6.RP.A.3Common CoreMathRatios and Proportional RelationshipsGrade 6
6.RP.A.3: Using Ratio and Rate Reasoning to Solve Problems
In plain English: 6.RP.A.3 is the Common Core grade 6 math standard that asks students to use ratio and rate reasoning to solve real-world problems with tables of equivalent ratios, tape diagrams, double number lines and equations. It covers ratio tables and their graphs, unit rates for prices and constant speed, percents as rates per 100, and converting measurement units. It prepares for proportional relationships in grade 7.
Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
a.Make tables of equivalent ratios relating quantities with whole number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.
b.Solve unit rate problems including those involving unit pricing and constant speed. For example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed?
c.Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.
d.Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.
Common Core State Standards for Mathematics · Domain: Ratios and Proportional Relationships (RP) · Cluster: Understand ratio concepts and use ratio reasoning to solve problems. Also written as 6.RP.3 · Official standard
Students use ratio and rate reasoning to solve everyday problems. A ratio compares two quantities, such as 3 cups of pretzels for every 2 cups of raisins. Students build tables of equivalent ratios (ratios that describe the same mix, such as 3 to 2 and 6 to 4), fill in missing values, plot the pairs on the coordinate plane, and use tables to decide which of two mixes is stronger.
Next, students find unit rates (the amount for 1 unit, such as dollars per pound or miles per hour) to compare prices and to solve constant speed problems, including the official example about mowing lawns. They treat a percent as a rate per 100, find a percent of a quantity, and find the whole when they know a part and its percent. Finally, they convert measurement units with ratio reasoning and watch how the units cancel when they multiply or divide. Students choose among four tools the standard names: ratio tables, tape diagrams, double number lines and simple equations. All numbers stay positive, and unit rates use whole numbers, decimals or simple fractions, as expected in grade 6.
Learning Objectives
By the end of this lesson, students will be able to:
Make a table of equivalent ratios, find missing values in it, and plot the pairs on the coordinate plane
Use ratio tables to compare two ratios and decide which mix is stronger or which rate is faster
Find unit rates to solve unit pricing and constant speed problems
Find a percent of a quantity as a rate per 100, and find the whole given a part and the percent
Convert measurement units with ratio reasoning and show how units cancel when multiplying or dividing
Prior Knowledge Required
Students should already be comfortable with:
Solving multiplication and division comparison problems 4.OA.A.2
Reading a fraction as division, such as 3/4 = 3 ÷ 4 5.NF.B.3
Converting units inside one system, such as feet to inches 5.MD.A.1
Graphing points in the first quadrant of the coordinate plane 5.G.A.2
Ratio language and the meaning of a unit rate 6.RP.A.16.RP.A.2
Adding, subtracting, multiplying and dividing decimals 6.NS.B.3
Show a lemonade recipe on the board and ask students to answer without a formal method:
Warm-Up Prompt
"A lemonade recipe uses 2 cups of lemon juice for every 5 cups of water. You have 6 cups of lemon juice. How much water do you need so the lemonade tastes the same? Name one more pair of amounts that would taste the same."
Collect answers. The recipe has been tripled (6 = 2 × 3), so the water must be tripled too: 5 × 3 = 15 cups. Some students may add 4 to both amounts and say 9 cups of water. Taste settles it: 6 cups of juice with 9 cups of water is much more sour than the recipe. Introduce the words: a ratio compares two quantities (2 cups of juice to 5 cups of water, written 2:5), and equivalent ratios (ratios that describe the same mix) come from multiplying or dividing both quantities by the same number. Other correct pairs include 4:10 and 1:2.5.
Direct Instruction20-25 minutes
Part 1: Ratio tables and graphs (standard a). A ratio table lists equivalent ratios in rows or columns. Build the trail mix table in Diagram 1 with the class, then plot each pair (pretzels, raisins) as a point. Ask: what do the points have in common? They line up on a straight line that would pass through (0, 0), because 0 cups of pretzels go with 0 cups of raisins. Then use tables to compare two ratios by making one quantity the same in both tables.
