Chapter 1 · Lesson 7 of 9 · Grade 7 · Ratios & Proportional Reasoning · MAT-07-RP-007
Solve Markup, Markdown, Tax, and Tip Problems
Represent and solve solve markup, markdown, tax, and tip problems problems using precise mathematical notation, models, and reasoning.
Math V4 · Course chapter
Chapter 1: Proportional Relationships, Percent, and Financial Reasoning
Ratios & Proportional Reasoning
Essential question: How can proportional relationships describe change, comparison, percent, and real decisions?
Where this lesson fits
Lesson 7 of 9. Each lesson builds on earlier chapter representations and ideas rather than resetting the topic.
Chapter destination
Create and defend a multi-step model using the central ideas of proportional relationships, percent, and financial reasoning, including a representation, calculation, interpretation, and independent check.
Learning objectives
What you should be able to do
- Represent and solve solve markup, markdown, tax, and tip problems problems using precise mathematical notation, models, and reasoning.
- Explain why a method for solve markup, markdown, tax, and tip problems works, interpret the result in context, and verify it independently.
Prerequisite check
Make sure the foundation is ready.
Grade 7 uses prerequisite knowledge as active reasoning.
Learn
Build the idea from meaning, not memorization.
Chapter 1: Proportional Relationships, Percent, and Financial Reasoning
Ratios & Proportional Reasoning. Lesson 7 of 9 extends earlier chapter reasoning.
Connection to the course
Use Grade 6 ratios, rates, fractions, and decimals as active knowledge. Connect today's representation to at least one earlier chapter representation.
Mathematical meaning
Solve Markup, Markdown, Tax, and Tip Problems is about multiplicative comparison. Equivalent representations preserve the same scale relationship, not merely the same difference.
Represent the relationship
The Math Box uses percent model to expose the structure. Change one input, predict the result, then connect the visual change to an equation, table, graph, number line, or geometric model.
Calculate, interpret, and verify
Verify with an inverse operation, equivalent representation, estimate, or second method so both the calculation and reasoning are auditable.
Math Box · Interactive lesson
Touch the math. Change it. See what stays true.
Use the interactive model before and after the worked examples. Change the inputs, make a prediction, then use the model to test whether your reasoning holds.
Percent Lab
Percent means “per 100.” Change the percent and the whole to see the part update.
- Model solve markup, markdown, tax, and tip problems in the Math Box.
- Change one input and predict the effect before testing it.
- Connect the visual, numerical, and symbolic representations.
- State what remains invariant and why.
Worked examples
Twenty different ways to see the concept work.
A 40 amount has a 5% percent increase. Find the change and resulting amount.
change = 5/100 × 40 = 2; result = 42An increase, tax, or tip adds the percent change to the original amount.
Represent before calculating. Use a ratio table, tape diagram, double number line, or equation for this task, then solve: A 45 amount has a 10% markdown. Find the change and resulting amount.
change = 10/100 × 45 = 4.5; result = 40.5The representation should show the same mathematical relationship as the calculation. A markdown reduces the original amount, so subtract the percent change.
Explain why a valid method works, then solve: A 50 amount has a 15% sales tax. Find the change and resulting amount.
change = 15/100 × 50 = 7.5; result = 57.5A complete explanation names the relationship or property being preserved. An increase, tax, or tip adds the percent change to the original amount.
Estimate or predict first, then calculate and decide whether the result is reasonable: A 55 amount has a 20% tip. Find the change and resulting amount.
change = 20/100 × 55 = 11; result = 66The estimate is a reasonableness check, not a replacement for the exact result. An increase, tax, or tip adds the percent change to the original amount.
Solve and verify the result with a second method, inverse operation, or equivalent representation: A 60 amount has a 25% percent increase. Find the change and resulting amount.
change = 25/100 × 60 = 15; result = 75Verification should independently support the result. An increase, tax, or tip adds the percent change to the original amount.
