Problem-solving method

The units and parts method, explained for parents

The step that turns a ratio into a real number. Here is what "one unit" means, the wording in a problem sum that calls for it, and how to tell whether your child has it.

Used from
P5
Taught at
P5–P6

What this method is

A ratio tells you how something is shared, not how much anything is worth. This method is the bridge: draw the quantities as equal boxes, find the line that pins a box to a real number, and the rest is multiplication.

The name comes from the two labels a child writes. When a question holds two sets of boxes that are not the same size, one set is called units and the other parts, so they never get added together.

When to reach for it

These are the words in a question that point at this method. You do not need to be able to solve it to spot them.

  • “in the ratio 3 : 4”

    A ratio, so nothing has a value yet. Draw 3 boxes and 4 boxes rather than writing numbers you do not have.

  • “23\frac{2}{3} of the buttons”

    The denominator picks the box count. Thirds means three boxes, and one of them is the thing worth finding.

  • “for every 5 red, there are 2 blue”

    A ratio written the long way round. Same drawing, same units, no new method needed.

  • “each of them used 160”

    One value pinned to two different box counts. This is the line that sizes a box, and it is usually the last sentence.

  • “How much did they have at first”

    You will be scaling back up to a whole at the end, so keep track of how many boxes the whole is worth.

Watch it work

23\frac{2}{3} of the buttons in a jar were shared between Nadia and Priya. Nadia used 23\frac{2}{3} of her buttons and Priya used 45\frac{4}{5} of hers. Each of them used 160 buttons. How many buttons were in the jar at first?

Nadia in units, Priya in parts. The boxes are different sizes, which is the whole reason they get different names, and the two shaded runs come out equal because both are 160.

  1. Give each girl her own counter

    Nadia = 3 units, Priya = 5 parts

    Nadia used two thirds of hers, so her buttons split into 3. Priya used four fifths, so hers split into 5. Different names because they are different sizes, and calling both of them units is where this question goes wrong.

  2. Size a unit

    2 units → 160, so 1 unit = 80 and 3 units = 240

    Here is the pin. The 160 belongs to 2 of Nadia's units, which makes one of them 80, and her whole share follows immediately.

  3. Size a part

    4 parts → 160, so 1 part = 40 and 5 parts = 200

    The same 160, but it covers 4 of Priya's boxes, so hers are half the size of Nadia's. Same sentence in the question, two different answers.

  4. Add what the two of them had

    240 + 200 = 440

    This is the shared buttons, not the jar. The question said two thirds were shared, so 440 is two thirds, and one step remains.

  5. Scale back to the whole jar

    2 thirds → 440, so 1 third = 220 and the jar = 3 × 220 = 660

    Units again, one last time, now on the jar itself. Children who stop at 440 have done every hard part of this question and answered a different one.

The one it gets confused with

All of these still draw as bars. What changes is what the boxes are counting, and that decision comes before any arithmetic.

  • The bar model

    Does the question already give real amounts? Then write the numbers straight into the bars. Counting boxes is for when nothing has a value yet.

  • Constant total, difference and part

    Does a ratio change partway through? Then the question is which thing stayed fixed, and that picks the method before you draw anything.

  • Branching

    Does a second fraction apply to what is left over rather than the whole? Then split the remainder first and count boxes afterwards.

  • Units and parts, the before-and-after version

    Some tuition notes use "units" for a ratio before a change and "parts" for the same ratio after it, for the same reason we do here. If your child's worksheet is doing that, this is the page you want.

How can you tell it has clicked?

Ask what one unit is worth. A child who has the method points straight at the line in the question that answers it. A child who has not starts multiplying fractions together and hopes.

Watch for the same letter written against both quantities. The working looks busy and correct, every line is a real calculation, and the answer is wrong by a margin nobody can trace afterwards.

One thing to try tonight

Take any question with a ratio in it, cover the numbers, and ask one thing: what is one unit worth?

If they cannot say, hunt together for the sentence that pins a box to a real amount. It is nearly always near the end. Finding it is the skill, and the multiplication after it is the easy half.

Where your child will meet it

Want the bigger picture on how these methods fit together? Maths heuristics, explained for parents

See which methods your child reaches for

StudyLah names the method in every worked solution, so you can see whether the problem was the method or the arithmetic.

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