Why it matters on the HSPT
Ratio and rate questions are the word problems most students get wrong on the Mathematics section, not because the math is hard but because the setup is. Speed, price per item, map scales, recipes, and mixing paint all use the same three tools: write the ratio, find the unit rate, or set up a proportion. Once the setup is right, the arithmetic is a single multiplication or division.
The rules
Writing a ratio. A ratio compares two numbers and can be written three ways: 3 to 2, 3:2, or 3/2. The order matters. If a class has 12 girls and 8 boys, the ratio of girls to boys is 12 to 8, which simplifies to 3 to 2, and the ratio of boys to girls is 2 to 3. Simplify ratios exactly as you simplify fractions.
Part to part and part to whole. The same class gives two kinds of ratio. Girls to boys, 3 to 2, is part to part. Girls to students, 12 to 20 or 3 to 5, is part to whole. Read the question carefully for which one it wants; the test builds wrong answers from the other one.
Rates and unit rates. A rate compares different units: 150 miles in 3 hours. A unit rate has 1 on the bottom: 50 miles per hour. Divide to get it. Unit rates make comparisons easy: a 12-ounce box for 3 dollars is 25 cents an ounce, and a 20-ounce box for 4 dollars is 20 cents an ounce, so the bigger box is the better deal.
Distance, rate, and time. Distance equals rate times time. Any one of the three comes from the other two: time is distance divided by rate, and rate is distance divided by time. Ninety miles at 30 miles per hour takes 3 hours.
Proportions. A proportion says two ratios are equal. Set it up with the same kind of quantity in the same position in both fractions, then cross-multiply. If 4 tickets cost 30 dollars, then 10 tickets cost x: 4 over 30 equals 10 over x, so 4x equals 300 and x is 75. Some proportions can be scaled without cross-multiplying: if 2 cups of flour make 12 muffins, then 6 cups make 36, because everything is tripled.
Splitting in a ratio. To divide a total in a ratio, add the ratio's parts to get the number of shares, find the size of one share, and multiply out. Sixty dollars split 1 to 2 between two people is 3 shares of 20, so 20 and 40.
Scale. A map scale of 1 inch to 25 miles is a ratio. Three and a half inches on the map is 3.5 times 25, which is 87.5 miles. Scale drawings of shapes work the same way for lengths, but areas scale by the square: doubling every side makes the area four times as big.
Worked examples
Example 1
A bag holds 15 red marbles and 25 blue marbles. What is the ratio of red marbles to all marbles?
- A.3:5
- B.3:8
- C.5:8
- D.5:3
Show answer
Answer: B: 3:8
All marbles means 15 plus 25, which is 40. Red to all is 15 to 40, which simplifies by 5 to 3 to 8. Choice A, 3 to 5, is red to blue, the part-to-part ratio, and it is the trap.
Example 2
A car travels 210 miles in 3.5 hours. What is its average speed?
- A.50 mph
- B.60 mph
- C.70 mph
- D.73.5 mph
Show answer
Answer: B: 60 mph
Rate is distance divided by time: 210 divided by 3.5. Doubling both gives 420 divided by 7, which is 60 miles per hour. Choice C, 70, comes from dividing by 3 instead of 3.5.
Example 3
If 6 notebooks cost $9, how much do 10 notebooks cost?
- A.$13
- B.$15
- C.$16
- D.$18
Show answer
Answer: B: $15
Find the unit rate first: 9 dollars for 6 notebooks is 1.50 each. Ten notebooks cost 15 dollars. The same thing as a proportion: 6 over 9 equals 10 over x, so 6x is 90 and x is 15.
Example 4
A recipe uses 3 eggs for every 2 cups of flour. How many eggs are needed for 8 cups of flour?
- A.6
- B.9
- C.12
- D.16
Show answer
Answer: C: 12
Eight cups is 4 times 2 cups, so multiply the eggs by 4 as well: 3 times 4 is 12. A proportion gives the same: 3 over 2 equals x over 8, so 2x is 24 and x is 12. Choice D, 16, scales the wrong number.
Example 5
Two friends split $84 in the ratio 3 to 4. How much does the friend with the larger share get?
- A.$36
- B.$42
- C.$48
- D.$63
Show answer
Answer: C: $48
Three plus 4 is 7 shares. Eighty-four divided by 7 is 12 dollars per share. The larger share is 4 times 12, which is 48 dollars. Choice A, 36, is the smaller share; choice B, 42, is an even split and ignores the ratio.
Example 6
On a map, 2 inches represent 50 miles. Two towns are 7 inches apart on the map. How far apart are they?
- A.140 miles
- B.175 miles
- C.200 miles
- D.350 miles
Show answer
Answer: B: 175 miles
Two inches is 50 miles, so 1 inch is 25 miles. Seven inches is 7 times 25, which is 175 miles. Choice D, 350, uses 50 miles per inch, forgetting that 50 miles goes with 2 inches.
Example 7
Which is the better buy: 8 ounces of cereal for $2.40 or 12 ounces for $3.00?
- A.8 ounces for $2.40
- B.12 ounces for $3.00
- C.They cost the same per ounce
- D.It cannot be determined
Show answer
Answer: B: 12 ounces for $3.00
Find the price per ounce. Two dollars and forty cents divided by 8 is 30 cents an ounce. Three dollars divided by 12 is 25 cents an ounce. The 12-ounce box is cheaper per ounce, so it is the better buy even though it costs more in total.
Common mistakes
- Reversing the order. Girls to boys is not boys to girls. Write the ratio in the order the question says.
- Mixing part-to-part with part-to-whole. If the question says out of all, the whole is the total of all parts.
- Setting up a proportion with quantities crossed. Both fractions must have the same kind of thing on top.
- Forgetting that the ratio's parts are shares, not amounts. A 3 to 4 split of 84 dollars is not 3 dollars and 4 dollars.
- Judging a better buy by total price. Compare per unit.
Practice
Ratios and rates live mostly in word problems, with unit prices and simple scaling in arithmetic. The fractions lesson covers the simplifying this page uses throughout.