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Branching Method

How a branch tree keeps successive fractions attached to the right remainder, with path multiplication and a full story check.

⏱ 9 min · 🎯 4 things to master

Branching Method: Which Whole Does the Fraction Mean? | PSLE Maths

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The most dangerous part of a fraction question can be the tiny phrase after of. One fraction may describe all the seedlings, while the next describes only the remainder. A connected branching tree keeps each fraction attached to its parent group.

Parents: ask your child to point to the parent box for each fraction and predict the fraction of the original total before revealing the path calculation.

By the end you will be able to label each parent whole, trace a two-edge fraction path, multiply the fractions, and use a known subgroup to recover the original total.

Start with one named parent

A is the group an outgoing fraction describes. Branching is useful when a group is split and one child is split again. It is a representation choice: it shows relationships clearly, whether you read forward or work backwards.

🤔 Predict first: In '13\frac{1}{3} of all seedlings are herbs', what is the parent?

Split the garden tree

Here is the full question:

An Eco Club has some seedlings. 13\frac{1}{3} are herbs. 25\frac{2}{5} of the remaining seedlings are flowers, and the rest are vegetables. There are 24 flower seedlings. How many seedlings are there altogether?

Begin with one root: All seedlings. 13\frac{1}{3} branches to herbs, so the complementary 23\frac{2}{3} branches to the Remaining seedlings group. The second split starts at Remaining, not at the root. 25\frac{2}{5} of Remaining are flowers; 35\frac{3}{5} of Remaining are vegetables.

Connected tree: All seedlings splits into Herbs at one-third of all and Remaining at two-thirds of all; Remaining splits into Flowers at two-fifths of remaining with 24 seedlings and Vegetables at three-fifths of remaining.
The second pair of branches leaves Remaining. Its fractions 25\frac{2}{5} and 35\frac{3}{5} describe that parent, not All seedlings.

Label every branch

Herbs=13 of all seedlings\text{Herbs} = \frac{1}{3}\text{ of all seedlings}
Remaining=1−13=23 of all seedlings\text{Remaining} = 1 - \frac{1}{3} = \frac{2}{3}\text{ of all seedlings}
Flowers=25 of remaining\text{Flowers} = \frac{2}{5}\text{ of remaining}
Vegetables=35 of remaining\text{Vegetables} = \frac{3}{5}\text{ of remaining}

🤔 Predict first: Which group is the parent of the flower branch?

Multiply along the flower path

To reach Flowers from All seedlings, follow two edges. First take 23\frac{2}{3} to reach Remaining. Then take 25\frac{2}{5} of that smaller parent to reach Flowers. A part of a part is found by multiplying along the connected path.

Follow the path

Flowers=25×23 of all seedlings\text{Flowers} = \frac{2}{5} \times \frac{2}{3}\text{ of all seedlings}
25×23=415\frac{2}{5} \times \frac{2}{3} = \frac{4}{15}
Flowers=415 of all seedlings\text{Flowers} = \frac{4}{15}\text{ of all seedlings}

Why is the result 415\frac{4}{15}? Imagine the original whole split into 15 equal shares. The 23\frac{2}{3} remainder occupies 10 shares. Dividing those 10 shares into five groups gives 2 shares per group. Flowers take two groups, or 4 of the 15 original shares.

🤔 Predict first: 13\frac{1}{3} of the whole is herbs. Flowers are 25\frac{2}{5} of the 23\frac{2}{3} remainder. What fraction of the original whole is flowers?

The question gives 24 flower seedlings. Flowers are 4 of 15 equal original shares. Use that known subgroup to find one share, then multiply by all 15 shares.

Recover the original total

4 shares=24 seedlings4\text{ shares} = 24\text{ seedlings}
1 share=24÷4=6 seedlings1\text{ share} = 24 \div 4 = 6\text{ seedlings}
15 shares=15×6=90 seedlings15\text{ shares} = 15 \times 6 = 90\text{ seedlings}

Branching path: change the original total

Predict first: 13\frac{1}{3} of the 34\frac{3}{4} remainder is what fraction of the original?

The activity uses the same path idea with stickers: 14\frac{1}{4} is used on Monday, so 34\frac{3}{4} remains; 13\frac{1}{3} of that remainder is 14\frac{1}{4} of the original. The number changes when the original total changes, but the path fraction stays fixed.

