If nothing enters or leaves then the total is your anchor
When things only move between two groups
Sometimes nothing is added and nothing is taken away: some of one share simply moves to the other. One friend gives marbles to another, or oil is poured from one container into another. Both shares change, but the total stays the same. The total is the anchor.
Suppose two amounts are in the ratio 3 : 5, which is 8 parts in total. After some of the second amount moves to the first, the ratio is 5 : 3. That is still 8 parts in total, so the two ratios already use parts of the same size: the first share gained 5 − 3 = 2 parts, and the second share lost the same 2 parts.
Before and after, the two bars together have the same length. Nothing entered and nothing left, so the total is still 8 parts.
Make the totals match first
When the two ratios have totals with different numbers of parts, rewrite them so the totals match. If the ratio was 5 : 7 before (12 parts) and 3 : 1 after (4 parts), multiply both numbers in the second ratio by 3: 3 : 1 = 9 : 3, which is also 12 parts. Now one part is the same amount in both ratios. The first share went from 5 parts to 9 parts, a gain of 4 parts, and the second share lost the same 4 parts. The real number in the question then tells you what those 4 parts are worth.
In the drawing below, the top bar is the ratio 2 : 5, which is 18 and 45, or 63 in total. The lower bar is the ratio 4 : 5, and you choose the size of its parts. Only one size keeps the total at 63, and that is the only case where the change could have been a move from one share to the other.
Make the total the quantity that did not move.
Drag the size of a part in the lower bar. At 9 the second share stays at 45 while the total grows; at 7 the total stays at 63 while 10 moves from the second share to the first.
Ratio and proportion · Marbles Passed from Daryl to Evan, with the Total Unchanged
QuestionDaryl and Evan had some marbles in the ratio 7 : 5. After Daryl gave 18 marbles to Evan, the ratio of Daryl's marbles to Evan's marbles became 1 : 2. (a) How many marbles did Daryl have at first? (b) How many marbles did Evan have after the gift?
1.Before, Daryl : Evan = 7 : 5, so the total is 7 + 5 = 12 units. After, Daryl : Evan = 1 : 2, which is 1 + 2 = 3 parts.
Before, Daryl : Evan = 7 : 5, which is 12 units. After, it is 1 : 2, only 3 parts, but the bar is the same length: no marbles are added or taken away. 2.No marbles are added or taken away, so the total bar is the same length before and after. Multiply the ratio after by 4 so that it also has 12 units: 1 : 2 = 4 : 8.
Multiply the ratio after by 4: 1 : 2 = 4 : 8, which is 12 units, like the bar before. 3.Daryl goes from 7 units to 4 units, so he gave away 7 − 4 = 3 units. He gave 18 marbles, so 1 unit is 18 ÷ 3 = 6 marbles.
Daryl goes from 7 units to 4 units. The 3 units he gave are 18 marbles, so 1 unit is 6. 4.(a) Daryl had 7 × 6 = 42 marbles at first.
(a) Daryl had 7 × 6 = 42 marbles at first. 5.(b) After the gift, Evan has 8 units, which is 8 × 6 = 48 marbles. Check: Daryl is left with 42 − 18 = 24 marbles, and 24 : 48 = 1 : 2.
(b) After the gift, Evan has 8 units: 8 × 6 = 48 marbles.
Answer: (a) 42 marbles; (b) 48 marbles
Watch out for
- Taking 7 − 1 = 6 units as Daryl's 18 marbles. The ratio before has 12 units and the ratio after has only 3 parts, so a part after is not the same size as a unit before. Make both totals 12 units first; then Daryl goes from 7 units to 4 units.
- Adding the 18 marbles to Daryl, or taking them from Evan. Daryl gave the marbles to Evan, so Daryl ends with 18 fewer and Evan with 18 more: 42 − 18 = 24 and 30 + 18 = 48.
Fractions · Oil Poured from One Container into the Other
QuestionContainer A and Container B hold 720 ml of oil altogether. At first, Container A holds 512 of the oil. Then 80 ml of oil is poured from Container B into Container A. (a) How much oil was in Container B at first? (b) What fraction of the oil is in Container B after the oil is poured?
1.Draw all the oil as a bar of 12 equal units. The 12 units are 720 ml, so 1 unit is 720 ÷ 12 = 60 ml.
All the oil as 12 equal units, because A's fraction is in twelfths: 720 ÷ 12 = 60 ml in each. 2.At first A holds 5 units, which is 5 × 60 = 300 ml.
At first A holds 5 units: 5 × 60 = 300 ml. 3.(a) B holds the other 12 − 5 = 7 units, which is 7 × 60 = 420 ml.
(a) B holds the other 7 units: 7 × 60 = 420 ml. 4.Move 80 ml from B's part of the bar to A's. The bar stays the same length, because the oil only moved: the total is still 720 ml, and B now holds 420 − 80 = 340 ml.
Pour 80 ml from B into A. The bar stays the same length: the total is still 720 ml, and B holds 420 − 80 = 340 ml. 5.(b) B holds 340720 of the oil, and dividing both numbers by 20 gives 1736. Check: A now holds 300 + 80 = 380 ml, and 380 + 340 = 720 ml.
(b) B holds 340720 = 1736 of the oil.
Answer: (a) 420 ml; (b) 1736 of the oil
Watch out for
- Giving 300 ml as the oil in Container B. 512 of the oil is in Container A; B holds the other 712, which is 420 ml.
- Writing B's share as 340640, as if the 80 ml had left the containers. The oil was poured from B into A, so the total is still 720 ml, and B holds 340720 of it.
Your turn
1.3 : 5 becomes 4 : 4 with nothing added. How many parts in total?