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.
Official wording from the Common Core State Standards for Mathematics (© 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers). View on thecorestandards.org
Once students understand ratios and unit rates, they put them to work. This is the problem-solving heart of sixth grade proportional reasoning, and it has four parts. Students build tables of equivalent ratios, find missing values, and plot the pairs as points on a coordinate grid, which line up through the origin. They solve unit rate problems about pricing and constant speed. They treat a percent as a rate per 100, so 30% of a number means 30/100 of it, and they work backwards to find a whole from a part and a percent. Finally, they use ratio reasoning to convert units, such as inches to feet.
The standard deliberately names several tools: ratio tables, tape diagrams, double number lines and equations. Students are expected to choose a sensible tool rather than apply one rule mechanically. A strong student can solve 'if 7 hours mows 4 lawns, how many lawns in 35 hours?' by scaling a table (35 is 5 times 7, so 20 lawns) and can also explain the same answer through a unit rate.
Extending 2:5 to 4:7 by adding 2 to each number breaks the ratio. Rows of a ratio table must be multiples of the same pair, such as 4:10 and 6:15.
In '12 is 30% of what number?' students often answer 30% of 12. Draw a tape diagram where 12 fills 30 out of 100 parts, then scale up to 100.
Changing 48 inches to feet by multiplying by 12 gives a huge number. Ask whether the answer should be more or fewer units, since feet are bigger than inches.
150% of a number is one and a half times that number. Percent is a rate per 100, so it can be more than 100.
A class has raised $45 for a field trip, which is 30% of its goal. What is the goal?
Answer: The goal is $150.
Let students meet the same problem in several representations side by side, for example a ratio table and a double number line for the lawn-mowing context. Asking 'which tool made this easiest?' builds flexibility. Percent problems become far less mysterious when students sketch a 100-part tape or a 0 to 100 percent number line.
Expect multi-step tasks on assessments: a unit price followed by a comparison, or a unit conversion inside a speed problem. Graphing a ratio table also links directly forward to proportional relationships in seventh grade.
Original questions written for this standard. Choose an option or type your answer, then press Check. Every question has a worked explanation.
Answer: 15 (also accepted: 15 cups)
10 cakes is 5 times 2 cakes, so multiply the flour by 5: 3 × 5 = 15 cups.
Answer: B) 20
35 hours is 5 times 7 hours, so 5 times as many lawns: 4 × 5 = 20 lawns.
Answer: 34
40% means 40/100. 40/100 × 85 = 34. Or: 10% of 85 is 8.5, and 4 × 8.5 = 34.
Answer: D) 72
25% is one quarter of the whole, so the whole is 4 times 18, which is 72. Check: 25% of 72 is 18.
Answer: 66 (also accepted: 66 inches)
Each foot is 12 inches, so 5.5 × 12 = 66 inches.
Answer: B) 12 pencils for $9
Multiply both quantities by 3: 4 × 3 = 12 pencils and 3 × 3 = $9. The other pairs do not scale both numbers by the same factor.
Answer: 200 (also accepted: 200 miles)
Distance = rate × time = 80 × 2.5 = 200 miles.
A full lesson with slides, activities and an exit ticket on solving ratio and rate problems, pitched to grade 6 and editable in PowerPoint or Google Slides.
Make a lesson →A printable, differentiated worksheet on 6.RP.A.3 with an answer key, ready in about a minute.
Make a worksheet →Turn solving ratio and rate problems into a quiz students answer online that marks itself, with a class summary for you.
Build a test →Tables of equivalent ratios, tape diagrams, double number line diagrams and equations. Students should be able to choose one that fits the problem.
Part c treats a percent as a rate per 100. Students find a percent of a quantity and find the whole when they know a part and its percent.
Yes. Part a asks students to plot pairs from a ratio table on the coordinate plane. The points line up through the origin, which prepares them for proportional relationships in grade 7.