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Introduction
Percent Applications helps you connect a mathematical rule to a result you can explain and check. Convert percent to a decimal fraction before calculating. Distinguish the amount of change from the final total. This lesson belongs to linear equations and builds toward independent Algebra I problem solving.
Alignment uses 19 TAC §111.39(c). Review lessons sample their listed standards. Readiness and supporting labels describe STAAR emphasis, not whether a topic is required.
Learning objectives
- Identify the quantities and representations used in percent applications.
- Apply the lesson's rule to solve a numerical or contextual problem.
- Explain each mathematical step using the correct vocabulary.
- Verify an answer and correct a likely error.
Prerequisite knowledge
Multiply decimals and interpret a rate.
Previous lesson: Unit Rates
Key mathematical vocabulary
- factor
- An expression multiplied by another to form a product.
- rate
- A ratio measured in units per unit.
- expression
- A mathematical phrase representing a value.
- solution
- A value making the original conditions true.
- input
- A value supplied to a rule.
Formulas and rules
- a(b+c)=ab+ac
- Equivalent expressions agree for all allowed inputs.
- A solution satisfies every original condition.
Percent applications
Convert percent to a decimal fraction before calculating. Distinguish the amount of change from the final total.
Percent means per hundred. Convert p% to p/100 before multiplying. A percentage of a quantity equals its decimal rate times that quantity. A discount subtracts a percentage of the original; a markup adds it.
A p% decrease uses the factor 1−p/100, while a p% increase uses 1+p/100. Successive percentages act on the updated quantity rather than the original one. State when money is rounded and distinguish prices before and after tax.
A method you can explain
- Read the question and identify the known information, unknown quantities, and any domain restrictions. Select a suitable representation and tool.
- Apply the relationship explained above. Convert the percentage to a decimal: 18%=0.18.
- Write each calculation or logical inference on a separate line. Keep units and signs visible. When two methods are possible, compare which one uses the given information most directly.
- Check the result in the ORIGINAL conditions, not merely the last simplified line. Sale price plus discount restores the original price.
- Interpret what the answer means. Decide whether its size, sign, and precision make sense; identify a restriction or assumption that would change the conclusion.
Five worked examples
Example 1 · Beginner
A $1200 item is discounted 18%. Find the sale price before tax.
- Convert the percentage to a decimal: 18%=0.18.
- Discount amount: 1200·0.18=$216.
- Subtract the discount: 1200−216=$984.
Answer: 984
Verify: Sale price plus discount restores the original price.
Avoid this mistake: Using 20 instead of 0.20 for 20%.
Example 2 · Basic
A $1300 item is discounted 18%. Find the sale price before tax.
- Convert the percentage to a decimal: 18%=0.18.
- Discount amount: 1300·0.18=$234.
- Subtract the discount: 1300−234=$1066.
Answer: 1066
Verify: Sale price plus discount restores the original price.
Avoid this mistake: Reporting the discount amount as the final price.
Example 3 · Intermediate
A $1400 item is discounted 18%. Find the sale price before tax.
- Convert the percentage to a decimal: 18%=0.18.
- Discount amount: 1400·0.18=$252.
- Subtract the discount: 1400−252=$1148.
Answer: 1148
Verify: Sale price plus discount restores the original price.
Avoid this mistake: Adding successive percentage rates without checking their bases.
Example 4 · Advanced
A $200 item is discounted 19%. Find the sale price before tax.
- Convert the percentage to a decimal: 19%=0.19.
- Discount amount: 200·0.19=$38.
- Subtract the discount: 200−38=$162.
Answer: 162
Verify: Sale price plus discount restores the original price.
Avoid this mistake: Using 20 instead of 0.20 for 20%.
Example 5 · Challenge
A $300 item is discounted 19%. Find the sale price before tax.
- Convert the percentage to a decimal: 19%=0.19.
- Discount amount: 300·0.19=$57.
- Subtract the discount: 300−57=$243.
Answer: 243
Verify: Sale price plus discount restores the original price.
Avoid this mistake: Reporting the discount amount as the final price.
Common mistakes and corrections
- Using 20 instead of 0.20 for 20%.
- Reporting the discount amount as the final price.
- Adding successive percentage rates without checking their bases.
Real-world applications
- Calculate a sale price and sales tax. Define the input and its units, apply the mathematical relationship, and check that the result fits the setting.
- Compare a pay increase with a change in monthly expenses. Define the input and its units, apply the mathematical relationship, and check that the result fits the setting.
Topic summary
Convert percent to a decimal fraction before calculating. Distinguish the amount of change from the final total.
Percent means per hundred. Convert p% to p/100 before multiplying. A percentage of a quantity equals its decimal rate times that quantity. A discount subtracts a percentage of the original; a markup adds it.
Frequently asked questions
What should I know before percent applications?
Multiply decimals and interpret a rate.
How can I check my work?
Sale price plus discount restores the original price. A numerical check can detect an error; when claiming an identity for all inputs, also explain the algebraic property that makes it true.
Which mistakes should I watch for?
Using 20 instead of 0.20 for 20%. Reporting the discount amount as the final price. Adding successive percentage rates without checking their bases.
Can I use another valid method?
Yes. Explain your assumptions, show enough intermediate work to justify the result, and verify it in the original problem. Equivalent expressions and valid alternative methods receive credit.
Mathematical process practice · A.1A–A.1G
- A.1A: Calculate a sale price and sales tax.
- A.1B: Plan, solve, check, and interpret the written response.
- A.1C: Select paper, a calculator, a table, or a graph; explain the choice.
- A.1D: Connect symbols with a verbal explanation and another representation.
- A.1E: Organize the solution with a table, diagram, or labeled equations.
- A.1F: Explain how the representation and the algebraic rule express the same relationship.
- A.1G: Justify each operation and the reasonableness of the conclusion.
Quick revision notes
- Convert percent to a decimal fraction before calculating. Distinguish the amount of change from the final total.
- Using 20 instead of 0.20 for 20%.
- Reporting the discount amount as the final price.
- Adding successive percentage rates without checking their bases.
Practice and explain
Try all four multiple-choice questions, then show your reasoning in the written response. Objective scores and AI feedback assess different skills.
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Original Cambridge Veritas practice. These are not official STAAR items. Cambridge Veritas is not affiliated with or endorsed by TEA.
Official Algebra I TEKS (§111.39) · TEA assessed curriculum · STAAR blueprint · TEA instructional materials