Proportion Ratios and Functions
This topic covers how to compare quantities using ratios and proportions. It is a foundational skill for solving problems involving mixtures, scaling, and interpreting linear relationships without a calculator.
Part of the ESAT Mathematics 1 syllabus — revision for the Engineering and Science Admissions Test (ESAT), the UAT-UK admissions test for Cambridge, Imperial, Oxford and UCL.
Key points
- A ratio `a:b` means for every `a` units of the first quantity, there are `b` units of the second. The total number of 'parts' is `a+b`.
- To convert a ratio to a fraction, place the part you are interested in over the total sum of the parts. For a ratio `a:b`, the fraction of the whole that is `a` is `a / (a+b)`.
- To combine two ratios with a common element (e.g., `A:B` and `B:C`), you must first scale the ratios to make the value for the common element (`B`) the same in both.
- A ratio `x:y = a:b` describes a direct proportion, which can be expressed as the linear function `y = (b/a)x`. This is the equation of a straight line passing through the origin with gradient `b/a`.
- The 'unitary method' is a powerful technique for proportion problems. Find the value of a single unit first (e.g., cost per item), then multiply to find the required value.
Diagram
Formulae
y = kx When two quantities, x and y, are in direct proportion. 'k' is the constant of proportionality. This is equivalent to stating that the ratio y:x is constant.
Total Amount / (a+b+c) To find the value of a single 'part' when a total amount is divided into a ratio of `a:b:c`. Multiply this value by `a`, `b`, or `c` to find the respective quantities.
Definitions
- Ratio
- A comparison of the relative sizes of two or more quantities. For example, `3:2` means the first quantity is 1.5 times the size of the second.
- Proportion
- A statement that two ratios are equal. It indicates that quantities scale with each other in a consistent way.
Worked example
An alloy is made from copper and tin in the ratio 5:3 by mass. A second alloy is made from tin and zinc in the ratio 4:1 by mass. If 40g of the first alloy and 100g of the second alloy are melted together, what is the final ratio of copper to tin to zinc in the new alloy?
- 1
Step 1:
Calculate the mass of each metal in the first 40g alloy.
The ratio is 5:3, so there are 5+3=8 parts.
One part is 40g / 8 = 5g.
So, Copper = 5 parts × 5g = 25g; Tin = 3 parts × 5g = 15g - 2
Step 2:
Calculate the mass of each metal in the second 100g alloy.
The ratio is 4:1, so there are 4+1=5 parts.
One part is 100g / 5 = 20g.
So, Tin = 4 parts × 20g = 80g; Zinc = 1 part × 20g = 20g - 3
Step 3:
Sum the masses for each metal.
Total Copper = 25g.
Total Tin = 15g (from first alloy) + 80g (from second alloy) = 95gTotal Zinc = 20g - 4
Step 4:
Write the final ratio of Copper:Tin:Zinc.
This is 25:95:20.
- 5
Step 5:
Simplify the ratio by dividing all parts by their highest common factor, which is 5.
The final ratio is (25/5) :
(95/5) :
(20/5), which simplifies to 5:19:4.
Answer: 5:19:4
Common mistakes
- ×Mistaking a part-to-part ratio for a part-to-whole fraction. For a 3:4 ratio, the fraction of the first component is 3/(3+4) = 3/7, not 3/4.
- ×When combining linked ratios (e.g., A:B and B:C), incorrectly adding or averaging the numbers for B instead of finding a common multiple.
- ×Failing to simplify a final ratio to its lowest terms, which may mean you don't recognise the correct answer in a multiple-choice list.
No-calculator tips
- ✓In proportion problems like `15x = 40 × 9`, don't multiply large numbers first. Instead, rearrange to `x = (40 × 9) / 15` and cancel common factors: `x = (40 × 3) / 5`, then `x = 8 × 3 = 24`.
- ✓When finding a common multiple to combine ratios, use the lowest common multiple (LCM) to keep the numbers as small and manageable as possible.
- ✓To simplify large ratios, test for divisibility by small prime numbers (2, 3, 5, 7, 11) systematically.