The Equal-at-the-End Method

Some problems give you a golden clue: at one moment, two people have the same amount. This equal stage tells you where to draw their bars. Then undo what happened to find the beginning.

Parents: ask your child to mark the equal stage before touching the starting numbers. The method tip names the reverse operation a marker wants to see.

By the end, you will be able to find a shared final amount, draw equal bars there, and work backwards carefully.

Start from the equal stage

The words equal, the same, or ended with the same tell you where to draw the bars. Do not split the initial total before accounting for what each person spent, received, or gave away.

Undo the changes

Hana and Mei have $90 altogether. Hana spends $24 and Mei spends $6. They then have the same amount. How much did each have at first?

After spending, their shared total is the original total minus the total spent. Split that shared amount equally, then add each person’s spending back.

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Match the final bars

🤔 Predict first: If the final amount is $60 split equally, what is one final bar?

$24$30$60

Work backwards

$$24 + 6 = 30 \text{ dollars spent}$$

$$90 - 30 = 60 \text{ dollars left}$$

$$\text{Each final amount} = 60 \div 2 = 30$$

$$\text{Hana before} = 30 + 24 = 54$$

$$\text{Mei before} = 30 + 6 = 36$$

The original total checks: Hana’s starting amount and Mei’s starting amount together make ninety. After each spend, both have thirty.

Equal at the start or end

The equal stage may be at the beginning or the end. If the pupils start equal, draw the equal bars first and move forwards. If they finish equal, work backwards as in Hana’s problem. The phrase tells you the stage; the operations tell you the direction.

Do not confuse the total

If one person spends more than the other, their starting amounts need not be equal. The final amounts are equal only after the changes. A constant-total transfer problem keeps the total unchanged; an equal-stage problem may involve spending or receiving money from outside.

Practice: work back from equality

1. Dana and Eli have $74 altogether. Dana gives away $10 and Eli gives away $2. They then have the same amount. Find their starting amounts.

Working

$$10 + 2 = 12 \text{ dollars given away}$$

$$74 - 12 = 62 \text{ dollars left}$$

$$\text{Each final amount} = 62 \div 2 = 31$$

$$\text{Dana before} = 31 + 10 = 41$$

$$\text{Eli before} = 31 + 2 = 33$$

2. Two teams have 68 pupils altogether. They receive bonuses of 14 and 6 pupils, then have equal totals. Find the starting team sizes.

Working

$$68 + 14 + 6 = 88 \text{ pupils after bonuses}$$

$$\text{Each final size} = 88 \div 2 = 44$$

$$\text{Team 1 before} = 44 - 14 = 30$$

$$\text{Team 2 before} = 44 - 6 = 38$$

3. Ivy spends $9. Joel receives $6 from outside. They then have the same amount, and they started with $87 altogether. Find the starting amounts.

Working

$$87 - 9 + 6 = 84 \text{ dollars after changes}$$

$$\text{Each final amount} = 84 \div 2 = 42$$

$$\text{Ivy before} = 42 + 9 = 51$$

$$\text{Joel before} = 42 - 6 = 36$$

Watch out — easily mixed up

Quick recap

🎯 Mastery check

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

1. What does an equal-stage clue tell you to draw?

   Equal-length barsA longer first barOnly the initial total

2. Hana and Mei spend money and then become equal. Which total should you split?

   The original totalThe shared final totalOnly Hana’s spending

3. To undo money spent before the equal stage, should you add it back or subtract it again?

   Add it backSubtract it againIgnore it

4. If one team receives a larger bonus than the other, must their starting amounts be equal?

   Yes, alwaysNo, only their final amounts matchOnly if the total is odd

5. Where should equal bars be drawn when two pupils start with the same number of cards?

   At the startAt the endAfter a random change

6. What is the final check after working backwards?

   Both original amounts follow their changes to the same final amountOnly one bar has a labelThe larger starting amount disappears

See my score

Ready to line up the bars? Find the equal stage first, then work backwards one action at a time.