Ratio table, missing values and graph
Trail mix uses 3 cups of pretzels for every 2 cups of raisins. How many cups of raisins go with 12 cups of pretzels? How many cups of pretzels go with 10 cups of raisins?
Equation: 3:2 = 12:8 = 15:10, so 8 cups of raisins and 15 cups of pretzels; plot (3, 2), (6, 4), (9, 6), (12, 8), (15, 10)
Comparing ratios with tables
Mix A uses 2 cups of blue paint for every 3 cups of yellow. Mix B uses 3 cups of blue for every 5 cups of yellow. Which mix is bluer?
Equation: With 15 cups of yellow: Mix A needs 10 cups of blue and Mix B needs 9, so Mix A is bluer
Unit rate (official example)
It took 7 hours to mow 4 lawns. At that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed?
Equation: 35 ÷ 7 = 5, so 4 × 5 = 20 lawns; the rate is 4/7 lawn per hour (or 7/4 = 1 3/4 hours per lawn)
Finding the whole from a part and a percent
15 students in a grade 6 class walk to school. That is 30% of the class. How many students are in the class?
Equation: 30% is 15, so 10% is 5 and 100% is 50 students
Converting units
A water cooler jug holds 5 gallons. How many quarts is that? (1 gallon = 4 quarts)
Part 2: Unit rates (standard b). A rate compares two quantities with different units, such as 24 miles in 2 hours. A unit rate tells how much for 1 unit: 12 miles per hour. Work the official lawn example above with a double number line (two number lines, one above the other, with matching marks lined up): 7 hours lines up with 4 lawns, and 35 hours lines up with 20 lawns. Then show two everyday uses:
Unit pricing: a 3-pound bag of oranges costs $4.50 and a 5-pound bag costs $7.00. Divide price by pounds: $4.50 ÷ 3 = $1.50 per pound and $7.00 ÷ 5 = $1.40 per pound. The 5-pound bag is the better buy because each pound costs less.
Constant speed: a cyclist rides 24 miles in 2 hours at a constant speed (the same speed the whole time). The unit rate is 24 ÷ 2 = 12 miles per hour. Diagram 2 shows the double number line: every hour adds 12 miles, so 5 hours gives 60 miles.
Part 3: Percents (standard c). A percent is a rate per 100: 25% means 25 out of every 100. To find 25% of a $60 jacket price, multiply by 25/100: (25/100) × 60 = 15, so the discount is $15. In the same way, 30% of a quantity means 30/100 times the quantity. To find the whole from a part, use a tape diagram (a bar split into equal parts), as in Diagram 2 and the fourth example: split 100% into ten 10% boxes, find one box, then multiply by 10.
Part 4: Converting units (standard d). A conversion like 1 gallon = 4 quarts is a ratio, so you can use a ratio table or multiply by a conversion rate written as a fraction. Write the units every time. In the fifth example, gallons appear on the top and the bottom, so they cancel and only quarts are left. The same thing happens when you multiply a rate by a time: 12 miles per hour × 3 hours = 36 miles, because hours cancel.
Guided Practice15 minutes
Pairs work through four problems, one at a time, and draw the tool they used (table, tape diagram or double number line) next to each answer. After each problem, one pair explains its drawing.
Problem 1: salt dough uses 4 cups of flour for every 3 cups of salt. Fill in the blanks, then plot the pairs (flour, salt) on grid paper.
Flour (cups)
4
8
12
?
20
Salt (cups)
3
?
9
15
?
Answers: 6 cups of salt for 8 cups of flour, 20 cups of flour for 15 cups of salt, and 15 cups of salt for 20 cups of flour. Problem 2: a train travels 150 miles in 3 hours at a constant speed. How far does it go in 5 hours? (50 miles per hour, so 250 miles.) Problem 3: what is 40% of 45 minutes? ((40/100) × 45 = 18 minutes.) Problem 4: how many feet are in 72 inches? (72 ÷ 12 = 6 feet.) Listen for students who add the same number to both quantities in the table instead of multiplying.
