The Before-and-After Model (BCA Table)

Some word problems hide the beginning behind a transfer, a gift, a loss, or a change. The Before-Change-After (BCA) table keeps the story in order. A change records what each person gains or loses. Write what each person had before, what changed, and what remained after. Then connect the final rows with the clue in the question.

Parents: ask your child to write the signs in the Change row before solving. A minus for the giver and a plus for the receiver makes the story visible.

By the end, you will be able to track a transfer, solve for one unit, and check the Before + Change = After relationship in every row.

Write the three stages

The three columns are Before, Change, and After. A transfer from Asha to Ben is negative for Asha and positive for Ben. The same amount moves, so the signed changes have equal size and opposite signs.

Use the after clue

Asha had $24 more than Ben. She gave $\frac{1}{4}$ of her money to Ben. They then had equal amounts. How much did each have before the transfer?

$\frac{1}{4}$ of Asha’s money is one unit. After giving it away, Asha has three units. Ben receives one unit and therefore has three units after the transfer. His before amount is two units. The original difference gives the value of one unit.

🔬

Replay the transfer

🤔 Predict first: If Asha gives money to Ben, what happens to their total?

It stays the sameIt doublesIt becomes zero

Build the BCA solution

$$\text{Asha before}=4u$$

$$\text{Ben before}=2u$$

$$\text{Transfer}=\frac{1}{4}\times4u=u$$

$$\text{Asha after}=4u-u=3u$$

$$\text{Ben after}=2u+u=3u$$

$$4u - 2u = 24$$

$$2u=24$$

$$u=12$$

$$\text{Asha before}=4\times12=48$$

$$\text{Ben before}=2\times12=24$$

$$\text{Asha after}=48-12=36$$

$$\text{Ben after}=24+12=36$$

The starting total is seventy-two and the after total is still seventy-two. The transfer changes who has the money; it does not change the total.

BCA also handles outside changes

BCA is broader than a transfer. If Kai gives Lina $6 from his own money, Kai’s Change is negative six and Lina’s Change is positive six. If a person loses $14 outside the comparison, write negative fourteen in that row. Only call the total constant when the change stays inside the two rows.

Practice: complete the table

1. Kai has $30 and gives $6 to Lina. They then have the same amount. How much did Lina have before?

Working

$$\text{Lina after}=30-6=24$$

$$\text{Lina before}=24-6=18$$

$$\text{Check: Kai after}=30-6=24$$

2. Mina has $18 more than Yusuf. She gives $\frac{1}{5}$ of her money to Yusuf, and they become equal. Find their starting amounts.

Working

$$\text{Mina}=5u$$

$$\text{Yusuf}=3u$$

$$\text{Mina after}=4u$$

$$\text{Yusuf after}=4u$$

$$5u-3u=18$$

$$2u=18$$

$$u=9$$

$$\text{Mina before}=5\times9=45$$

$$\text{Yusuf before}=3\times9=27$$

$$\text{Mina after}=45-9=36$$

$$\text{Yusuf after}=27+9=36$$

3. A box has $\frac{3}{5}$ star badges and the rest are moon badges. $\frac{1}{2}$ the stars and 14 moon badges are given away; 42 badges remain. How many badges were there at first?

Working

$$\text{Stars}=6u,\quad \text{moons}=4u$$

$$\text{After stars}=3u$$

$$\text{After moons}=4u-14$$

$$3u+4u-14=42$$

$$7u=56,\quad u=8$$

$$\text{Total}=10\times8=80\text{ badges}$$

Watch out — easily mixed up

Quick recap

🎯 Mastery check

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

1. Kai gives money to Lina. What sign belongs in Kai’s Change row?

   PositiveNegativeNo sign

2. What equation should every BCA row satisfy?

   Before + Change = AfterBefore − After = Change onlyBefore × Change = After

3. Asha has $48 and Ben has $24. Asha gives $\frac{1}{4}$ of her money to Ben. What relationship holds after the transfer?

   Asha still has four times BenBen gives the money backTheir after amounts are equal

4. What does an internal transfer do to the pair’s combined total?

   Adds the transfer twiceKeeps it constantRemoves the whole total

5. If a person receives a gift from outside, what should the BCA table show?

   No change anywhereA negative change for the giver inside the pairA positive change in that row

6. What is the best final check for a BCA solution?

   Run each Before value through its signed ChangeCheck only the larger Before valueIgnore the After row

See my score

Ready to make the story visible? Lay out Before, Change, and After, then let each row balance itself.