If one share never changes you can anchor the whole problem to it
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.
Ratio and proportion · Non-Fiction Books Added to a Shelf, with the Fiction Unchanged
QuestionThe ratio of the number of fiction books to the number of non-fiction books on a shelf was 5 : 3. The librarian then added 54 non-fiction books and no fiction books, and the ratio became 2 : 3. (a) How many fiction books were on the shelf? (b) How many non-fiction books were on the shelf after the 54 were added?
1.Before, fiction : non-fiction = 5 : 3. After, fiction : non-fiction = 2 : 3. The fiction books do not change, so the fiction bar is the same length in both.
Before, fiction : non-fiction = 5 : 3; after, 2 : 3. No fiction book is added, so the fiction bar is the same length in both, cut into 5 parts and into 2. 2.The least common multiple of 5 and 2 is 10, so make the fiction 10 units in both. Multiply the first ratio by 2: 5 : 3 = 10 : 6. Multiply the second ratio by 5: 2 : 3 = 10 : 15.
Make the fiction 10 units in both: 5 : 3 = 10 : 6 and 2 : 3 = 10 : 15. Now one unit is the same number of books in both bars. 3.The non-fiction grows from 6 units to 15 units, so the 54 books added are 15 − 6 = 9 units. 1 unit is 54 ÷ 9 = 6 books.
The non-fiction grows from 6 units to 15 units. The 9 units are the 54 books, so 1 unit is 54 ÷ 9 = 6 books. 4.(a) The fiction is 10 units, which is 10 × 6 = 60 books.
(a) The fiction is 10 × 6 = 60 books. 5.(b) After the 54 are added, the non-fiction is 15 units, which is 15 × 6 = 90 books. Check: before, there were 6 × 6 = 36 non-fiction books, and 60 : 36 = 5 : 3; after, 60 : 90 = 2 : 3.
(b) After the 54 are added, the non-fiction is 15 × 6 = 90 books.
Answer: (a) 60 fiction books; (b) 90 non-fiction books
Watch out for
- Thinking the non-fiction did not change because it is 3 in both ratios. The fiction is 5 units before and 2 parts after, so a unit is a different number of books in each ratio. Once the fiction is matched, the non-fiction goes from 6 units to 15 units.
- Taking the fiction from 5 down to 2 and calling the 3 lost units the 54 books. No fiction books were taken away; the 54 books were added to the non-fiction. Match the fiction first, and then the growth in the non-fiction, 9 units, is the 54 books.
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.
Percentages · Mushrooms Dried in the Sun: Water Is Lost and the Pulp Stays
QuestionA crate of fresh mushrooms has a mass of 40 kg, and 90% of the mass is water. The mushrooms are dried in the sun until water makes up 60% of their mass. Only water is lost. (a) What is the mass of the dried mushrooms? (b) How much water was lost?
1.Only water is lost, so the pulp, the part that is not water, has the same mass before and after drying. Fresh, the pulp is 100% − 90% = 10% of the mass: 10100 × 40 = 4 kg.
Only water is lost, so the pulp stays the same. Fresh, the pulp is 10% of 40 kg: 4 kg. 2.Dried, the water is 60% and the pulp is 40%. Draw the dried mushrooms as 5 equal units: water is 60% = 35, 3 units, and pulp is 40% = 25, 2 units.
Dried, the pulp is 40% = 25 of the mass: 2 of 5 units. The water is the other 3. 3.The pulp is still 4 kg, so 2 units are 4 kg, and 1 unit is 4 ÷ 2 = 2 kg.
The pulp is still 4 kg, so 2 units are 4 kg and 1 unit is 2 kg. 4.(a) The dried mushrooms are 5 units: 5 × 2 = 10 kg.
(a) The dried mushrooms are 5 units: 5 × 2 = 10 kg. 5.(b) Only water was lost, so the water lost is 40 − 10 = 30 kg. Check: the fresh mushrooms held 0.9 × 40 = 36 kg of water and the dried ones 3 × 2 = 6 kg, and 36 − 6 = 30.
(b) The water lost is 40 − 10 = 30 kg: the fresh water, 36 kg, less the 6 kg still in the dried mushrooms.
Answer: (a) 10 kg; (b) 30 kg
Watch out for
- Taking 90% − 60% = 30% of the 40 kg away, to get 28 kg. The 60% is a percentage of the dried mass, which is less than 40 kg, so the two percentages are of different wholes and cannot be subtracted.
- Taking 60% of 40 kg, 24 kg, as the dried mass. The 60% is the share of water in the dried mushrooms, not the share of the fresh mass that is left. The part that stays the same is the pulp, 4 kg, and it is 40% of the dried mass.
Your turn
1.6 : 3, and the first share is 24. What is the second share?