7.RP.A.3Common CoreMathRatios and Proportional RelationshipsGrade 7
7.RP.A.3: Multistep Ratio and Percent Problems
In plain English: 7.RP.A.3 is the Common Core grade 7 math standard that asks students to use proportional relationships to solve multistep ratio and percent problems. Typical problems are sales tax, tips, commissions, fees, markups and markdowns, simple interest, percent increase and decrease, and percent error. The key idea is that a percent of an amount is that amount times the percent over 100.
Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.
Common Core State Standards for Mathematics · Domain: Ratios and Proportional Relationships (RP) · Cluster: Analyze proportional relationships and use them to solve real-world and mathematical problems. Also written as 7.RP.3 · Official standard
Students solve problems that take more than one step, where each step uses a ratio or a percent. A percent is a rate per 100, so 8% of an amount is 8/100 times the amount. Because the part is always the same fraction of the whole, the part and the whole are in a proportional relationship (their ratio stays the same). The standard names the everyday problems students should meet: simple interest, tax, markups and markdowns, gratuities (tips) and commissions, fees, percent increase and decrease, and percent error. Each of these words is defined in the Direct Instruction table.
The lesson builds one habit for every problem type: find the base (the amount the percent is taken of), find the percent of it, then decide whether to add it, subtract it or use it on its own. Students see that the base can change in the middle of a problem, for example when a discount is taken from a price that was already marked up. They also learn a shortcut: a 7% tax means you pay 107% of the price, so you can multiply by 1.07 in one step. Compound interest (interest paid on earlier interest) and other repeated percent growth are left for high school.
Learning Objectives
By the end of this lesson, students will be able to:
Solve multistep ratio problems, such as splitting a total in a given ratio and then finding a cost
Find sales tax, tips, commissions and fees, and use them to find a total or to work backward to an unknown amount
Find prices after markups and markdowns, including two percent changes in a row on different bases
Find a percent increase, a percent decrease and a percent error, always dividing by the right base
Find simple interest with I = P × r × t and explain why it grows in proportion to time
Prior Knowledge Required
Students should already be comfortable with:
Finding a percent of a quantity as a rate per 100, and finding the whole from a part and the percent 6.RP.A.3
Writing equations such as y = kx for proportional relationships, where k is the constant of proportionality 7.RP.A.2
Adding, subtracting, multiplying and dividing decimals 6.NS.B.3
Absolute value as the distance of a number from 0 6.NS.C.7
Remind students that sales tax is a percent of the price that the store adds to your bill and passes on to the government. Then show this problem:
Warm-Up Prompt
"Sneakers cost $50. They are 20% off, and then 8% sales tax is added to the sale price. Maya says this is the same as paying 12% less than $50. Is she right? Find what you would pay."
Let pairs argue for 2 minutes, then collect answers. The sale price is 50 - 0.20 × 50 = $40. The tax is 0.08 × 40 = $3.20, so you pay $43.20. Paying 12% less than $50 would be $44.00. Maya is not right: the tax is 8% of $40, not 8% of $50, because the base changed after the discount. Tell students that this is a multistep problem (one that takes two or more steps), and that today's question for every step is: "a percent of what?"
Direct Instruction20 minutes
Part 1: The percent equation. In a proportional relationship, part = percent × base, where the base is the amount you take the percent of. Write the percent as a decimal or a fraction first: 8% = 0.08 = 8/100. The same fact can be written as a proportion (an equation that sets two ratios equal): part/base = percent/100. A tape diagram (a bar split into parts to show amounts) helps students see the base as 100%.
Go through the table with the class. Every row uses the same percent equation, and only the name and the base change.
