Using Fractions in Ratio Problems
This topic involves converting between ratios and fractions to solve problems, often by combining two or more ratios to find a new relationship between quantities.
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 written as a: b is equivalent to the fraction a / b.
- To combine two ratios, such as a: b and b: c, you can make the value for the common term 'b' the same in both ratios.
- An alternative method for finding a: c is to express the ratios as fractions (a/b and b/c) and multiply them: (a / b) × (b / c) = a / c.
- Final ratios should always be simplified to their lowest whole number terms by dividing all parts by their highest common factor.
- Pay close attention to the order of terms. The ratio a: b is not the same as b: a.
Formulae
If a : b = p : q, then a / b = p / q To convert a ratio into a fractional form, which is essential for algebraic manipulation and combining different ratios.
Definitions
- Ratio
- A comparison of two or more quantities, showing their relative sizes. 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. For example, 2:4 is in proportion to 1:2.
Worked example
In a box of chocolates, the ratio of dark to milk chocolates is D : M = 2 : 5. The ratio of white to dark chocolates is W : D = 3 : 4. What is the ratio of milk to white chocolates (M : W)?
- 1
Step 1:
Express the given ratios as fractions.
D / M = 2 / 5 and W / D = 3 / 4 - 2
Step 2:
Identify the target ratio, M :
W, which corresponds to the fraction M / W.
- 3
Step 3:
To find M / W, we need to combine the given fractions.
Notice that M / W = (M / D) × (D / W).
- 4
Step 4:
We have D / M = 2 / 5, so M / D = 5 / 2.
We have W / D = 3 / 4, so D / W = 4 / 3.
- 5
Step 5:
Multiply the two fractions:
M / W = (5 / 2) × (4 / 3) = 20 / 6 - 6
Step 6:
Simplify the resulting fraction:
20 / 6 = 10 / 3 - 7
Step 7:
Convert the fraction back into a ratio:
M :
W = 10 :3.
Answer: 10 : 3
Common mistakes
- ×Incorrectly inverting fractions: When given a : b, writing a / b but then using b / a in a calculation by mistake. Double-check which term is the numerator.
- ×Arithmetic errors in multiplication: Simple mistakes when multiplying fractions, especially under time pressure. For example, calculating 5 × 4 = 10 instead of 20.
- ×Finding a common term for the wrong quantity: When combining a : b and b : c, accidentally modifying the ratios to match 'a' or 'c' instead of the linking term 'b'.
No-calculator tips
- ✓Use the 'common term' method to avoid fraction multiplication. For the example (D:M=2:5, W:D=3:4), the common term is D. Its values are 2 and 4. The LCM is 4. So, scale D:M by 2 to get 4:10. Now you have W:D=3:4 and D:M=4:10. This gives W:D:M = 3:4:10, so M:W = 10:3.
- ✓Cancel before multiplying fractions. If you were calculating (8/15) × (5/4), cancel the 5 into the 15 (leaving 3) and the 4 into the 8 (leaving 2). The calculation becomes a much simpler (2/3) × (1/1) = 2/3.