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Multi-Step Ratio &
Percent Problems

Master tax, tips, markups, discounts, simple interest, and more: solve real-world problems that combine ratios and percents in multiple steps.

7th Grade Math

What Is a Percent?

A percent means "per hundred." Writing \( 25\% \) is the same as writing the fraction \( \frac{25}{100} \) or the decimal \( 0.25 \). Percents let us compare parts to a whole using a common scale of 100.

\( \frac{1}{2} \)
 10 10 = 25%

The Percent Equation

Most percent problems boil down to one core equation with three quantities. Know any two and you can find the third.

Part Percent Whole
\( \text{Part} = \text{Percent} \times \text{Whole} \)
 or equivalently
\( \text{Percent} = \text{Part} \div \text{Whole} \)

Where Do Percents Show Up?

Tax
Added to the price of items you buy: sales tax, VAT.
Tip / Gratuity
Extra amount added at restaurants, usually \(15\%\) to \(20\%\).
Markup
A store raises the price above its cost to earn profit.
Discount
A reduction from the original price: sales, coupons.

Markup vs. Markdown

MARKUP
\( \text{Cost} + \text{Markup} = \text{Selling Price} \)
 Price goes UP MARKDOWN
\( \text{Original} - \text{Discount} = \text{Sale Price} \)
 Price goes DOWN
Markup Formula
\( \text{Selling Price} = \text{Cost} \times (1 + r) \) where \( r \) is the markup rate as a decimal.
Markdown Formula
\( \text{Sale Price} = \text{Original} \times (1 - r) \) where \( r \) is the discount rate as a decimal.

Percent Increase & Decrease

When a quantity changes, we often describe the change as a percent of the original. The formula works the same whether the value goes up or down.

Simple Interest Formula

Simple interest is charged (or earned) only on the original principal: the starting amount. It does not compound.

\( I = P \times r \times t \)
 I Interest P Principal r Rate (decimal) t Time (years)

Tax + Tip: Two Steps

In many states, you pay sales tax on the food and then add a tip on the pre-tax total. These are two separate percent calculations.

Food Bill: $40.00
+ 8% Tax: \( \$40 \times 0.08 = \$3.20 \)
 Subtotal: $43.20
+ 20% Tip on $40: \( \$40 \times 0.20 = \$8.00 \)
Total: \( \$43.20 + \$8.00 = \$51.20 \)

Percent Error

Percent error measures how far off an estimate or measurement is from the actual value, expressed as a percent of the actual value.

Example
You estimate a jar holds 80 marbles. It actually holds 92. Percent error: \( \frac{|80 - 92|}{92} \times 100 \approx 13.0\% \).
Why It Matters
Scientists and engineers use percent error to evaluate the accuracy of their measurements.

Solving Multi-Step Problems

1
Read carefully. Identify all the percent operations: tax, tip, discount, interest, etc.
2
Convert each percent to a decimal. Divide by 100: \( 15\% \to 0.15 \).
3
Determine the base for each calculation. Tax is on the price; tip may be on the pre-tax amount.
4
Compute each step in order. Write intermediate results before combining.
5
Check your answer. Does the final number make sense? A discount should make the price smaller, not larger.

Example 1: Discount Then Tax

A jacket costs $60. It is on sale for \(25\%\) off. Sales tax is \(6\%\). What is the total cost?

1
Discount: \( \$60 \times 0.25 = \$15 \). Sale price: \( \$60 - \$15 = \$45 \).
2
Tax: \( \$45 \times 0.06 = \$2.70 \).
3
Total: \( \$45 + \$2.70 = \$47.70 \).

Example 2: Commission

A real-estate agent earns a \(3\%\) commission on every sale. She also pays a \(\$200\) monthly fee to her agency. If she sells a house for \(\$250{,}000\), what is her net earnings for the month?

1
Commission: \( \$250{,}000 \times 0.03 = \$7{,}500 \).
2
Subtract fee: \( \$7{,}500 - \$200 = \$7{,}300 \).

Example 3: Simple Interest

You deposit \(\$1{,}200\) into a savings account that pays \(4\%\) simple interest per year. How much total will be in the account after \(3\) years?

1
Find interest: \( I = P \times r \times t = 1200 \times 0.04 \times 3 = \$144 \).
2
Total amount: \( A = P + I = \$1{,}200 + \$144 = \$1{,}344 \).

Common Mistakes

✕ Wrong Base
Calculating tax on the discounted + tax total instead of on the sale price alone.
✕ Forgetting to Convert
Using \( 8 \) instead of \( 0.08 \) when multiplying by a percent. Always divide by 100 first.
✕ Adding Instead of Multiplying
Adding 20% to the price as "\( + 20 \)" dollars instead of computing \( \text{price} \times 0.20 \).

Summary Table

Scenario Formula Multiplier
Tax / Tip / Markup \( \text{Price} + \text{Price} \times r \) \( \text{Price} \times (1 + r) \)
Discount / Markdown \( \text{Price} - \text{Price} \times r \) \( \text{Price} \times (1 - r) \)
Percent Change \( \frac{|\text{New}-\text{Old}|}{|\text{Old}|} \times 100 \) Compare to 1
Simple Interest \( I = Prt \) \( A = P(1 + rt) \)
Percent Error \( \frac{|\text{Est}-\text{Act}|}{|\text{Act}|} \times 100 \) N/A

Question 1 of 5

A shirt costs $35. It is marked down \(20\%\), and then \(7\%\) sales tax is applied. What is the total cost?

Question 2 of 5

A restaurant bill is $52. You leave a \(18\%\) tip on the pre-tax amount. Tax is \(5\%\). What is the total you pay?

Question 3 of 5

You invest $800 at \(5\%\) simple interest per year. How much interest do you earn after \(4\) years?

Question 4 of 5

A store buys a lamp for $24 and marks it up \(75\%\). What is the selling price?

Question 5 of 5

A student estimates a plant is \(30\) cm tall. The actual height is \(36\) cm. What is the percent error (rounded to the nearest tenth)?