The problem types named in 7.RP.A.3
Word
What it means
How to find it
Sales tax
A percent of the price that is added to what you pay
tax = tax rate × price
Tip (gratuity)
Extra money for service, as a percent of the bill
tip = tip rate × bill
Commission
Money a salesperson earns as a percent of what they sell
commission = rate × sales
Fee
A charge for a service: a flat amount (the same every time) or a percent
percent fee = rate × amount
Markup
The amount a store adds to its cost (what it paid) to set its price
price = cost + rate × cost
Markdown (discount)
The amount a store takes off a price
sale price = price - rate × price
Simple interest
Money paid for the use of money: a percent of the starting amount, the principal, for each year
I = P × r × t (principal × yearly rate × years)
Percent increase or decrease
A change written as a percent of the original (starting) amount
change ÷ original × 100%
Percent error
How far an estimate or measurement is from the actual value, as a percent of the actual value
|estimate - actual| ÷ actual × 100%, where the absolute value keeps the error from being negative
Multistep ratio: part of a total, then a cost
A pink paint mix uses 3 parts red to 5 parts white. Kai needs 12 liters of the mix. Red tempera paint costs $6 per liter. How much does the red paint cost?
Equation: 3 + 5 = 8 parts, so each part is 12 ÷ 8 = 1.5 liters. Red: 3 × 1.5 = 4.5 liters. Cost: 4.5 × $6 = $27.
Tax and tip (gratuity)
A family's dinner bill is $36 before tax. Sales tax is 7.5%, and they leave a 20% tip on the bill before tax. What do they pay in all?
Equation: Tax: 0.075 × 36 = $2.70. Tip: 0.20 × 36 = $7.20. Total: 36 + 2.70 + 7.20 = $45.90. Both percents use the same base, so 36 × 1.275 = $45.90 in one step.
Markup, then markdown
A store pays $48 for a jacket and marks it up 75%. Later it marks the jacket down 30%. What is the sale price? Is it more or less than what the store paid?
Equation: Markup: 0.75 × 48 = $36, so the price is 48 + 36 = $84 (or 1.75 × 48). Markdown: 0.30 × 84 = $25.20, so the sale price is 84 - 25.20 = $58.80 (or 0.70 × 84). That is $10.80 more than the store paid.
Percent increase and percent decrease
A school had 640 students last year and 720 this year. A bike that cost $250 now costs $210. Find each percent change.
Equation: Increase: 720 - 640 = 80, and 80 ÷ 640 = 0.125 = 12.5%. Decrease: 250 - 210 = 40, and 40 ÷ 250 = 0.16 = 16%. Divide by the original amount each time.
Simple interest
Nia puts $500 in a savings account that pays 3% simple interest per year. How much interest does she earn in 4 years, and what is her balance (the amount in the account) then?
Equation: I = P × r × t = 500 × 0.03 × 4 = $60. Balance: 500 + 60 = $560. Each year adds the same $15, so the interest is proportional to the time.
Diagram 1 shows the jacket example as tape diagrams: the markup is 75% of $48, but the markdown is 30% of $84, a bigger base. Diagram 2 graphs Nia's interest year by year. The points lie on a straight line through (0, 0), which is what a proportional relationship looks like. Point out the shortcut in the examples: adding a percent p to an amount is the same as multiplying by 1 + p, and taking p off is the same as multiplying by 1 - p.
Part 2: Commissions, fees and percent error. Work these three short cases on the board:
Commission: Dev sells shoes and earns a 5% commission. He sold $2,400 of shoes this week, so he earns 0.05 × 2,400 = $120.
Fee: a ticket website adds a 10% service fee and a flat $3 order fee to a $55 order. The service fee is 0.10 × 55 = $5.50, so the total is 55 + 5.50 + 3 = $63.50.
Percent error: Rosa estimates that her backpack weighs 5 pounds, and a scale shows 8 pounds. The error is |5 - 8| = 3 pounds, and 3 ÷ 8 = 0.375, so the percent error is 37.5%. Divide by the actual value, not by the estimate.
Guided Practice15 minutes
Pairs solve four problems. Before computing, each pair writes the base of every percent in words, for example "18% of the haircut price". Circulate and ask: "A percent of what?"
Guided practice problems with answers
Problem
Answer
A haircut costs $25, and you leave an 18% tip. What do you pay in all?
0.18 × 25 = $4.50 tip, so $29.50 (or 1.18 × 25)
Lena earns a 4% commission. She sold $3,150 of phones this month. What is her commission?
0.04 × 3,150 = $126
A video game's price went from $60 to $51. What is the percent decrease?