Check the garden story

13×90=30 herbs\frac{1}{3} \times 90 = 30\text{ herbs}
Remaining=90−30=60 seedlings\text{Remaining} = 90 - 30 = 60\text{ seedlings}
25×60=24 flowers\frac{2}{5} \times 60 = 24\text{ flowers}
Vegetables=60−24=36 seedlings\text{Vegetables} = 60 - 24 = 36\text{ seedlings}
30+24+36=90 seedlings30 + 24 + 36 = 90\text{ seedlings}

The intermediate Remaining group is a subtotal, not an extra terminal category. Add herbs, flowers and vegetables once.

Branching and bars have different jobs

The branching tree shows who came from whom. A bar model can show equal lengths to scale. For a nested fraction, use the tree first when the parent relationship is easy to lose; convert to equal shares when you need a numerical answer. The tree's line length does not represent amount.

🤔 Predict first: What does a short tree branch tell you?

Watch out — easily mixed up

Quick recap

Graduated practice

1. Stickers used on two days

I used 14\frac{1}{4} of my stickers on Monday. On Tuesday, I used 13\frac{1}{3} of the remaining stickers. What fraction of my original stickers did I use on Tuesday?

Show solution

After Monday, 34\frac{3}{4} of the original stickers remain. Tuesday takes 13\frac{1}{3} of that parent.

Practice 1

Remainder after Monday=1−14=34\text{Remainder after Monday} = 1 - \frac{1}{4} = \frac{3}{4}
Tuesday=13×34=14 of the original\text{Tuesday} = \frac{1}{3} \times \frac{3}{4} = \frac{1}{4}\text{ of the original}

Tuesday used 14\frac{1}{4} of the original stickers. The final remainder is 12\frac{1}{2}, and 14\frac{1}{4} + 14\frac{1}{4} + 12\frac{1}{2} = 1.

2. Money left after two branches

Nadia spent 14\frac{1}{4} of her money on lunch. She then spent 23\frac{2}{3} of the remaining money on a book. She had $18 left. How much money did she have at first?

Show solution

After lunch, 34\frac{3}{4} remains. Spending 23\frac{2}{3} leaves 13\frac{1}{3} of that remainder, which is 14\frac{1}{4} of the original.

Practice 2

Money left=13×34=14 of original\text{Money left} = \frac{1}{3} \times \frac{3}{4} = \frac{1}{4}\text{ of original}
14 of original=$18\frac{1}{4}\text{ of original} = \$18
Original money=18×4=$72\text{Original money} = 18 \times 4 = \$72

Check: lunch is $18, leaving $54; the book costs 23\frac{2}{3} × $54 = $36, leaving $18.

3. Science books and adventure books

25\frac{2}{5} of the books on a shelf are science books. The rest are fiction. 23\frac{2}{3} of the fiction books are adventure stories. There are 24 adventure books. How many books are on the shelf?

Show solution

Fiction is 35\frac{3}{5} of all books. Adventure is 23\frac{2}{3} of fiction, so the path to adventure is 23\frac{2}{3} times 35\frac{3}{5}.

Practice 3

Fiction=1−25=35\text{Fiction} = 1 - \frac{2}{5} = \frac{3}{5}
Adventure=23×35=25 of all\text{Adventure} = \frac{2}{3} \times \frac{3}{5} = \frac{2}{5}\text{ of all}
2 shares=24 books2\text{ shares} = 24\text{ books}
1 share=24÷2=12 books1\text{ share} = 24 \div 2 = 12\text{ books}
Whole shelf=5×12=60 books\text{Whole shelf} = 5 \times 12 = 60\text{ books}

Check: 24 science books and 36 fiction books; 23\frac{2}{3} of 36 is 24 adventure books.

🎯 Mastery check

Answer all 7 — your progress is saved on this device.

  1. In '13\frac{1}{3} of all seedlings are herbs', which is the parent?

  2. 25\frac{2}{5} of which group are flowers in the garden problem?

  3. 25\frac{2}{5} of the 23\frac{2}{3} remainder is what fraction of all seedlings?

  4. Four original shares are 24 flower seedlings. What is one share?

  5. If one share is 6 seedlings and the whole has 15 shares, what is the total?

  6. Why is 13\frac{1}{3} plus 25\frac{2}{5} not the correct flower-and-herb total?

  7. In the garden check, why is Remaining not added to herbs, flowers and vegetables?