Independent Practice10-15 minutes
Students solve five problems on their own and show one tool for each. (1) Sam reads 45 pages in 30 minutes and Lee reads 32 pages in 20 minutes. Who reads faster? (Sam reads 1.5 pages per minute and Lee reads 1.6, so Lee. A table to 60 minutes gives 90 pages and 96 pages.) (2) A 16-ounce jar of peanut butter costs $3.20 and a 28-ounce jar costs $5.04. Which is the better buy? ($0.20 and $0.18 per ounce, so the 28-ounce jar.) (3) A soccer team won 60% of its 25 games. How many games did it win? (15.) (4) Kai has saved $48, which is 80% of the price of a bike helmet. What does the helmet cost? (80% is $48, so 10% is $6 and 100% is $60.) (5) A race is 5 kilometers long. How many meters is that? (1 kilometer = 1,000 meters, so 5,000 meters.)
Closure5 minutes
Exit ticket: (1) A car uses 4 gallons of gas to go 112 miles. What is the unit rate in miles per gallon? (28 miles per gallon.) (2) 13 is 20% of what number? (20% is 13, so 100% is 5 × 13 = 65.) (3) Convert 3 meters to centimeters, and show which units cancel. (3 meters × (100 centimeters / 1 meter) = 300 centimeters; meters cancel.)
Differentiation Strategies
For Struggling Students
Give a ratio table template with arrows labeled "× 2", "× 3" between rows, so students multiply both quantities instead of adding
Use a pre-drawn tape diagram with 10 equal boxes for every percent problem, and have students write 10% under each box first
Provide a conversion card (1 foot = 12 inches, 1 yard = 3 feet, 1 gallon = 4 quarts, 1 pound = 16 ounces, 1 meter = 100 centimeters) and let students build a two-row table from it
For Advanced Students
Ask for a price pair where the bigger package is NOT the better buy, and explain how the store could fix its price
Ask students to convert a walking speed of 3 miles per hour into feet per minute (1 mile = 5,280 feet) and show every unit that cancels
Give two points from a ratio table graph, such as (3, 12) and (5, 20), and ask for three more points and the unit rate
Assessment Guidance
What to Look For
Check that students multiply or divide both quantities by the same number when they build a ratio table, and that they can say why adding the same number changes the mix. In unit rate problems, look for the unit written with the answer (dollars per ounce, miles per hour) and for the correct order of division: price ÷ amount gives a price per unit. In percent problems, students should be able to draw a tape diagram or double number line that matches their work. In conversions, ask students to write the units and cross out the ones that cancel.
02
Classroom Activities
3 Activities
1
Recipe Ratio Tables on Grid Paper
20 minPairs
Each pair gets one recipe card, builds a ratio table, answers the missing-value question on the card, and plots the pairs on grid paper. Then two pairs with the same kind of recipe meet and use their tables and graphs to compare ratios (standard a).
Recipe Cards (4 cards)
Card A, fruit punch: 3 cups of juice for every 2 cups of sparkling water. How much sparkling water goes with 12 cups of juice? (8 cups)
Card B, fruit punch: 4 cups of juice for every 3 cups of sparkling water. How much juice goes with 9 cups of sparkling water? (12 cups)
Card C, orange paint: 1 cup of red for every 3 cups of yellow. How much yellow goes with 4 cups of red? (12 cups)
Card D, orange paint: 2 cups of red for every 5 cups of yellow. How much red goes with 20 cups of yellow? (8 cups)
Procedure
Make a table with at least 5 rows by multiplying both amounts on the card by 2, 3, 4 and 5
Answer the question on the card and circle that row
On grid paper, put the second amount (sparkling water or yellow) on the x-axis and the first amount (juice or red) on the y-axis, then plot every row as a point
Pair A meets pair B, and pair C meets pair D. Extend both tables until one amount matches, then decide which mix is stronger
Discussion Questions
With 6 cups of sparkling water, Card A uses 9 cups of juice and Card B uses 8. Which punch tastes more like juice? (Card A)
With 15 cups of yellow, Card C uses 5 cups of red and Card D uses 6. Which orange is redder? (Card D)
On your grid paper, which punch has the steeper line? What does a steeper line mean in this recipe? (Card A: more juice for each cup of sparkling water)
Why do all your points line up with (0, 0)?
