Ratios & proportions
Scale ratios, split amounts, and compare unit prices.
In your head, jot if needed · no calculator why?
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the idea
A ratio like 3 : 4 compares two amounts: for every 3 of one, there are 4 of the other. Equal ratios come from multiplying both terms by the same number, never adding: 3 : 4 is the same as 6 : 8 and 21 : 28.
Prices and rates work the same way. Compare packs by the price per unit, and scale a rate from the amount for one unit.
techniques
Find the scale factor
- Find the pair of matching terms that are both known, like 4 and 28.
- Divide the bigger by the smaller to get the scale factor: 28 ÷ 4 = 7.
- Scale the other term by the same factor: 3 × 7 = 21.
- If the missing term sits with the bigger numbers, multiply. With the smaller numbers, divide.
worked example
5 : 8 = x : 56. What is x?
- 56 ÷ 8 = 7, so the scale factor is 7.
- x = 5 × 7 = 35.
Answer: 35
Count the parts
- Add the ratio numbers to get the total number of parts.
- Divide the amount by the total parts to get one part.
- Multiply one part by the share you want.
worked example
A bakery splits a $360 profit between two owners in the ratio 5 : 3. What is the smaller share?
- 5 + 3 = 8 parts.
- One part: $360 ÷ 8 = $45.
- The smaller share: 3 × $45 = $135.
Answer: $135
Price or amount per one
- Divide to get the amount for one unit: dollars per ounce, pages per minute.
- Packs: the lowest price per unit wins, whatever the size.
- Rates: multiply the amount per unit by the new time.
worked example
Coffee comes in two sizes: 12 oz for $3.60 and 20 oz for $5.40. What is the lower price per ounce?
- 12 oz: $3.60 ÷ 12 = $0.30 an ounce.
- 20 oz: $5.40 ÷ 20 = $0.27 an ounce.
- The 20 oz pack is the better value.
Answer: $0.27
watch out for
- Adding the same amount to both terms. Equal ratios multiply: 3 : 4 scales to 21 : 28, not 27 : 28.
- Dividing by the larger ratio number instead of the total parts. For 2 : 3, one part is the amount divided by 5.
- Stopping at the rate per minute when the question asks about a longer time.
practice
Missing term
worked example
1 : 2 = 6 : x. What is x?
Answer: 12
- 1 × 6 = 6, so scale the 2 by 6 as well: x = 2 × 6 = 12.
Best buy
worked example
Almonds come in four sizes. Which is the best value per ounce?
- 16 oz for $6.72
- 8 oz for $3.68
- 24 oz for $9.60
- 48 oz for $21.60
Answer: 24 oz for $9.60
- Price per ounce: 8 oz → $0.46, 16 oz → $0.42, 24 oz → $0.40, 48 oz → $0.45.
- So 24 oz for $9.60 is the best value.
Scale a rate
worked example
A machine fills 380 bottles in 5 hours. At the same rate, how many bottles does it fill in 10 hours?
Answer: 760
- 380 ÷ 5 = 76 bottles per hour, and 76 × 10 = 760.