Interpret the answer in context after solving. What does the result mean here? A 65 amount has a 30% markdown. Find the change and resulting amount.
change = 30/100 × 65 = 19.5; result = 45.5State the result with its meaning, units, direction, or comparison—not only a number. A markdown reduces the original amount, so subtract the percent change.
Error analysis: a student says, "Treating a ratio as additive." Explain the mistake, then solve this related task correctly: A 70 amount has a 5% sales tax. Find the change and resulting amount.
Check for a common scale factor or constant unit rate. Correct solution: change = 5/100 × 70 = 3.5; result = 73.5Error analysis requires identifying the broken idea, not merely replacing the final answer. An increase, tax, or tip adds the percent change to the original amount.
Compare two related cases and explain what changes and what stays mathematically consistent. Case A: A 75 amount has a 10% tip. Find the change and resulting amount. Case B: A 120 amount has a 25% percent increase. Find the change and resulting amount.
Case A: change = 10/100 × 75 = 7.5; result = 82.5 Case B: change = 25/100 × 120 = 30; result = 150Compare the structure, representation, units, and result rather than only the numbers. An increase, tax, or tip adds the percent change to the original amount.
A 80 amount has a 15% percent increase. Find the change and resulting amount.
change = 15/100 × 80 = 12; result = 92An increase, tax, or tip adds the percent change to the original amount.
Represent before calculating. Use a ratio table, tape diagram, double number line, or equation for this task, then solve: A 85 amount has a 20% markdown. Find the change and resulting amount.
change = 20/100 × 85 = 17; result = 68The representation should show the same mathematical relationship as the calculation. A markdown reduces the original amount, so subtract the percent change.
Explain why a valid method works, then solve: A 90 amount has a 25% sales tax. Find the change and resulting amount.
change = 25/100 × 90 = 22.5; result = 112.5A complete explanation names the relationship or property being preserved. An increase, tax, or tip adds the percent change to the original amount.
Estimate or predict first, then calculate and decide whether the result is reasonable: A 95 amount has a 30% tip. Find the change and resulting amount.
change = 30/100 × 95 = 28.5; result = 123.5The estimate is a reasonableness check, not a replacement for the exact result. An increase, tax, or tip adds the percent change to the original amount.
Solve and verify the result with a second method, inverse operation, or equivalent representation: A 100 amount has a 5% percent increase. Find the change and resulting amount.
change = 5/100 × 100 = 5; result = 105Verification should independently support the result. An increase, tax, or tip adds the percent change to the original amount.
Interpret the answer in context after solving. What does the result mean here? A 105 amount has a 10% markdown. Find the change and resulting amount.
change = 10/100 × 105 = 10.5; result = 94.5State the result with its meaning, units, direction, or comparison—not only a number. A markdown reduces the original amount, so subtract the percent change.
Error analysis: a student says, "Treating a ratio as additive." Explain the mistake, then solve this related task correctly: A 110 amount has a 15% sales tax. Find the change and resulting amount.
Check for a common scale factor or constant unit rate. Correct solution: change = 15/100 × 110 = 16.5; result = 126.5Error analysis requires identifying the broken idea, not merely replacing the final answer. An increase, tax, or tip adds the percent change to the original amount.
Compare two related cases and explain what changes and what stays mathematically consistent. Case A: A 115 amount has a 20% tip. Find the change and resulting amount. Case B: A 160 amount has a 5% percent increase. Find the change and resulting amount.
Case A: change = 20/100 × 115 = 23; result = 138 Case B: change = 5/100 × 160 = 8; result = 168Compare the structure, representation, units, and result rather than only the numbers. An increase, tax, or tip adds the percent change to the original amount.
A 120 amount has a 25% percent increase. Find the change and resulting amount.
change = 25/100 × 120 = 30; result = 150An increase, tax, or tip adds the percent change to the original amount.
Represent before calculating. Use a ratio table, tape diagram, double number line, or equation for this task, then solve: A 125 amount has a 30% markdown. Find the change and resulting amount.
change = 30/100 × 125 = 37.5; result = 87.5The representation should show the same mathematical relationship as the calculation. A markdown reduces the original amount, so subtract the percent change.
