Let's solve this problem by working backwards! Working backwards is one of the most reliable strategies for internal transfer questions because the total number of cards never changes.
Always include units (game cards) in answers and show all working steps.
Step 1 — PLAN:
Total cards at the beginning and end = 144 game cards (this is an internal transfer, so the total stays constant).
In the end, Ahmad and Bala had an equal number of game cards.
Bala gave 1/3 of what he had to Ahmad, leaving him with 2/3 of his amount.
Ahmad had previously given 1/4 of his original cards to Bala, leaving Ahmad with 3/4 of his original amount.
We will work backwards from the final state to find Ahmad's starting amount.
Step 2 — EXECUTE:
Find each person's cards in the end:
Since they ended up with equal cards:
144 / 2 = 72 game cards each.
In the end:
Ahmad had 72 game cards.
Bala had 72 game cards.
Work backwards through the second transfer (Bala gave 1/3 of his cards to Ahmad):
Before giving 1/3 away, Bala's cards represented 3 parts.
After giving away 1 part (1/3), Bala had 2 parts left:
2 parts = 72 game cards
1 part = 72 / 2 = 36 game cards
Bala gave 36 game cards to Ahmad.
Before this transfer took place:
Bala had: 72 + 36 = 108 game cards
Ahmad had: 72 - 36 = 36 game cards
Work backwards through the first transfer (Ahmad gave 1/4 of his cards to Bala):
Ahmad gave away 1/4 of his original cards, so he had 3/4 left.
These remaining 3/4 equal the 36 game cards Ahmad had before the second transfer:
3 units = 36 game cards
1 unit = 36 / 3 = 12 game cards
Ahmad's original cards (4 units) = 4 × 12 = 48 game cards.
Step 3 — VALIDATE:
Let's check forward to confirm every step works:
Ahmad started with 48 game cards.
Bala started with: 144 - 48 = 96 game cards.
Ahmad gave 1/4 of 48 to Bala:
1/4 of 48 = 12 game cards.
Ahmad had left: 48 - 12 = 36 game cards.
Bala now had: 96 + 12 = 108 game cards.
Bala gave 1/3 of 108 to Ahmad:
1/3 of 108 = 36 game cards.
Bala had left: 108 - 36 = 72 game cards.
Ahmad now had: 36 + 36 = 72 game cards.
Both ended with 72 game cards, which matches the problem perfectly!
🎯 Model Answer
Ahmad had 48 game cards at first.
📖 Why it works
Total Unchanged Concept: Because the cards were only passed between Ahmad and Bala, the total remained 144 throughout every step.
Working Backwards: When the final amounts are known (equal shares of 144), reversing each fraction transfer from the end to the start avoids complicated algebra.
Fraction of a Set: When Bala gave away 1/3 of his cards, his remaining cards were 2/3 of his total, meaning 72 cards = 2 parts. When Ahmad gave away 1/4 of his cards, his remaining cards were 3/4 of his starting cards, meaning 36 cards = 3 units.
⚠️ Common mistakes to avoid
Assuming 1/3 was based on the start: Taking 1/3 of Bala's original cards instead of 1/3 of his cards after receiving Ahmad's cards.
Subtracting from the wrong person: Adding cards back to the wrong person during the backward-working step.
Forgetting that the total stays 144: Changing the total sum along the way.
💡 Pro tip
In internal transfer word problems where the final state is given ("in the end they had equal cards"), always divide the total first and work backwards step-by-step!
📚 Source: MOE Curriculum PSLE Mathematics Syllabus 2026 (Topic Fractions: dividing a proper fraction by a whole number without calculator)