Mathematics II · Calculator allowed

Percent Problems in Financial Contexts

Tips, discounts, sales tax, commission, and interest are all the same idea wearing different hats: a percent of an amount of money. Learn two reliable methods and the money math stops being scary.

Percent just means "out of 100." When you see 25%, think 25 out of every 100, or 25⁄100, which is the same as the decimal 0.25. Almost every money question on the CAEC, a 15% tip, a 20% discount, 13% sales tax, is really asking you to find a percent of a dollar amount.

This is the calculator part of the test, so you may use a calculator and you are given a formula sheet. Your job is to set up the calculation correctly. We'll use two methods, and you can pick whichever one clicks for you.

Two methods that always work

You only need one of these, but it helps to see both. They give the exact same answer.

Method 1 · Proportion
is ⁄ of = % ⁄ 100

The amount you want is the "is," the whole (total) is the "of," and the percent sits over 100. Fill in the three numbers you know, then cross-multiply to find the missing one.

Method 2 · Decimal multiplier
percent × whole = part

Turn the percent into a decimal (move the point two places left, so 15% becomes 0.15), then multiply by the whole amount. Fast and great on a calculator.

Turning a percent into a decimal: divide by 100, which is the same as moving the decimal point two spots to the left. 7% = 0.07, 13% = 0.13, 150% = 1.5, and 6.5% = 0.065.

Worked example #1: a 15% tip on a $48 meal

Your restaurant bill is $48 and you want to leave a 15% tip. How much is the tip? This is simply "find 15% of 48." Here it is both ways.

  • Proportion: set up is⁄of = %⁄100, so x⁄48 = 15⁄100. Cross-multiply: 100x = 15 × 48 = 720, then x = 720 ÷ 100 = 7.2.
  • Decimal multiplier: 15% = 0.15, then 0.15 × 48 = 7.2.
Proportion:   x/48 = 15/100
              100x = 15 × 48 = 720
              x = 720 ÷ 100 = 7.20

Multiplier:   0.15 × 48 = 7.20
Answer: the tip is $7.20. The total you'd pay is 48 + 7.20 = $55.20.

Worked example #2: a 20% discount, then 13% sales tax

A jacket is marked $80 with 20% off. After the discount, 13% sales tax is added. What do you pay at the till? Do it in order: discount first, then tax on the new price.

  • Find the discount: 20% of 80 = 0.20 × 80 = 16. So you save $16.
  • Subtract it: 80 − 16 = 64. The sale price is $64.
  • Add the tax: 13% of 64 = 0.13 × 64 = 8.32, then 64 + 8.32 = 72.32.
Discount:  0.20 × 80   = 16.00
Sale price: 80 − 16     = 64.00
Tax:       0.13 × 64    =  8.32
Total:     64 + 8.32    = 72.32
Answer: you pay $72.32. Shortcut for next time: taking 20% off leaves 80% of the price, so 0.80 × 80 = 64 in one step. Adding 13% tax means paying 113%, so 1.13 × 64 = 72.32.

Percent increase and percent decrease

A markup, a raise, or sales tax is a percent increase. A discount, a markdown, or depreciation is a percent decrease. The shortcut is to adjust 100%:

  • Increase by r%: multiply by (1 + r⁄100). A 13% increase means × 1.13.
  • Decrease by r%: multiply by (1 − r⁄100). A 20% decrease means × 0.80.
Original100%−20%80%gone+13%100%+13113%

A decrease takes a slice off; an increase adds a slice on. Either way you are scaling the original 100%.

To find a percent change from two amounts, use: change ⁄ original × 100. If a price rises from $40 to $50, the change is 10, so 10 ÷ 40 = 0.25 = a 25% increase. Always divide by the original amount, not the new one.

Worked example #3: commission on a sale

A salesperson earns 6% commission on what they sell. Last week they sold $12,500 worth of goods. How much commission did they earn? Commission is just a percent of the sales total.

  • Set it up: 6% of 12,500 = 0.06 × 12,500.
  • Calculate: 0.06 × 12,500 = 750.
Commission = 6% of 12,500
           = 0.06 × 12,500
           = 750
Answer: $750 in commission. Quick sense check: 6% is a bit more than 5% (which would be $625), so $750 looks right.

Worked example #4: simple interest (I = Prt)

Good news: the simple interest formula I = Prt is given to you on the CAEC formula sheet, so you don't have to memorize it. P is the principal (starting money), r is the yearly rate as a decimal, and t is the time in years.

You deposit $2,000 in an account that pays 4% simple interest per year. How much interest do you earn in 3 years, and what is the total?

  • Identify the parts: P = 2000, r = 4% = 0.04, t = 3.
  • Plug into I = Prt: I = 2000 × 0.04 × 3.
  • Calculate: 2000 × 0.04 = 80 (one year), then 80 × 3 = 240.
  • Find the total: 2000 + 240 = 2240.
I = P r t
I = 2000 × 0.04 × 3
I = 240            (interest earned)

Total = P + I
      = 2000 + 240 = 2240
Answer: $240 interest, $2,240 total. Watch the units: r must be a decimal (0.04, not 4) and t must be in years. If a question gave you 6 months, that would be t = 0.5.

Tips that keep money problems simple

  • Decide what is the "whole." The percent is always taken of something. Find that total first, it is the "of" in the proportion.
  • Convert the percent to a decimal early. Move the point two places left. This one habit prevents most percent mistakes.
  • Read "of" as multiply. "15% of $48" literally means 0.15 × 48.
  • Do steps in order for tax and discounts. Apply the discount to get the new price, then apply tax to that new price.
  • Sanity-check the size. 10% of an amount is just the amount with the decimal moved one place left, so 10% of $48 = $4.80. Use that to judge whether your answer is reasonable.

Your turn: practice problems

A calculator is allowed. Set up each one carefully, then check yourself. No peeking until you have tried.

  1. What is 18% of $250?
  2. A $60 pair of shoes is 25% off. What is the sale price (before tax)?
  3. A used car's price drops from $9,000 to $7,200. What is the percent decrease?
  4. You invest $1,500 at 5% simple interest for 2 years. How much interest do you earn, and what is the total?
Tap to reveal the answers
  • 1. 18% of 250 = 0.18 × 250 = $45.
  • 2. Discount = 0.25 × 60 = 15, so sale price = 60 − 15 = $45. (Or take 75% directly: 0.75 × 60 = 45.)
  • 3. Change = 9000 − 7200 = 1800. Then 1800 ÷ 9000 = 0.20 = a 20% decrease (divide by the original $9,000).
  • 4. I = Prt = 1500 × 0.05 × 2 = $150 interest. Total = 1500 + 150 = $1,650.

Why this matters for the CAEC

The calculator part of the CAEC math test is full of practical, money-based word problems: shopping discounts, tips, tax, paycheques, and interest. They all rest on the same percent skills you just practised. Get comfortable setting up "percent of an amount," and a whole category of questions becomes routine.

Want more practice like this? Our CAEC math guide and the CAEC Ready Workbook are packed with worked examples and practice questions, or start with a free math sample to test yourself.

Disclaimer

This article is a general math tutorial for study purposes. CAEC Ready is an independent study resource and is not affiliated with or endorsed by any government, ministry of education, or official CAEC testing provider.

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