Modification for Distance Learning
Share the cards as images and have pairs plot points in a free online graphing tool or on a shared grid slide. Pairs post a screenshot of both lines on one graph before the comparison discussion.
2
Best Buy and Speed Card Sort
20 minGroups of 3-4
Groups receive 8 cards: 6 price cards that form 3 pairs of the same product, and 2 constant speed cards. Groups find each unit rate, decide the better buy, and solve the speed problems (standard b).
Price Cards (6 cards)
Cereal: a 12-ounce box for $3.60 ($0.30 per ounce) and an 18-ounce box for $4.86 ($0.27 per ounce)
Juice boxes: 8 for $4.00 ($0.50 each) and 10 for $5.50 ($0.55 each)
Apples: 3 pounds for $4.47 ($1.49 per pound) and 5 pounds for $7.25 ($1.45 per pound)
Speed Cards (2 cards)
A family drives 165 miles in 3 hours at a constant speed. How far do they drive in 5 hours? (55 miles per hour, so 275 miles)
Ana walks 6 miles in 2 hours. Ben walks 10 miles in 4 hours. Both walk at a constant speed. Who walks faster? (Ana: 3 miles per hour; Ben: 2.5 miles per hour)
Procedure
One student reads a card, a second finds the unit rate with a calculator or by hand, and a third checks by multiplying back; roles rotate
Sort each price pair into "better buy" and "not the better buy" and write the unit price on a sticky note on each card
For the speed cards, draw a double number line and mark the unit rate
Discussion Questions
In which pair is the bigger package NOT the better buy? (Juice boxes: the 8-pack costs less per box)
Why is a total price not enough to decide the better buy?
On the family's double number line, which mark tells you the speed?
Challenge Variation
Groups bring or invent a price pair from a store flyer, write it on a blank card, and trade with another group, which must find the better buy and explain it with unit prices.
3
Tape Diagrams and Measure-and-Convert
20 minPairs
Pairs work at two stations. At the first, they draw tape diagrams on grid paper to solve 8 percent cards (standard c). At the second, they measure classroom objects and convert the measurements with ratio tables (standard d).
Station 1: Percent Cards (8 cards)
Percent of a quantity: 20% of 35 (7), 50% of 68 (34), 75% of 40 (30), 10% of 90 (9)
Finding the whole: 9 is 30% of what number? (30), 14 is 20% of what number? (70), 45 is 75% of what number? (60), 18 is 40% of what number? (45)
Station 2: Measure and Convert
Measure the height of the classroom door and the height of a desk in inches with a yardstick or tape measure
Convert each height to feet with a ratio table (1 foot : 12 inches). A typical classroom door is about 80 inches tall, which is 6 feet 8 inches, or 6 2/3 feet
Measure a new pencil in centimeters with a meter stick, then convert to meters (100 centimeters : 1 meter). A new pencil is about 19 centimeters, or 0.19 meters
Write each conversion as a multiplication with units, for example 80 inches × (1 foot / 12 inches), and cross out the units that cancel
Discussion Questions
On a tape diagram, why do you find 10% first?
When you convert inches to feet, does the number get bigger or smaller? Why does that make sense?
Which percent cards could you solve with 25% boxes instead of 10% boxes?
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: A Ratio Table and Its Graph
The trail mix table lists equivalent ratios of pretzels to raisins, 3:2 through 15:10. Each row becomes a point (pretzels, raisins) on the coordinate plane, drawn to scale. The points lie on one straight line through (0, 0), and each step of 3 cups of pretzels adds 2 cups of raisins.
