percent problems

Part = Percent × Whole   (percent as a decimal)

Find the part

What is 35% of 240?

35% = 0.35convert0.35(240) = 84multiply

Find the percent

42 out of 50 correct — what percent?

4250 = 0.84part ÷ whole84%convert

Find the whole

30 is 20% of what number?

0.20x = 30let x be the wholex = 300.20 = 150divide

Percent increase and decrease

Percent change = amount of changeoriginal amount × 100%

⚠ Watch out

Always divide by the original amount, not the new amount.

Increase

A ticket goes from $40 to $50.

50 − 40 = 10the change1040 = 0.25÷ original25% increase

Decrease

A jacket goes from $80 to $60.

80 − 60 = 20the change2080 = 0.25÷ original25% decrease

The multiplier method

To find the new amount directly, multiply by a single factor.

p% increase

× (1 + p100)

$50 fee, up 12%: 50(1.12) = $56

p% decrease

× (1 − p100)

$90 item, 20% off: 90(0.80) = $72

The multiplier method is especially useful when several percent changes happen in a row.

Discounts, markups, and tax

Each can be solved two ways: compute the change and add or subtract it, or multiply by the right factor.

Discount

$120 shoes, 25% off.

0.25(120) = 30the discount120 − 30 = $90subtract

Or: 75% remains, so 120(0.75) = $90

Markup

Cost $60, marked up 40%.

0.40(60) = 24the markup60 + 24 = $84add

Or: 60(1.40) = $84

Sales tax

$75 item, 8% tax.

0.08(75) = 6the tax75 + 6 = $81add

Or: 75(1.08) = $81

⚠ Watch out

A 40% markup means the price increases by 40%. That’s different from a selling price that is 40% of the cost.

Discount followed by tax

ProblemA $200 item is discounted by 15%, then 6% sales tax is added. What is the final cost?
200(0.85) = 170discount first 170(1.06) = $180.20tax on the discounted price
⚠ Watch out

Don’t subtract 15% and add 6% to get a net 9% decrease. The two percents are applied to different amounts.

Profit and loss

Profit

Selling price greater than cost.Profit = Selling price − Cost

Loss

Selling price less than cost.Loss = Cost − Selling price

ProblemA store buys an item for $45 and sells it for $60. What is the percent profit?
60 − 45 = 15the profit 1545 × 100% ≈ 33.3%relative to the original cost

Simple interest

Simple interest — calculated only on the original principal.
I = PrtI  interestP  principalr  annual rate as a decimalt  time in years
Problem$2,000 is invested at 5% simple interest for 3 years. How much interest is earned?
I = 2000(0.05)(3)substitute I = $300total in the account: 2000 + 300 = $2,300

Compound interest

Compound interest — interest is added to the balance, and future interest is calculated on that new balance.
A = P(1 + r)tA  final amountP  principalr  annual rate as a decimalt  years (compounded annually)
Problem$1,000 is invested at 4% compounded annually for 3 years. What is the balance, and how much interest is earned?
A = 1000(1.04)3substitute A ≈ 1000(1.124864) A ≈ $1,124.86interest: 1124.86 − 1000 = $124.86

Simple vs. compound

Simple

Interest is always based on the original principal.

Compound

The balance grows, so later interest is calculated on principal plus earlier interest. For the same principal, rate, and time, compound interest produces the larger final balance.

Repeated percent change

Percent changes can describe growth or decline over several periods. A car loses 10% of its value each year, starting at $20,000:

20000(0.90) = 18000after year 1 18000(0.90) = 16200after year 2 20000(0.90)2 = 16200same thing in one step
⚠ Watch out

A 10% decrease each year does not mean subtracting the same dollar amount each year. The percent is applied to the new value each time.

This idea returns in Exponential, Radical, and Rational Equations as exponential growth and decay.

TSI strategy: identify the original amount

The most important question in a percent problem

What amount is the percent based on?

Percent change

The original amount.

Discount

The original price.

Sales tax

The taxable purchase price.

Simple interest

The original principal.

Identify the correct base first and the calculation usually becomes easy.

🔑 Key tip: translate percent language into multiplication

“25% of 80”

0.25(80)

“increase 80 by 25%”

80(1.25)

“decrease 80 by 25%”

80(0.75)

Recognizing the right multiplier is one of the fastest ways to solve percent problems accurately.

Percents and Finance Review Quiz

Next: Linear Equations and Functions

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