60 - 51 = 9, and 9 ÷ 60 = 0.15: a 15% decrease
Omar estimated that the classroom is 9 m long. It is 10 m long. What is his percent error?
|9 - 10| = 1, and 1 ÷ 10 = 0.10: 10%
Listen for these errors: dividing by the new amount in a percent change, dividing by the estimate in a percent error, and writing 4% as 0.4 instead of 0.04. Ask one pair to explain why the tip and the total are different answers.
Independent Practice15 minutes
Students solve six problems on their own. For each one, they write the base, the percent as a decimal, and a sentence with the answer and its units.
Independent practice problems with answers
Problem
Answer
A backpack costs $40, and sales tax is 6.5%. What is the total?
0.065 × 40 = $2.60 tax, so $42.60
A $120 scooter is 15% off. What is the sale price?
0.85 × 120 = $102
A shop pays $8 for a mug and marks it up 125%. What is its price?
8 + 1.25 × 8 = 8 + 10 = $18
A town's yearly rainfall went from 20 inches to 23 inches. What is the percent increase?
3 ÷ 20 = 0.15: a 15% increase
Find the simple interest on $2,000 at 2% per year for 5 years.
2,000 × 0.02 × 5 = $200
Oatmeal cookies use 2 cups of oats for every 3 cups of flour. A baker uses 7.5 cups of flour, and oats cost $0.60 per cup. What do the oats cost?
7.5 × 2/3 = 5 cups of oats, and 5 × 0.60 = $3.00
Closure5 minutes
Exit ticket: (1) A $30 shirt is 40% off, and then 5% sales tax is added to the sale price. What do you pay? (Answer: 0.60 × 30 = $18, and 1.05 × 18 = $18.90.) (2) A pack of paper cost $5.00 and now costs $4.40. What is the percent decrease? (Answer: 0.60 ÷ 5.00 = 0.12, a 12% decrease.) (3) In one sentence: in a percent error, why do you divide by the actual value?
Differentiation Strategies
For Struggling Students
Draw a tape diagram split into 10 equal parts of 10% before each problem, as in Diagram 1, and find one part first
Keep a card with the question "a percent of what?" and the words base, part and percent on it
Start with friendly percents (10%, 25%, 50%), then move to percents such as 7.5% or 18% with a calculator
For Advanced Students
Ask whether a 20% markup followed by a 20% markdown returns to the cost, and have students explain the answer with a tape diagram
Work backward: a price after 8% tax is $54. What was the price before tax?
Compare 3% simple interest with an account that adds 3% of the new balance each year, and name this as an extension beyond this standard (compound interest, in high school)
Assessment Guidance
What to Look For
Check that students name the base of every percent before they compute, and that they change the decimal correctly (4% is 0.04, 125% is 1.25). In multistep problems, look for the second percent taken of the new amount, not the original. For percent change, students divide by the original amount, and for percent error, by the actual value. Final answers should be rounded to the cent for money and given in a sentence with units.
02
Classroom Activities
3 Activities
1
Receipt Check
15 minPairs
Pairs get four printed receipt cards. Each card was made up for this activity. Three receipts contain one mistake, and one is correct. Pairs check every line, find the mistake, explain the error, and write the correct total.
Receipt Cards
Receipt A (restaurant): food $48.00, tax 8% $3.84, tip 20% of the food $9.60, total $61.44
Receipt B (store sale): hoodie $55.00, 20% off -$11.00, sale price $44.00, tax 6% $3.30, total $47.30
Receipt C (online order): order $70.00, service fee 5% $5.00, flat shipping $6.00, total $81.00
A is correct: 0.08 × 48 = 3.84, 0.20 × 48 = 9.60, and 48 + 3.84 + 9.60 = 61.44
B taxed the full $55 instead of the $44 sale price. The tax should be 0.06 × 44 = $2.64, so the total is $46.64
C treated 5% as $5. The fee should be 0.05 × 70 = $3.50, so the total is 70 + 3.50 + 6 = $79.50
D used 0.6% instead of 6%. The commission should be 0.06 × 1,850 = $111.00
Discussion Questions
Receipts B and C charged the customer too much. Which one overcharged by more, and by how much? (C, by $1.50; B overcharged by $0.66.)