Explain why a valid method works, then solve: A 130 amount has a 5% sales tax. Find the change and resulting amount.
change = 5/100 × 130 = 6.5; result = 136.5A complete explanation names the relationship or property being preserved. An increase, tax, or tip adds the percent change to the original amount.
Estimate or predict first, then calculate and decide whether the result is reasonable: A 135 amount has a 10% tip. Find the change and resulting amount.
change = 10/100 × 135 = 13.5; result = 148.5The estimate is a reasonableness check, not a replacement for the exact result. An increase, tax, or tip adds the percent change to the original amount.
Practice
Guided practice with hints
01Solve and verify the result with a second method, inverse operation, or equivalent representation: A 140 amount has a 15% percent increase. Find the change and resulting amount.
Hint: Choose a representation before computing.
Answer: change = 15/100 × 140 = 21; result = 161
02Interpret the answer in context after solving. What does the result mean here? A 145 amount has a 20% markdown. Find the change and resulting amount.
Hint: Explain why the method works and name a verification strategy.
Answer: change = 20/100 × 145 = 29; result = 116
03Error analysis: a student says, "Treating a ratio as additive." Explain the mistake, then solve this related task correctly: A 150 amount has a 25% sales tax. Find the change and resulting amount.
Hint: Choose a representation before computing.
Answer: Check for a common scale factor or constant unit rate. Correct solution: change = 25/100 × 150 = 37.5; result = 187.5
04Compare two related cases and explain what changes and what stays mathematically consistent. Case A: A 155 amount has a 30% tip. Find the change and resulting amount. Case B: A 200 amount has a 15% percent increase. Find the change and resulting amount.
Hint: Explain why the method works and name a verification strategy.
Answer: Case A: change = 30/100 × 155 = 46.5; result = 201.5 Case B: change = 15/100 × 200 = 30; result = 230
05A 160 amount has a 5% percent increase. Find the change and resulting amount.
Hint: Choose a representation before computing.
Answer: change = 5/100 × 160 = 8; result = 168
06Represent before calculating. Use a ratio table, tape diagram, double number line, or equation for this task, then solve: A 165 amount has a 10% markdown. Find the change and resulting amount.
Hint: Explain why the method works and name a verification strategy.
Answer: change = 10/100 × 165 = 16.5; result = 148.5
Practice
Independent practice
01Explain why a valid method works, then solve: A 170 amount has a 15% sales tax. Find the change and resulting amount.
Answer: change = 15/100 × 170 = 25.5; result = 195.5
02Estimate or predict first, then calculate and decide whether the result is reasonable: A 175 amount has a 20% tip. Find the change and resulting amount.
Answer: change = 20/100 × 175 = 35; result = 210
03Solve and verify the result with a second method, inverse operation, or equivalent representation: A 180 amount has a 25% percent increase. Find the change and resulting amount.
Answer: change = 25/100 × 180 = 45; result = 225
04Interpret the answer in context after solving. What does the result mean here? A 185 amount has a 30% markdown. Find the change and resulting amount.
Answer: change = 30/100 × 185 = 55.5; result = 129.5
05Error analysis: a student says, "Treating a ratio as additive." Explain the mistake, then solve this related task correctly: A 190 amount has a 5% sales tax. Find the change and resulting amount.
Answer: Check for a common scale factor or constant unit rate. Correct solution: change = 5/100 × 190 = 9.5; result = 199.5
06Compare two related cases and explain what changes and what stays mathematically consistent. Case A: A 195 amount has a 10% tip. Find the change and resulting amount. Case B: A 240 amount has a 25% percent increase. Find the change and resulting amount.
Answer: Case A: change = 10/100 × 195 = 19.5; result = 214.5 Case B: change = 25/100 × 240 = 60; result = 300
07A 200 amount has a 15% percent increase. Find the change and resulting amount.
Answer: change = 15/100 × 200 = 30; result = 230
08Represent before calculating. Use a ratio table, tape diagram, double number line, or equation for this task, then solve: A 205 amount has a 20% markdown. Find the change and resulting amount.