Diagram 2: A Double Number Line and a Tape Diagram
Top: a double number line for a cyclist who rides 24 miles in 2 hours at a constant speed. Equal spaces on both lines show that each hour adds 12 miles, the unit rate. Bottom: a tape diagram with ten 10% boxes. If 30% of a class is 15 students, each box is 5 students, and the whole class is 50 students.
04
Homework Assignment
~30 min
6.RP.A.3 Homework: Ratios, Rates, Percents and Units
Directions: Show your work with a ratio table, tape diagram, double number line or equation for every problem. Write the units with every answer. For conversions, show which units cancel.
Part 1: Ratio Tables and Graphs (Problems 1-2)
A smoothie recipe uses 5 strawberries for every 2 bananas. Make a ratio table with at least 4 rows. How many strawberries go with 8 bananas? How many bananas go with 30 strawberries? Plot the pairs (bananas, strawberries) on the coordinate plane and describe the pattern.
Jada mixes 3 scoops of drink powder with 8 cups of water. Omar mixes 4 scoops with 10 cups of water. Make a ratio table for each drink and use your tables to decide whose drink is stronger. Explain how the tables show it.
Part 2: Unit Rates (Problems 3-4)
A 6-pack of yogurt cups costs $4.20 and a 4-pack costs $3.00. Find the unit price of each pack. Which is the better buy? How much would 18 yogurt cups cost if you buy only the better-buy packs?
Lin rides her bike 28 miles in 2 hours at a constant speed. Find her speed in miles per hour. How long does it take her to ride 35 miles? How far does she ride in 45 minutes? Draw a double number line.
Part 3: Percents and Units (Problems 5-6)
(a) A school has 480 students, and 35% of them ride the bus. How many students ride the bus? (b) On a library shelf, 11 books are mysteries. That is 25% of the books on the shelf. How many books are on the shelf? Draw a tape diagram for part (b).
(a) A class party needs 3 gallons of punch. How many cups is that? (1 gallon = 16 cups) If each of 45 students drinks 1 cup, is there enough punch? (b) Leo is 5 feet 2 inches tall. What is his height in inches? For both parts, write the conversion as a multiplication and show which units cancel.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Ratio Tables and Graphs
Tables are correct, missing values found, points plotted correctly
One error in a table or graph
Tables built by adding, or missing
Unit Rates
Unit rates correct with units, better buy and speed answers justified
Correct rates but a missing unit or explanation
Rates missing or divided in the wrong order
Percents
Percent of a quantity and the whole found correctly, with a tape diagram
One answer correct, or a diagram that does not match
Both answers incorrect
Unit Conversions
Conversions correct, units written and canceled
Correct numbers but units missing
Conversions incorrect
05
Quiz: 20 Questions
Interactive, with answers
Instructions
Work through the questions in order. 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 recipe uses 4 eggs for every 5 cups of milk. Which table shows equivalent ratios of eggs to milk?
Answer: B
In choice B, each row multiplies both 4 and 5 by the same number (2, 3 and 4), so every ratio is equivalent to 4:5. Choice A adds 1 to both amounts, which changes the mix: 5:6 is not equivalent to 4:5. Choice C multiplies the eggs but adds 3 cups of milk each time. Choice D adds 2 eggs each time but multiplies the milk.
Question 2 of 20 · Multiple Choice
Four cups of rice serve 10 people. At the same rate, how many cups of rice serve 35 people?
Answer: D
35 people is 3.5 times 10 people, so multiply the rice by 3.5 too: 4 × 3.5 = 14 cups. A table works as well: 4 cups for 10 people, 2 cups for 5 people, 14 cups for 35 people. Choice A adds 25 to the rice because 35 - 10 = 25. Choice B multiplies 35 by 10/4, which uses people per cup instead of cups per person. Choice C is the rice for 30 people.
Question 3 of 20 · Multiple Choice
The points (2, 8), (4, 16) and (6, 24) come from a table of equivalent ratios. Which point also belongs on the graph?