Receipt D paid the salesperson too little. How can a quick estimate catch the mistake? (10% of $1,850 is $185, so 6% must be more than half of that.)
On Receipt B, why must the tax come after the discount?
Modification for Distance Learning
Share the four receipts as images on a slide. Pairs mark each line as correct or wrong in a shared table and type the corrected totals.
2
Multiplier Match
15 minPairs
A multiplier is the single number you multiply the original amount by to get the new amount, for example 1.07 for adding 7% tax. Pairs match 8 situation cards with 8 multiplier cards (16 cards in all), then use every card on a $40 price.
Cards (answer key)
Add 7% sales tax: ×1.07
Take 30% off: ×0.70
Mark up 30%: ×1.30
Pay the bill plus an 18% tip: ×1.18
A price drops 15%: ×0.85
A seller's 5% commission (what the seller earns): ×0.05
Balance after 1 year of 3% simple interest: ×1.03
Mark up 200%: ×3
Procedure
Pairs deal the situation cards face up in one row and the multiplier cards in a second row
They match each pair and say the percent sentence out loud, for example "take 30% off means you pay 70%"
They use each multiplier on $40 and list the 8 results from smallest to largest: $2, $28, $34, $41.20, $42.80, $47.20, $52, $120
Discussion Questions
Three of the multipliers are less than 1. Which ones, and what does that tell you about the new amount?
Use ×1.30 and then ×0.70 on $40. Why do you get $36.40 and not $40?
Why is the multiplier for a 200% markup 3 and not 2?
Challenge Variation
Pairs write two new cards of their own, one with a multiplier greater than 2 and one less than 0.5, and trade them with another pair.
3
Estimate, Measure, Compare
20 minGroups of 3
Each student estimates the length of four classroom objects in centimeters, then the group measures them with a ruler or meter stick and finds each percent error. Students then compare their errors to see who estimated best.
Procedure
Write your estimates for a pencil, a notebook, a desk top (width) and a classroom door (height) before anyone measures
Measure each object to the nearest centimeter
Find each percent error: |estimate - actual| ÷ actual × 100%, rounded to the nearest tenth of a percent
Circle your smallest and your largest percent error
Sample Results (invented data for one student)
Pencil: estimate 12 cm, actual 16 cm, error 4 cm, percent error 25.0%
Notebook: estimate 30 cm, actual 28 cm, error 2 cm, percent error 7.1%
Desk width: estimate 50 cm, actual 60 cm, error 10 cm, percent error 16.7%
Door height: estimate 180 cm, actual 203 cm, error 23 cm, percent error 11.3%
Discussion Questions
In the sample, the door estimate missed by the most centimeters, but the pencil has the largest percent error. Why?
The desk estimate was 10 cm off. How far off would a door estimate have to be for the same percent error?
Is it possible to have a percent error over 100%? Give an example.
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: Tape Diagrams for a Markup and a Markdown
The store pays $48 for the jacket (100%). A 75% markup adds $36, so the price is $84. The 30% markdown is taken from the $84 price: split it into 10 equal parts of $8.40, take 3 parts off, and 7 parts, $58.80, are left. The bars are drawn to scale, 6.5 pixels per dollar.
Diagram 2: Simple Interest Grows in Proportion to Time
Nia's $500 earns 3% simple interest, which is $15 each year. The interest after t years is 15t dollars, so the points (0, 0), (1, 15), (2, 30), (3, 45) and (4, 60) lie on a straight line through (0, 0). Drawn to scale.
04
Homework Assignment
~30 min
7.RP.A.3 Homework: Multistep Ratio and Percent Problems
Directions: Show all work. For every percent, write its base in words before you compute. Round money to the nearest cent, and answer each question in a sentence with units.
Part 1: Ratios, Tax and Tips (Problems 1-2)
Ms. Lee makes 4 kg of trail mix with peanuts and raisins in a ratio of 5 to 3 by weight. Peanuts cost $9 per kg and raisins cost $7 per kg. How many kilograms of each does she use, and what does the whole batch cost?
A family's restaurant bill is $64 before tax. Sales tax is 8.5%, and they leave an 18% tip on the bill before tax. Find the tax, the tip and the total they pay.
