A ratio counts parts, not amounts
A ratio tells you how something is shared, not how much each share is. If Ann and Ben share some money in the ratio 2 : 1, Ann has 2 parts and Ben has 1 part, so there are 2 + 1 = 3 equal parts altogether. The ratio does not yet tell you how much one part is worth.
The same bar, counted in two ways. Cutting every part in two doubles both numbers in the ratio, but no one's share changes.
Cutting every part in two
Now cut every part in two. Ann's share is the same length of bar as before, but it is now counted as 4 smaller parts, and Ben's share is counted as 2. So the ratio 2 : 1 can also be written as 4 : 2, or 6 : 3, and so on. Multiplying both numbers in a ratio by the same number renames the ratio without moving anything from one share to the other.
This matters when two ratios describe the same situation at two different times. Before you can compare them, a share that did not change must be written with the same number of parts in both ratios. Rewrite both ratios so that this share has the same number of parts. The lowest common multiple of its two numbers is usually the easiest number to choose.
3 × 1 = 3 but 2 × 1 = 2: the first share is a different number of parts in each ratio, so the parts are different sizes and cannot be counted against each other
Multiply both ratios until the first share has the same number of parts in each
Blue to white is 3 : 4 before and 2 : 5 after. No blue paint is added, so the blue bar is the same length in both; only the number of parts it is cut into changes.
Worked example: Paint Mixed Too Dark: White Added Until the Shade Is Lighter
Question A painter mixes 14 liters of paint from blue paint and white paint in the ratio 3 : 4. The shade is too dark, so she adds white paint only, until the ratio of blue paint to white paint is 2 : 5. (a) How many liters of white paint does she add? (b) White paint is sold only in 2-liter cans, at $9.50 a can. White paint is sold only in whole cans. How much does she pay for the cans of white paint she needs?
1.No blue paint is added, so the blue share is the same before and after. Before, blue : white = 3 : 4; after, blue : white = 2 : 5.
Before, blue : white = 3 : 4; after, 2 : 5. No blue is added, so the blue bar is the same length in both. 2.Rewrite both ratios so the blue share matches. The least common multiple of 3 and 2 is 6: 3 : 4 = 6 : 8 and 2 : 5 = 6 : 15.
Make the blue shares match: 3 : 4 = 6 : 8 and 2 : 5 = 6 : 15. Now one unit is the same paint in both bars. 3.Before, the 14 liters are 6 + 8 = 14 units, so one unit is 14 ÷ 14 = 1 liter.
Before, 14 units hold 14 liters, so one unit is 1 liter. 4.(a) The white goes from 8 units to 15 units, so she adds 15 − 8 = 7 units, which is 7 liters.
(a) The white grows from 8 units to 15: 7 units, which is 7 liters added. 5.(b) 3 cans hold only 6 liters, so she buys 4 cans and pays 4 × $9.50 = $38.00. Check: blue 6 liters, white 15 liters, and 6 : 15 = 2 : 5.
(b) 7 liters needs 4 cans: 4 × $9.50 = $38.00.
Answer: (a) 7 liters; (b) $38.00
Common mistakes
- Adding 5 − 4 = 1 unit of white, 2 liters. The blue share is 3 units before and 2 units after, so a unit is a different amount of paint in the two ratios; they must be rewritten so the blue shares match before the white shares are compared.
- Paying for 3.5 cans, or for 3 cans. Paint is sold only in whole cans, and 3 cans hold 6 liters, 1 liter short of the 7 liters she needs.