Percent Problems

Percent problems are everywhere — sales tax, tips, test scores, pay raises, medical dosages, discounts. Every percent problem boils down to a relationship among three quantities: the part, the whole (or base), and the percent. Once you can identify which quantity is missing, you can solve any percent problem using the same core equation. For the foundational introduction to percentages, see Percentages in the Arithmetic section.

The Core Equation

Every percent problem can be written as:

$\text{Part} = \text{Percent} \times \text{Whole}$

Or equivalently, using the decimal form of the percent:

$P = r \times W$

where $r$ is the percent written as a decimal (for example, $25$% becomes $0.25$).

Depending on which value is unknown, you rearrange:

Type 1: Finding the Part (Percent of a Number)

Question pattern: “What is $r$% of $W$?”

Example 1: What is $15$% of $240$?

Convert the percent to a decimal and multiply:

$0.15 \times 240 = 36$

Answer: $36$

Example 2: What is $6.5$% of $800$?

$0.065 \times 800 = 52$

Answer: $52$

Mental Math Shortcut

To find $10$% of any number, move the decimal one place left. Then adjust:

• $10$% of $240 = 24$
• $5$% of $240 = 12$ (half of $10$%)
• $15$% of $240 = 24 + 12 = 36$

This technique is especially useful for estimating tips.

Type 2: Finding the Percent

Question pattern: ”$P$ is what percent of $W$?”

Example 3: $27$ is what percent of $180$?

$r = \frac{27}{180} = 0.15$

Convert to percent: $0.15 \times 100 = 15$%.

Answer: $15$%

Example 4: A student scores $42$ out of $56$ on a quiz. What is the percentage score?

$r = \frac{42}{56} = 0.75$

$0.75 \times 100 = 75$%

Answer: $75$%

Type 3: Finding the Whole

Question pattern: ”$P$ is $r$% of what number?”

Example 5: $18$ is $30$% of what number?

$W = \frac{18}{0.30} = 60$

Answer: $60$

Example 6: A sale price of $35 represents $70$% of the original price. What was the original price?

$W = \frac{35}{0.70} = 50$

Answer: The original price was $50.

The Proportion Method

An alternative approach that many students prefer: set up a proportion with 100 as the denominator.

$\frac{\text{Part}}{\text{Whole}} = \frac{\text{Percent}}{100}$

Example 7: $24$ is what percent of $60$?

$\frac{24}{60} = \frac{x}{100}$

Cross-multiply: $60x = 2{,}400$, so $x = 40$.

Answer: $40$%

Example 8: What is $85$% of $300$?

$\frac{x}{300} = \frac{85}{100}$

Cross-multiply: $100x = 25{,}500$, so $x = 255$.

Answer: $255$

Both the equation method and the proportion method always give the same result. Use whichever feels more natural to you.

Percent Increase and Decrease

Percent change measures how much a value has grown or shrunk, expressed as a percentage of the original.

$\text{Percent Change} = \frac{\text{New} - \text{Original}}{\text{Original}} \times 100$

If the result is positive, it is an increase. If negative, it is a decrease.

Example 9: Percent Increase

A store raised the price of a tool from $40 to $52. What is the percent increase?

$\frac{52 - 40}{40} \times 100 = \frac{12}{40} \times 100 = 30$%

Answer: $30$% increase

Example 10: Percent Decrease

A laptop originally priced at $600 is marked down to $480. What is the percent decrease?

$\frac{480 - 600}{600} \times 100 = \frac{-120}{600} \times 100 = -20$%

The negative sign tells us this is a decrease. Answer: $20$% decrease

Finding the New Value After a Percent Change

You can also work forward from a percent change:

• After an increase: New $= \text{Original} \times (1 + r)$
• After a decrease: New $= \text{Original} \times (1 - r)$

Example 11: A $250 rent payment increases by $4$%. What is the new rent?

$250 \times (1 + 0.04) = 250 \times 1.04 = 260$

Answer: $260

Discount and Tax Calculations

Discounts and taxes are the most common real-world percent problems.

Discounts

A discount reduces the price. If an item is $25$% off:

$\text{Sale Price} = \text{Original Price} \times (1 - 0.25) = \text{Original Price} \times 0.75$

Example 12: A jacket originally costs $80 and is $35$% off. What is the sale price?

$80 \times (1 - 0.35) = 80 \times 0.65 = 52$

Answer: $52

Sales Tax

Tax increases the price. If the tax rate is $8$%:

$\text{Total Price} = \text{Subtotal} \times (1 + 0.08) = \text{Subtotal} \times 1.08$

Example 13: A meal costs $24.50 before $8$% tax. What is the total?

$24.50 \times 1.08 = 26.46$

Answer: $26.46

Discount Then Tax (Combined)

When a discounted item is then taxed, apply the discount first, then the tax.

Example 14: A $120 pair of shoes is $20$% off, and the tax rate is $7$%. What is the total?

Step 1 — Discount: $120 \times 0.80 = 96$

Step 2 — Tax: $96 \times 1.07 = 102.72$

Answer: $102.72

Note: You cannot simply combine the discount and tax into a single percent. $20$% off then $7$% tax is not the same as $13$% off.

Real-World Application: Retail — Tipping

A restaurant bill is $45.60. You want to leave a $20$% tip. What is the tip, and what is the total?

$\text{Tip} = 45.60 \times 0.20 = 9.12$

$\text{Total} = 45.60 + 9.12 = 54.72$

Mental math approach: $10$% of $45.60 is $4.56. Double that for $20$%: $9.12.

Answer: $9.12 tip, $54.72 total

Real-World Application: Nursing — Concentration Changes

A saline solution is $0.9$% sodium chloride (NaCl). A nurse has 500 mL of this solution. How many grams of NaCl are in the bag?

Since $0.9$% means $0.9$ grams per $100$ mL:

$\frac{0.9}{100} \times 500 = 0.009 \times 500 = 4.5 \text{ grams}$

The bag contains $4.5$ grams of NaCl. Accurate percent calculations are critical in clinical settings for preparing IV solutions and verifying medication concentrations.

Common Mistakes to Avoid

1. Dividing by the new value instead of the original for percent change. Percent change is always relative to the original value, not the new one.

2. Confusing “percent of” with “percent off.” $25$% of $80 is $20 (the discount amount). $25$% off $80 means you pay $60 (the remaining $75$%).

3. Adding discount and tax percents together. $20$% off plus $8$% tax does not equal $12$% off. You must apply them sequentially.

4. Forgetting to convert percent to a decimal before computing. $15$% of $200$ is $0.15 \times 200 = 30$, not $15 \times 200 = 3{,}000$.

5. Assuming a percent increase followed by the same percent decrease returns to the original. A $20$% increase on $100 gives $120. A $20$% decrease on $120 gives $96, not $100. The base changes between the two calculations.

Practice Problems

Test your skills. Click to reveal each answer.

Key Takeaways

• Every percent problem involves three quantities: part, whole, and percent — identify which is missing, then use $P = r \times W$
• The proportion method ($\frac{\text{Part}}{\text{Whole}} = \frac{\text{Percent}}{100}$) is an equally valid approach
• Percent change is always calculated relative to the original value
• For discounts and taxes, apply them sequentially — never add or subtract the percentages
• The mental math shortcut (find $10$%, then adjust) is invaluable for quick estimates and tipping
• These same percent relationships appear throughout algebra (solving equations like $0.15x = 45$) and statistics (probabilities expressed as percentages)

Return to Pre-Algebra for more topics in this section.