Part 2: Markups, Markdowns and Commissions (Problems 3-4)
A bike shop pays $220 for a bike and marks it up 60%. During a sale, the bike is marked down 25%. Find the price before the sale and the sale price. The sale price is what percent of the $220 the shop paid?
Jamal earns $400 per week plus a 3% commission on his sales. Last week he earned $616 in all. What were his sales last week? Explain each step.
Part 3: Percent Change and Simple Interest (Problems 5-6)
A town had 2,500 people. The next year its population rose to 2,800, and the year after that it fell by 10%. Find the percent increase in the first year, the population after the second year, and the percent change over the two years from 2,500.
Ana puts $1,200 in a savings account that pays 2.5% simple interest per year. How much interest does she earn in 3 years? After how many years will she have earned $150 in interest?
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Base of Each Percent
Base named for every percent and used correctly, including a new base after a first change
Base named but one percent taken of the wrong amount
Bases missing or mostly wrong
Percents and Ratios
Percents written as correct decimals and ratios split correctly
One conversion or ratio error
Several conversion or ratio errors
Computation
All steps shown and correct, money rounded to the cent
One arithmetic error
Most answers incorrect
Answer in Context
Each answer given in a sentence with units
Answers correct but no units or sentences
Answers missing
05
Quiz: 20 Questions
Interactive, with answers
Instructions
Work through the questions in order. A calculator is allowed. Your score updates as you answer, and Reset quiz clears everything so you or your students can try again.
Multiple choice: pick an option to check it. Short answer: write your answer, then reveal the model answer.
0 of 20 answered · 0 correct
Question 1 of 20 · Multiple Choice
A $45 game has 6% sales tax. What is the total cost?
Answer: B
The tax is 0.06 × 45 = $2.70, so the total is 45 + 2.70 = $47.70 (or 1.06 × 45). Choice A is the tax alone, not the total. Choice C adds 0.06 dollars instead of 6% of the price. Choice D treats 6% as $6.
Question 2 of 20 · Multiple Choice
A lunch bill is $52, and Mia leaves a 15% tip. How much is the tip?
Answer: A
The tip is 0.15 × 52 = $7.80. Choice B is a 10% tip. Choice C is the bill plus the tip, not the tip. Choice D divides 52 by 15 instead of multiplying by 0.15.
Question 3 of 20 · Multiple Choice
Jeans cost $60 and are 35% off. What is the sale price?
Answer: C
The discount is 0.35 × 60 = $21, so the sale price is 60 - 21 = $39 (or 0.65 × 60). Choice A is the discount, not the sale price. Choice B subtracts 35 dollars instead of 35%. Choice D adds 35% instead of taking it off.
Question 4 of 20 · Multiple Choice
A store pays $30 for a lamp and marks it up 80%. What is the store's price for the lamp?
Answer: D
The markup is 0.80 × 30 = $24, so the price is 30 + 24 = $54 (or 1.80 × 30). Choice A is the markup alone. Choice B adds 80 dollars instead of 80%. Choice C divides 30 by 0.80 instead of multiplying by 1.80.
Question 5 of 20 · Multiple Choice
A salesperson earns a 6% commission. She sells $3,500 of furniture. What is her commission?
Answer: A
The commission is 0.06 × 3,500 = $210. Choice B writes 6% as 0.006. Choice C writes 6% as 0.6. Choice D divides 3,500 by 6 instead of multiplying by 0.06.
Question 6 of 20 · Multiple Choice
Three concert tickets cost $42 each. The website adds a 10% service fee on the ticket total and a flat $2.50 order fee. What is the total cost?
Answer: C
The tickets cost 3 × 42 = $126. The service fee is 0.10 × 126 = $12.60, and the order fee is $2.50 once: 126 + 12.60 + 2.50 = $141.10. Choice A takes the 10% fee on one ticket only (126 + 4.20 + 2.50). Choice B forgets the $2.50 order fee. Choice D charges the $2.50 order fee on every ticket (3 × (42 + 4.20 + 2.50)).
Question 7 of 20 · Multiple Choice
The price of a movie ticket went from $16 to $20. What is the percent increase?
