Primary Fraction Remainders: Keep Track of the Changing Whole

Write down what each fraction belongs to

In a public discussion about Primary maths workings, a contributor asked about a method called branching and how to present solutions. The useful mathematical idea behind a branch diagram is to keep track of parts of parts. For successive-remainder questions, the most important label is the quantity that each fraction refers to.

A fraction of the original amount and the same fraction of a remainder usually represent different amounts. Before calculating, write the words “of the original” or “of the remainder” beside each fraction. A diagram is useful when it preserves that distinction. It is less useful when it merely collects several fractions around an unlabelled line.

Follow an original worked example forward

Consider this original practice problem: a pupil spends one quarter of some money on a book, then spends two fifths of the remaining money on stationery. The pupil has $54 left. How much money was there at first? The purchase details and numbers are teaching examples, not facts about the online contributor.

First identify the part remaining after the book. One whole minus one quarter leaves three quarters of the original money. Next identify the part remaining after stationery. One whole remainder minus two fifths of that remainder leaves three fifths of that remainder. The final amount is therefore three fifths of three quarters of the original.

Multiply the two remaining fractions: 3/5 × 3/4 = 9/20. Thus $54 represents nine twentieths of the original money. Nine equal units are worth $54, so one unit is worth $6. Twenty units are worth $120. Notice that the nine units here are twentieths of the original amount. Naming the unit prevents the final multiplication from becoming a guess.

Make a branch diagram readable

Start the diagram with a box labelled “original money”. Split it into two branches: “book: 1/4 of original” and “first remainder: 3/4 of original”. Split only the remainder branch again. Its two new labels are “stationery: 2/5 of first remainder” and “final money: 3/5 of first remainder”.

Along the final path, multiply 3/4 by 3/5. Do not add the labels along a path: each successive label acts on the amount reaching that branch. Addition belongs elsewhere, when combining separate parts expressed relative to the same whole. This is why a diagram needs words as well as fraction symbols.

The stationery spending can also be expressed as 2/5 × 3/4 = 3/10 of the original money. Now all three final portions share the same reference whole: the book is 1/4, stationery is 3/10, and money left is 9/20. Converting to twentieths gives 5/20 + 6/20 + 9/20 = 1 whole.

Check by rebuilding the story

Begin with the proposed $120. The book costs one quarter of that, or $30, leaving $90. Stationery costs two fifths of $90, or $36, leaving $54. Both purchases and the final amount agree with the question. This check tests the meaning of the fractions as well as the arithmetic.

A common incorrect approach subtracts 1/4 and 2/5 directly from one whole. That treats the stationery fraction as if it also referred to the original money. Ask the child to say the full phrase aloud: “two fifths of the money remaining after the book”. Then return to the $90 stage. The words identify why the direct subtraction changes the problem.

Practise one change at a time

On the next question, first ask only for the fraction remaining, without finding the original amount. Once that is clear, add a final dollar value and work back to the start. Later, contrast a problem saying “two fifths of the original money” with one saying “two fifths of the remainder”. This comparison makes the reference whole visible.

Use the Primary Maths collection to find appropriate fraction word problems and check their wording carefully. Keep the MOE Primary Mathematics syllabus as the official curriculum reference. For a particular school's presentation requirements, ask the teacher to review one completed solution; a forum reply cannot establish a universal marking rule.