Answer: change = 20/100 × 205 = 41; result = 164
Common mistakes
Learn to catch the error, not just the answer.
Check for a common scale factor or constant unit rate.
Convert both to a common unit rate first.
Applications & challenge
Use the idea beyond a single exercise.
Applications
- Compare prices, speeds, recipes, maps, taxes, discounts, tips, and rates using common units.
- Scale designs and predict related quantities while preserving multiplicative relationships.
Challenge problems
Explain why a valid method works, then solve: A 210 amount has a 25% sales tax. Find the change and resulting amount.
change = 25/100 × 210 = 52.5; result = 262.5
Estimate or predict first, then calculate and decide whether the result is reasonable: A 215 amount has a 30% tip. Find the change and resulting amount.
change = 30/100 × 215 = 64.5; result = 279.5
Solve and verify the result with a second method, inverse operation, or equivalent representation: A 220 amount has a 5% percent increase. Find the change and resulting amount.
change = 5/100 × 220 = 11; result = 231
Interpret the answer in context after solving. What does the result mean here? A 225 amount has a 10% markdown. Find the change and resulting amount.
change = 10/100 × 225 = 22.5; result = 202.5
Flashcards
Retrieve it from memory.
Flashcards
Card 1 of 4
Game mode
Grade 7 Chapter 1 Reasoning Lab
Solve solve markup, markdown, tax, and tip problems tasks using calculation, representation, explanation, interpretation, verification, error analysis, and comparison.
Practice
Mastery check
01Error analysis: a student says, "Treating a ratio as additive." Explain the mistake, then solve this related task correctly: A 230 amount has a 15% sales tax. Find the change and resulting amount.
Answer: Check for a common scale factor or constant unit rate. Correct solution: change = 15/100 × 230 = 34.5; result = 264.5
02Compare two related cases and explain what changes and what stays mathematically consistent. Case A: A 235 amount has a 20% tip. Find the change and resulting amount. Case B: A 280 amount has a 5% percent increase. Find the change and resulting amount.
Answer: Case A: change = 20/100 × 235 = 47; result = 282 Case B: change = 5/100 × 280 = 14; result = 294
03A 240 amount has a 25% percent increase. Find the change and resulting amount.
Answer: change = 25/100 × 240 = 60; result = 300
04Represent before calculating. Use a ratio table, tape diagram, double number line, or equation for this task, then solve: A 245 amount has a 30% markdown. Find the change and resulting amount.
Answer: change = 30/100 × 245 = 73.5; result = 171.5
05Explain why a valid method works, then solve: A 250 amount has a 5% sales tax. Find the change and resulting amount.
Answer: change = 5/100 × 250 = 12.5; result = 262.5
06Estimate or predict first, then calculate and decide whether the result is reasonable: A 255 amount has a 10% tip. Find the change and resulting amount.
Answer: change = 10/100 × 255 = 25.5; result = 280.5
07Solve and verify the result with a second method, inverse operation, or equivalent representation: A 260 amount has a 15% percent increase. Find the change and resulting amount.
Answer: change = 15/100 × 260 = 39; result = 299
08Interpret the answer in context after solving. What does the result mean here? A 265 amount has a 20% markdown. Find the change and resulting amount.
Answer: change = 20/100 × 265 = 53; result = 212
Terminology
Words to know
- ratio
- A multiplicative comparison between two quantities.
- unit rate
- A rate expressed per one unit of the second quantity.
- equivalent ratios
- Ratios made by scaling both quantities by the same nonzero factor.
- constant of proportionality
- The unit rate k in a proportional equation y = kx.
Continue learning
Connected concepts
Curriculum references
Standards and instructional references
Grade 7 scope, sequence, and public task structure
Coherent sequencing, representations, and reasoning-rich tasks.Eureka Math² Grade 7 program structure
Module coherence, cumulative practice, and fluency.enVision Mathematics Grade 7 instructional model
Problem-based entry points, visual learning, modeling, and application.Common Core State Standards for Mathematics — Grade 7
Grade-level content and mathematical-practice expectations.