Answer: B
The change is 20 - 16 = $4, and 4 ÷ 16 = 0.25, so the increase is 25%. Choice A divides by the new price, 4 ÷ 20. Choice C writes the $4 change as 4%. Choice D is 16 ÷ 20, the old price as a percent of the new one.
Question 8 of 20 · Multiple Choice
A library had 1,250 visitors in March and 1,000 in April. What is the percent decrease?
Answer: D
The change is 1,250 - 1,000 = 250, and 250 ÷ 1,250 = 0.20, so the decrease is 20%. Choice A divides by the new amount, 250 ÷ 1,000. Choice B is 1,000 ÷ 1,250, the April visitors as a percent of March. Choice C is 1,250 ÷ 1,000, the original divided by the new amount.
Question 9 of 20 · Multiple Choice
Leo estimated that a jar held 40 marbles. It held 50. What is his percent error?
Answer: A
The error is |40 - 50| = 10 marbles, and 10 ÷ 50 = 0.20, so the percent error is 20%. Choice B divides by the estimate, 10 ÷ 40. Choice C writes the 10-marble error as 10%. Choice D is 40 ÷ 50, the estimate as a percent of the actual count.
Question 10 of 20 · Multiple Choice
Sam puts $800 in an account that pays 4% simple interest per year. How much interest does he earn in 3 years?
Answer: C
I = P × r × t = 800 × 0.04 × 3 = $96. Choice A is the interest for 1 year only. Choice B is the balance after 3 years, $800 + $96, not the interest. Choice D uses 4 instead of 0.04 for the rate.
Question 11 of 20 · Multiple Choice
In a class of 28 students, the ratio of students who walk to school to students who ride the bus is 3 to 4. How many students walk?
Answer: B
There are 3 + 4 = 7 parts, so each part is 28 ÷ 7 = 4 students, and 3 parts are 12 students. Choice A takes 3/4 of 28, which treats the ratio as a fraction of the whole class. Choice C is the number who ride the bus. Choice D is the size of one part, not the 3 parts that walk.
Question 12 of 20 · Multiple Choice
A $90 coat is 25% off. At the register, you get another 10% off the sale price. What do you pay?
Answer: D
The first discount leaves 0.75 × 90 = $67.50. The second takes 10% of $67.50, which is $6.75, so you pay $60.75 (or 0.75 × 0.90 × 90). Choice A adds the percents and takes 35% off $90. Choice B stops after the first discount. Choice C uses only the 10% discount.
Question 13 of 20 · Multiple Choice
Sales tax is 8%. Which expression gives the total cost of an item with price x dollars, including tax?
Answer: C
The total is the price plus 8% of the price: x + 0.08x = 1.08x. Choice A adds 8 cents, not 8% of x. Choice B is the tax alone. Choice D adds $8 no matter what the price is.
Question 14 of 20 · Multiple Choice
After a 20% discount, a sweater costs $36. What was the price before the discount?
Answer: A
The sale price is 80% of the original, so 0.80p = 36 and p = 36 ÷ 0.80 = $45. Check: 45 - 0.20 × 45 = 36. Choice B adds 20% of $36 to $36, but the discount was 20% of the original. Choice C takes 20% off again. Choice D adds 20 dollars.
Question 15 of 20 · Short Answer
A store pays $12 for a phone case and marks it up 150%. A customer pays 5% sales tax on the store's price. What does the customer pay?
Markup: 1.50 × 12 = $18, so the price is 12 + 18 = $30 (or 2.50 × 12). Tax: 0.05 × 30 = $1.50. The customer pays $31.50.
Question 16 of 20 · Short Answer
A bean plant was 40 cm tall on Monday and 52 cm tall on Friday. What is the percent increase in its height?
The change is 52 - 40 = 12 cm, and 12 ÷ 40 = 0.30, so the height increased by 30%. Divide by the original height, 40 cm.
Question 17 of 20 · Short Answer
Kiara measured the length of a room as 4.6 m. The actual length is 5 m. What is her percent error?
The error is |4.6 - 5| = 0.4 m, and 0.4 ÷ 5 = 0.08, so the percent error is 8%.
Question 18 of 20 · Short Answer
Tomas borrows $600 at 5% simple interest per year and pays it all back after 2 years. How much does he pay back in all?