Answer: A
In every pair, the second number is 4 times the first, so (8, 32) fits: 32 = 4 × 8. Choice B adds 2 to both numbers of (6, 24), which breaks the ratio. Choice C reverses the order: the pair (1, 4) belongs, but (4, 1) does not. Choice D adds 4 to the first number of (6, 24) and 8 to the second.
Question 4 of 20 · Multiple Choice
Juice A uses 2 cups of concentrate for every 5 cups of water. Juice B uses 3 cups of concentrate for every 7 cups of water. Which juice tastes stronger?
Answer: C
Make the water the same in both tables. Juice A: 2:5 = 14:35. Juice B: 3:7 = 15:35. With the same water, Juice B has more concentrate, so it is stronger. Choice A has the right numbers but reads them backward: with the same 6 cups of concentrate, Juice A needs more water, so Juice A is the weaker juice. Choice B compares differences (5 - 2 = 3 and 7 - 3 = 4) instead of ratios. Choice D is true for both juices, but it does not compare how much concentrate goes with the water.
Question 5 of 20 · Multiple Choice
Maria scored 12 goals in 16 shots. Dev scored 15 goals in 20 shots. Who is more accurate?
Answer: D
Both ratios simplify to 3 goals for every 4 shots: 12:16 = 3:4 and 15:20 = 3:4. A table shows it too: at 80 shots, both would score 60 goals. Choice A compares only the goals and ignores the shots. Choice B compares misses (4 and 5), which is an additive comparison, not a ratio.
Question 6 of 20 · Multiple Choice
An 8-ounce block of cheese costs $3.60. What is the price per ounce?
Answer: A
Price per ounce = price ÷ ounces = $3.60 ÷ 8 = $0.45. Choice B divides in the wrong order: 8 ÷ 3.60 ≈ 2.22 is ounces per dollar, not dollars per ounce. Choice C multiplies instead of dividing. Choice D is the price of 4 ounces.
Question 7 of 20 · Multiple Choice
Brand X sells 5 notebooks for $6.25. Brand Y sells 3 notebooks for $3.90. Which is the better buy?
Answer: B
Unit prices: $6.25 ÷ 5 = $1.25 per notebook and $3.90 ÷ 3 = $1.30 per notebook, so Brand X costs less for each notebook. Choice A compares total prices, but the packs have different numbers of notebooks. Choice D divides 5 by 6.25, which gives notebooks per dollar.
Question 8 of 20 · Multiple Choice
What is 30% of 70?
Answer: C
30% means 30/100, so 30% of 70 is (30/100) × 70 = 21. A tape diagram gives the same answer: 10% of 70 is 7, so 30% is 21. Choice A subtracts 30 from 70. Choice B multiplies 30 × 70 and forgets to divide by 100. Choice D divides 70 by 0.3.
Question 9 of 20 · Multiple Choice
Which expression finds 45% of 80?
Answer: A
A percent is a rate per 100, so 45% of 80 is 45/100 times 80, which equals 36. Choice B flips the fraction and gives more than 80, which cannot be 45% of 80. Choice C forgets the "per 100" and gives 3,600.
Question 10 of 20 · Multiple Choice
24 is 60% of what number?
Answer: D
60% is 24, so 10% is 24 ÷ 6 = 4, and 100% is 4 × 10 = 40. Check: (60/100) × 40 = 24. Choice A finds 60% of 24 instead of the whole. Choice B adds 24 and 60. Choice C multiplies 24 by 60.
Question 11 of 20 · Multiple Choice
Ray has paid $45 toward a skateboard. That is 30% of its price. What is the price of the skateboard?
Answer: B
30% is $45, so 10% is $15, and 100% is $150. Check: (30/100) × 150 = 45. Choice A finds 30% of $45, but $45 is the part, not the whole. Choice C adds 45 and 30. Choice D adds 30% of $45 to $45.
Question 12 of 20 · Multiple Choice
How many inches are in 7 feet? (1 foot = 12 inches)
Answer: C
Each foot is 12 inches, so 7 feet × 12 inches per foot = 84 inches. Choice A adds 7 and 12. Choice B divides 7 by 12, which would convert 7 inches into feet. Choice D uses 3 per foot, the number of feet in a yard.
