The share that does not change
In many ratio problems, one quantity stays exactly as it was while another changes. A librarian adds books to a shelf, but the fiction books are not touched. Mushrooms dry in the sun and lose water, but the solid part of each mushroom stays. The quantity that does not change is the same real amount before and after, and that makes it the anchor for the whole problem.
Suppose the ratio of Ann's stickers to Ben's is 4 : 3, and Ann still has the 12 stickers she started with. Ann's share is the anchor: we know both its number of parts, 4, and its real amount, 12.
Ann's 12 stickers fill her 4 parts. Every part, hers and Ben's, is the same amount.
Find one part, then count parts
Ann's 12 stickers fill her 4 parts, so one part is worth 12 ÷ 4 = 3 stickers. Every share in the ratio is made of parts of this same size, so any other share is just a number of these parts. Ben has 3 parts, so Ben has 3 × 3 = 9 stickers.
This works only when the unchanged share has the same number of parts in the ratio before and in the ratio after. In a before-and-after problem the two ratios usually give it different numbers, and then one part is a different amount in each ratio. So first rewrite both ratios until the unchanged share has the same number of parts in both, as in Rewriting a Ratio to Match a Share. Then the change in the other share can be counted in parts, and the one real number in the question tells you how much a part is worth.
12 ÷ 4 = 3 in each of Ann's parts, so each of Ben's 3 parts is 3 as well: 3 × 3 = 9.
Worked example: Constant One Part (Single Unchanged Quantity)
Question The ratio of the number of fiction books to non-fiction books on a shelf was 5 : 3. After the librarian added 54 non-fiction books to the shelf and no fiction books were added or removed, the ratio of fiction books to non-fiction books became 2 : 3. How many fiction books were on the shelf?
1.Before: Fiction has 5 units, Non-fiction has 3 units.
Before: fiction : non-fiction = 5:3. 2.After: Fiction has 2 units, Non-fiction has 3 units.
After: 2:3. The fiction bar has the same length in both rows because no fiction book moved; its units are just cut differently. 3.Equalize Fiction bars by subdividing: 5 units × 2 = 10 units; 2 units × 5 = 10 units.
Give fiction one cut: 5 × 2 = 10 and 2 × 5 = 10 units. 4.Adjust Non-fiction bars accordingly: Before = 3 × 2 = 6 units; After = 3 × 5 = 15 units.
Non-fiction follows the same cuts: 3 × 2 = 6 before, 3 × 5 = 15 after. 5.Change in Non-fiction = 15 − 6 = 9 units = 54.
Non-fiction grew by 15 − 6 = 9 units, and 54 books were added. 6.1 unit = 54 ÷ 9 = 6.
u = 54 ÷ 9 = 6. 7.Fiction books = 10 × 6 = 60.
Fiction = 10u = 60 books.
Answer: 60 fiction books
Common mistakes
- Assuming non-fiction units did not change because both ratios show the digit '3'.
- Adding 54 to the total of the initial ratio units (5 + 3 = 8) without equalizing the unchanged component.
The same idea in percentages
The anchor does not have to be a person's share, and the question does not have to be written as a ratio. In the next problem, mushrooms lose water as they dry, but the solid part of the mushrooms, the pulp, does not change. The pulp is the anchor, even though the question gives percentages.
Worked example: Percentage with Single Unchanged Quantity (Moisture Loss)
Question A crate of fresh mushrooms weighed 40 kg. Water made up 90% of the total mass of the fresh mushrooms. After being dried under the sun, water made up only 60% of the mass of the dried mushrooms. (a) What was the mass of the mushrooms after drying? (b) What mass of water was evaporated during the drying process?
1.Fresh state: Water : Pulp = 90% : 10% = 9 : 1. Total = 10 units = 40 kg.
Fresh: water 9 units, pulp 1, total 40 kg. 2.Value of 1 pulp unit = 40 ÷ 10 = 4 kg.
1 unit is 4 kg, so the pulp is 4 kg, and drying does not change it. 3.Dried state: Water : Pulp = 60% : 40% = 3 : 2.
Dried: water 3 : pulp 2. 4.Since pulp is 4 kg, 2 pulp units = 4 kg ⟹ 1 unit = 2 kg.
The pulp is still 4 kg, now 2 units: 1 unit is 2 kg. 5.Total dried units = 3 (water) + 2 (pulp) = 5 units.
(a) 5 units of 2 kg: 10 kg. 6.(a) Total dried mass: 5 × 2 = 10 kg.
(a) 5 units of 2 kg: 10 kg. 7.(b) Evaporated water: 40 − 10 = 30 kg.
(b) Water lost: 40 − 10 = 30 kg.
Answer: (a) 10 kg; (b) 30 kg
Common mistakes
- Assuming the mass drops by 30% because water drops from 90% to 60%, yielding 40 × 0.70 = 28 kg (ignoring base change).
- Calculating 60% of the initial 40 kg (24 kg) as the dried mass.