Branching: fraction-of-the-remainder problems

Branching problems are among the most reliably missed questions in P6 maths, and almost always for one reason: a fraction of the remainder is not a fraction of the total.

If someone spends ⅓ of their money, then ¼ of the remainder, they have not spent ⅓ + ¼. The second fraction is measured against a smaller amount.

Two ways to solve it

The fraction way works but invites slips. The units way avoids fractions almost entirely, and it’s what most Singapore students find safer under exam pressure.

The trick with units: choose a starting number of units that divides cleanly by every denominator in the question. If you see thirds and quarters, start with 12 units.

Worked example

Siti had some money. She spent ⅓ of it on a book, then ¼ of the remainder on a pen. She had $36 left. How much did she have at first?

The denominators are 3 and 4, so let the starting amount be 12 units.

Stage 1 — the book. ⅓ of 12u = 4u spent. Remainder = 12u − 4u = 8u.

Stage 2 — the pen. ¼ of the remainder = ¼ of 8u = 2u spent. Left = 8u − 2u = 6u.

Stage 3 — match to the real number.

6u = 36
1u = 6

Starting amount = 12u = 12 × 6 = $72.

Check by walking it forwards: $72 → spends $24 on the book, $48 left → spends ¼ of 48 = $12 on the pen → $36 left ✓.

Notice she spent ⅓ + ⅙ of the original, not ⅓ + ¼. Working in units meant you never had to notice that at all.

Drawing the branch

The name comes from the diagram. Each stage splits what remains:

Total (12u)
├── book        4u
└── remainder   8u
    ├── pen     2u
    └── left    6u

Draw this and the structure is unambiguous — you can see that the pen’s fraction hangs off the 8u branch, not off the 12u total.

Common mistakes

• Applying the second fraction to the total. The error the question is built to catch.
• Choosing too few starting units. If 12u doesn’t divide cleanly at some stage, use a bigger multiple — 24u or 36u. Fractions of units mean you picked badly; start again rather than pushing through.
• Stopping at the remainder. The question usually asks for the original amount, not what’s left.
• Losing track of which remainder. With three stages there are three different remainders. Label them.

Quick check

Marcus had some stickers. He gave ¼ to his sister, then ⅖ of the remainder to his friend. He had 36 stickers left. How many did he start with?

Sister    = ¼ of 20u  = 5u
Remainder = 20u − 5u  = 15u

Friend    = ⅖ of 15u  = 6u
Left      = 15u − 6u  = 9u

9u = 36
1u = 4

Start = 20u = 20 × 4