Question 13 of 20 · Multiple Choice
Which calculation converts 4 pounds to ounces so that the units cancel? (1 pound = 16 ounces)
Answer: A
In choice A, pounds appear on the top and the bottom, so they cancel and ounces are left: 4 × 16 = 64 ounces. In choice B, the pounds do not cancel (the result would be pounds squared per ounce), and the answer is far too small. Choice C adds two different units. Choice D divides the conversion number by 4 instead of multiplying.
Question 14 of 20 · Multiple Choice
A drone flies 450 meters in 30 seconds at a constant speed. How far does it fly in 2 minutes?
Answer: C
The unit rate is 450 ÷ 30 = 15 meters per second. 2 minutes is 120 seconds, so the drone flies 15 × 120 = 1,800 meters. Choice A multiplies 15 by 2 and treats 2 minutes as 2 seconds. Choice B doubles 450, as if 2 minutes were 60 seconds. Choice D multiplies 450 by 30 instead of dividing.
Question 15 of 20 · Multiple Choice
A runner jogs 5 miles in 50 minutes at a constant speed. How many minutes does she take for each mile?
Answer: B
Minutes per mile = minutes ÷ miles = 50 ÷ 5 = 10 minutes per mile. Choice A divides 5 by 50: 0.1 is her speed in miles per minute, so the units are swapped. Choice C multiplies instead of dividing. Choice D subtracts 5 from 50.
Question 16 of 20 · Short Answer
A bracelet uses 2 blue beads for every 7 red beads. Copy and complete the table, then list the pairs (blue, red) you would plot on the coordinate plane. Blue: 2, 4, ?, 10. Red: 7, ?, 21, ?.
Multiply both numbers by the same factor. Blue 6 goes with 21 red (× 3), red 14 goes with 4 blue (× 2) and red 35 goes with 10 blue (× 5). The points to plot are (2, 7), (4, 14), (6, 21) and (10, 35). They lie on a straight line through (0, 0).
Question 17 of 20 · Short Answer
A 2-pound bag of grapes costs $5.00 and a 3-pound bag costs $6.90. Use unit prices to decide which bag is the better buy, and explain.
Unit prices: $5.00 ÷ 2 = $2.50 per pound and $6.90 ÷ 3 = $2.30 per pound. The 3-pound bag is the better buy because each pound costs $0.20 less. The total price alone does not decide it, because the bags hold different amounts.
Question 18 of 20 · Short Answer
It took 6 hours to paint 4 rooms. At that rate, how many rooms could be painted in 15 hours? At what rate were rooms being painted?
15 hours is 2.5 times 6 hours, so 4 × 2.5 = 10 rooms. A double number line works too: 3 hours for 2 rooms, so 15 hours for 10 rooms. The rate is 4/6 = 2/3 room per hour, or 6/4 = 1 1/2 hours per room.
Question 19 of 20 · Short Answer
In a survey, 42 students chose pizza as their favorite school lunch. That is 35% of the students surveyed. How many students were surveyed? Show a tape diagram or double number line.
35% is 42 students, so 5% is 42 ÷ 7 = 6 students, and 100% is 6 × 20 = 120 students. On a double number line, 0% to 100% lines up with 0 to 120, and 35% lines up with 42. Check: (35/100) × 120 = 42.
Question 20 of 20 · Short Answer
Sofia rides her bike at a constant speed of 15 miles per hour. How far does she ride in 40 minutes? Show how the units work.
15 miles per 60 minutes is 1 mile per 4 minutes, so in 40 minutes she rides 40 ÷ 4 = 10 miles. With units: 40 minutes × (1 mile / 4 minutes) = 10 miles, and minutes cancel. A common error is to multiply 15 × 40 = 600, which treats 40 minutes as 40 hours.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does 6.RP.A.3 mean?
