Working with Ratios

Number

Two quantities are in direct proportion when multiplying one by a factor multiplies the other by the same factor
Their ratio stays fixed, so doubling one doubles the other
Where the factor is not stated, it is recovered by comparing the two known values of the same quantity, then applied to the other

Finding the value of a single unit first turns most proportion questions into one division followed by one multiplication
Divide to reach one unit, then multiply up to the number required

Worked example

Scaling with the unitary method

Six identical books have a total mass of 2.7 kg. Work out the mass of eleven of these books.

Solution:

The books are identical, so mass and number are in direct proportion: double the books, double the mass
The unitary method goes down to one first, then up to the number wanted
Mass of one book = 2.7 ÷ 6 = 0.45 kg
Write that line down on its own — it is where the method mark sits when the final figure is wrong
Mass of eleven books = 0.45 × 11 = 4.95 kg
Check the direction: eleven books is nearly twice six, and 4.95 kg is nearly twice 2.7 kg
Multiplying 2.7 by 11 and then dividing by 6 gives the same answer, because the two steps can be done in either order

Two quantities are in inverse proportion when multiplying one by a factor divides the other by that same factor
More workers means less time, and more people sharing means a smaller share each, which is what makes the relationship inverse rather than direct
Find the factor by dividing the new quantity by the old one, then divide the other quantity by it
- If 2 machines take 15 hours, then 6 machines is three times as many, so the time is 15 ÷ 3 = 5 hours
The unitary method still works, provided the scaling is reversed at each step
- 4 taps fill a tank in 90 minutes, so one tap would take 4 × 90 = 360 minutes
- Six taps then take 360 ÷ 6 = 60 minutes
Check the direction of the answer against the situation: a larger crew finishing in a longer time is a sign the two steps were the wrong way round

The context decides the direction of rounding, not the usual half-way rule
Where a quantity must be sufficient, round up even when the nearer whole number is lower
- A job needing 2.4 tins of paint requires 3 tins
Where a quantity is limited by what is available, round down instead