Units and parts: solving ratio problems without algebra

Units and parts is the workhorse of P6 maths. Once ratios appear, most questions reduce to the same three moves: write the ratio as units, find one unit, answer the actual question. Nearly every lost mark comes from skipping the third.

The method

A ratio of 5 : 3 means the quantities are made of 5 equal units and 3 equal units — the same size unit in both. That single sentence does most of the work, because it means you can add, subtract and compare units freely.

From any ratio you get three things for free:

• the total in units (5 + 3 = 8u)
• the difference in units (5 − 3 = 2u)
• each quantity in units

Match whichever one the question gives you a real number for, and you can find 1 unit.

Worked example

The ratio of red marbles to blue marbles in a box is 5 : 3. There are 24 more red marbles than blue. How many marbles are there altogether?

Red = 5u, Blue = 3u. Drawn, the structure is immediate:

The question gives a difference, so use the difference in units:

5u − 3u = 24
     2u = 24
     1u = 12

Now answer what was asked — the total, not one unit:

Total = 8u = 8 × 12 = 96 marbles

Check: red = 60, blue = 36. Difference 24 ✓, and 60 : 36 simplifies to 5 : 3 ✓.

Fractions are ratios in disguise

P6 questions often hide a ratio inside a fraction, and the wording decides which one you have.

“Ali has ⅔ as many stamps as Bala” → Ali : Bala = 2 : 3

“Ali has ⅔ of the total” → Ali : Bala = 2 : 1

These are completely different situations. Read whether the fraction compares the quantity to another quantity or to the whole — getting this wrong makes every later step wrong, and it’s one of the most common errors in the paper.

Three-part ratios

The method doesn’t change with three quantities; the arithmetic just gets heavier.

Three friends share a sum of money in the ratio 2 : 3 : 7. The largest share is $60 more than the smallest. How much was shared?

Largest − smallest = 7u − 2u = 5u = 60, so 1u = 12. Total = 12u = $144.

Common mistakes

• Answering “1 unit”. You found 1u = 12 and wrote 12. The question asked for the total. Underline what’s being asked before you start.
• Mixing unit sizes. Units from one ratio are not the same size as units from a different ratio in the same question. Use u and p to keep them apart.
• Using the total when given the difference. Check which one the real number attaches to.
• Not simplifying at the end. If the answer is a ratio, simplify it.

Quick check

The ratio of boys to girls in a hall is 4 : 7. There are 132 children in total. How many more girls than boys are there?

Total = 4u + 7u = 11u

11u = 132
 1u = 12

Difference = 7u − 4u = 3u
           = 3 × 12