6.RP.A.3 means students use ratio and rate reasoning to solve real problems. It has four parts: (a) ratio tables and their graphs, (b) unit rates, including unit prices and constant speed, (c) percents as a rate per 100, and (d) converting measurement units. Students can use tables, tape diagrams, double number lines or equations, whichever fits the problem.
Is 6.RP.A.3 taught in grade 6 or grade 7?
It is a grade 6 standard. In grade 7, students build on it with proportional relationships (7.RP.A.2) and multistep ratio and percent problems such as tax, tips and percent increase (7.RP.A.3). In grade 6, problems usually take one or two steps, and all numbers are positive.
What is the difference between a ratio, a rate and a unit rate?
A ratio compares two quantities, such as 4 dogs to 6 cats. A rate is a ratio of quantities with different units, such as 100 words in 2 minutes. A unit rate gives the amount for 1 unit of the second quantity: 50 words per minute. Unit rates make comparisons easy because both rates are measured against the same 1 unit.
What are tape diagrams and double number lines?
A tape diagram is a bar split into equal parts. It helps with ratio and percent problems: for a ratio of 1:4, draw 1 box and 4 boxes of the same size. A double number line is two number lines, one above the other, with matching marks lined up. It works well for rates, such as dollars and pounds, or miles and hours. Both are drawn to show equal groups, so the spaces must be equal.
How do you find the whole when you know a part and the percent?
Find the value of a small, easy percent first, then scale up to 100%. For example, if 16 is 20% of a number, 10% is 8, so 100% is 80. A tape diagram with ten 10% boxes shows this clearly. Students can also think of it as a ratio table with the rows "percent" and "amount", where 100% is the missing value.
What are common mistakes with ratio tables?
A common mistake is adding the same number to both quantities. For 2:3, adding 2 to each gives 4:5, which is not equivalent: the correct next row is 4:6. Another mistake is mixing up the order, for example plotting (raisins, pretzels) when the table lists pretzels first. Ask students to label every row and every axis.
How do students find the better buy?
They find the unit price of each option by dividing the price by the amount, then compare. A 12-ounce bottle for $1.80 costs $0.15 per ounce, and a 20-ounce bottle for $2.60 costs $0.13 per ounce, so the 20-ounce bottle is the better buy. The bigger package is often, but not always, the better buy, so students should always check.
How should students convert units with ratio reasoning?
They treat each conversion fact as a ratio, such as 1 yard : 3 feet. Then they use a ratio table or multiply by the conversion written as a fraction, placing the unit they want to remove on the bottom. For 2 yards: 2 yards × (3 feet / 1 yard) = 6 feet, and yards cancel. Writing the units every time is the part of standard d that says to "manipulate and transform units".
Do students need equations for 6.RP.A.3?
Equations are one of the tools the standard lists, but not the only one. In grade 6, simple equations such as 5 × ? = 35 or cost = 1.20 × pencils fit well, and they connect to 6.EE.C.9. Students should also be able to solve problems with tables, tape diagrams and double number lines, and explain how each tool matches their equation.
How can parents help with ratios and percents at home?
Everyday tasks give good practice. At the store, ask which size is the better buy and compare unit prices on the shelf labels. When cooking, double or halve a recipe. On a car trip, estimate how far you will travel in 2 hours at the current speed. With sales, ask what 20% off a price would save.
07
Related Standards
6 standards
These standards connect to 6.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.1Prerequisite
Understand ratios and use ratio language to describe two related quantities
Lesson coming soon
5.MD.A.1Prerequisite
Convert among different-sized units within one measurement system
Lesson coming soon
Alongside
6.RP.A.2Parallel
Understand a unit rate a/b for a ratio a:b and use rate language
Lesson coming soon
6.EE.C.9Parallel
Use variables, tables and graphs for two quantities that change together
Lesson coming soon
After this lesson
7.RP.A.2Next step
Recognize and represent proportional relationships between quantities
Lesson coming soon
7.RP.A.3Next step
Use proportional relationships to solve multistep ratio and percent problems
