The Constant-Total Method

When something moves from one person to another, the two amounts change but their combined total stays constant. This constant total method keeps the whole visible. Think of a fixed tray of counters: moving counters between two boxes changes the split, never the whole tray.

Parents: ask your child to write the total before and after the transfer. If the transfer stays inside the two people, both totals must match.

By the end, you will be able to spot an internal transfer, find one unit from the change, and check the total before and after.

Keep the whole fixed

An internal transfer has opposite changes: one amount loses what the other gains. A gift from outside, a purchase, or a loss does change the combined total, so use a BCA table to record it first.

Equalise by moving one unit

Priya and Sam have money in the ratio 3 : 5. Sam gives Priya $12, and they then have equal amounts. Find their starting amounts.

After the transfer the eight units are split equally into four units and four units. Priya gains one unit, so one unit is twelve dollars. The original bars are three units and five units.

🔬

Move the transfer

🤔 Predict first: After an internal transfer, what must happen to the combined total?

It stays fixedIt doublesIt becomes the transferred amount

Find one unit

$$3+5=8\text{ units}$$

$$\text{After equal: }8\div2=4\text{ units each}$$

$$4-3=1\text{ unit}=\$12$$

$$\text{Priya before}=3\times12=\$36$$

$$\text{Sam before}=5\times12=\$60$$

$$\text{Priya after}=36+12=48$$

$$\text{Sam after}=60-12=48$$

The total check is twelve dollars transferred within the pair: the starting total and final total are both ninety-six dollars.

Do not confuse constant total and constant part

Constant total describes a transfer between two quantities. Constant part describes a piece that stays unchanged while another piece changes. If the question says a fixed number is removed from each amount, compare the unchanged piece; if it says one amount is given to the other, keep the total fixed.

Practice: transfer inside the pair

1. Red and blue counters are in the ratio 2 : 6. Twelve blue counters move to red, and the amounts become equal. Find the starting amounts and the final amount in each group.

Working

$$\text{After equal: }(2+6)\div2=4\text{ units each}$$

$$4-2=2\text{ units}=12\text{ counters}$$

$$1\text{ unit}=12\div2=6\text{ counters}$$

$$\text{Red start}=2\times6=12$$

$$\text{Blue start}=6\times6=36$$

$$\text{Red after}=12+12=24$$

$$\text{Blue after}=36-12=24$$

2. Mo and Lin have money in the ratio 1 : 3. Mo receives $18 from Lin and they become equal. Find their starting amounts.

Working

$$\text{After equal: }(1+3)\div2=2\text{ units each}$$

$$2-1=1\text{ unit}=\$18$$

$$\text{Mo before}=1\times18=\$18$$

$$\text{Lin before}=3\times18=\$54$$

$$\text{Mo after}=18+18=36$$

$$\text{Lin after}=54-18=36$$

3. Two sticker collections are in the ratio 3 : 7. Nine stickers move from the larger collection to the smaller. The new ratio is 4 : 6. Find the original piles.

Working

$$\text{Total}=3+7=10\text{ units}$$

$$\text{Moved}=9=1\text{ unit}$$

$$\text{After smaller}=3+1=4\text{ units}$$

$$\text{After larger}=7-1=6\text{ units}$$

$$\text{One unit}=9\text{ stickers}$$

$$\text{Original smaller}=3\times9=27$$

$$\text{Original larger}=7\times9=63$$

$$\text{After amounts}=36\text{ and }54$$

Watch out — easily mixed up

Quick recap

🎯 Mastery check

Answer all 6 — your progress is saved on this device.

1. Sam gives money to Priya from his own amount. What happens to their combined total?

   It stays fixedIt increases by the giftIt becomes zero

2. A ratio is 3 : 5 and an internal transfer makes it 4 : 4. How many units move into the smaller part?

   One unitThree unitsFour units

3. If one transferred unit is worth $12, what is a starting part of three units worth?

   $12$36$48

4. Which change can make the combined total change?

   A gift from outsideA transfer between the pairA swap inside the two bars

5. What should the before and after totals do in a constant-total problem?

   MatchDoubleBe unrelated

6. What is the safest first step when a transfer is described?

   Draw the two starting parts and total unitsAdd the transfer to the wholeIgnore the recipient

See my score

Ready to keep the whole steady? Follow the transfer from one part to the other and let the unchanged total anchor your model.