Interest: 600 × 0.05 × 2 = $60. He pays back the $600 he borrowed plus the interest: $660.
Question 19 of 20 · Short Answer
On a map, 2 cm stands for 15 km. Two towns are 9 cm apart on the map. How far apart are they? A bus drives between them at an average speed of 45 km per hour. How long is the trip?
Each centimeter stands for 15 ÷ 2 = 7.5 km, so 9 cm stands for 9 × 7.5 = 67.5 km. The trip takes 67.5 ÷ 45 = 1.5 hours (1 hour 30 minutes).
Question 20 of 20 · Short Answer
A salesperson earns an 8% commission. She wants to earn $600 in commission this month. How much must she sell? Show how you know.
0.08 × s = 600, so s = 600 ÷ 0.08 = $7,500 of sales. Check: 8% of $7,500 is 0.08 × 7,500 = $600.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does 7.RP.A.3 mean?
7.RP.A.3 means students can solve ratio and percent problems that take several steps. Each step uses a proportional relationship: the part is the same percent of the base every time. The standard lists the kinds of problems: simple interest, tax, markups and markdowns, tips and commissions, fees, percent increase and decrease, and percent error.
What grade is 7.RP.A.3, and what comes after it?
It is a grade 7 standard in the Ratios and Proportional Relationships domain. In grade 8, students graph proportional relationships and read the unit rate as the slope (8.EE.B.5). In high school, repeated percent growth, such as compound interest, becomes exponential functions (HSF.LE.A.1).
What is the difference between a markup and a markdown?
A markup raises a price and a markdown lowers it, and both are percents of the amount they start from. A store marks up its cost to set a selling price, then may mark that price down in a sale. Because the markdown is taken from the bigger, marked-up price, a markup and a markdown of the same percent do not cancel. For example, $50 marked up 20% is $60, and $60 marked down 20% is $48.
Do you figure the tip before or after tax?
Either can be correct, so a problem should say which one. Many people tip on the bill before tax, and that is what the problems on this page do unless they say otherwise. The math is the same both ways: tip = tip rate × the amount the problem names as the base.
How do you find percent error?
Divide the size of the error by the actual value and write the result as a percent. For example, if you guess that a hallway is 22 m long and it is really 25 m, the error is 3 m, and 3 ÷ 25 = 0.12, so the percent error is 12%. The difference is always taken as a positive number, so a guess that is too high and a guess that is too low by the same amount have the same percent error.
Is simple interest the same as the interest a bank pays?
Not usually: most savings accounts pay compound interest, which is beyond this standard. Simple interest is always a percent of the original principal, so it adds the same amount every year and grows in proportion to time. Compound interest also pays interest on earlier interest, which students study with exponential functions in high school.
How does 7.RP.A.3 connect to 7.RP.A.2?
7.RP.A.2 is about recognizing proportional relationships and writing them as y = kx, and 7.RP.A.3 uses those relationships to solve problems. For example, with 6% sales tax, the total is y = 1.06x, where x is the price, and the tax alone is y = 0.06x. Both are proportional relationships with constants 1.06 and 0.06.
Why do students need to know which amount is the base?
A percent means nothing until you know what it is a percent of. In a percent change the base is the original amount, in a percent error it is the actual value, and after a discount the tax is taken of the new price. Many wrong answers on these problems come from using the right percent with the wrong base.
How can parents help with 7.RP.A.3 at home?
Use real receipts and sale signs. At a store, ask your child to estimate a sale price or a total with tax before you pay, then compare with the receipt. At a restaurant, let them work out a tip: 10% is easy to find, and 20% is twice that. Ask them to say what the percent is taken of each time.
What mistakes should teachers watch for?
A common mistake is adding percents that have different bases, for example treating 25% off followed by 10% off as 35% off. Other frequent errors are writing 3% as 0.3, dividing by the new amount in a percent change, dividing by the estimate in a percent error, and giving the tax or the discount when the question asks for the total or the sale price.
07
Related Standards
5 standards
These standards connect to 7.RP.A.3: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
6.RP.A.3Prerequisite
Use ratio and rate reasoning, including percents, to solve real-world problems
