Set up and solve proportions, work with unit rates, and convert units without losing track of what each number counts.
You compare quantities and scale them. Questions give you a ratio, a rate, or a scale and ask for a missing amount. Unit conversion appears often, sometimes chained across two or three units, and sometimes mixing time units with distance or volume units. Other questions give a part-to-part ratio and ask for a part-to-whole amount, or give a recipe or map scale and ask what a change in one quantity does to another. Density, speed, and price per unit are common wrappers.
Problem-Solving and Data Analysis is about 15 percent of the 44 math questions, and ratios and rates usually take up around 2 to 3 of them.
Written for this skill, with the answer explained. 86 more are in the app.
A car travels 60 miles per gallon on average. At this rate, how many miles does the car travel in 4 gallons?
In a class, the ratio of boys to girls is 1:4. If there are 25 students in total, how many are boys?
Why: A 1:4 ratio splits the students into 1 + 4 = 5 equal parts. With 25 students, each part is 25/5 = 5. Boys are 1 part, so 5 students. 20 is 4(5), the number of girls. 6 is close to 25/4 = 6.25, which divides by the 4 girls' parts instead of the 5 total parts. 12 is about half of 25, the split for a 1:1 ratio.
A recipe uses 2 cups flour for every 3 cups sugar. For 12 cups sugar, how much flour?
Why: Set up the proportion flour/sugar = 2/3 = x/12. The 12 cups of sugar is 4 times the 3 in the ratio, so scale the flour by 4 as well. That gives x = 2(4) = 8 cups. 6 cups is 12/2, which divides the sugar by the 2 in the ratio instead of scaling both parts. 9 cups comes from subtracting the 3 in the ratio from the 12 rather than scaling. 18 cups is (3/2)(12), which flips the ratio and makes flour the larger amount, but this recipe uses less flour than sugar.
A lab solution contains acid and water in a ratio of 4 to 6. If 10 more units of water are added and no acid is added or removed, what is the new ratio of acid to water?
Why: The acid stays at 4 units and the water becomes 6 + 10 = 16, so the new ratio is 4 to 16, which reduces to 1 to 4. "2 to 3" is the ORIGINAL ratio 4 to 6 reduced, which ignores the added water. "4 to 6" is the original ratio unchanged. "14 to 6" adds the 10 units to the acid instead of the water.
Put the same type of quantity in the same position on both sides, for example miles on top and hours on the bottom in both fractions. Then cross multiply and solve. If the units line up, the setup is right.
A part-to-part ratio compares two groups to each other, like 3 red to 5 blue. A part-to-whole ratio compares one group to the total, which here would be 3 out of 8. Read carefully which one the question gives and which one it wants.
Multiply by conversion fractions arranged so the unit you want to remove cancels diagonally. Chain as many fractions as you need in one line, then multiply the tops and divide by the bottoms once at the end.
Questions on ratios, rates, and proportions alone, with explanations on every one, and the app tracks which ones you keep missing.
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