Applying Ratios to Problems
This topic covers how to use ratios to compare and scale quantities in practical scenarios like mixing solutions or converting units. Mastering ratios is crucial for solving multi-step problems where you need to relate different quantities to each other 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 like A:B compares the relative sizes of quantities. It means for every 'A' units of one thing, there are 'B' units of another.
- To find the actual amount of a substance, first calculate the total number of 'parts' in the ratio. Then, divide the total quantity by the number of parts to find the value of one part.e.g., for 2:3, there are 2+3=5 parts
- When comparing quantities that share a common element (e.g., A:B and B:C), you must make the common part (B) equal in both ratios by finding a common multiple before you can establish a direct ratio (A:C).
- Ratios are essential for scaling. A map scale of 1:50000 means any length on the map is 50000 times smaller than the real-world length. Be vigilant with unit conversions (e.g., cm to m to km).
- A ratio A:B can be expressed in fractions. The fraction of the whole that is A is A/(A+B), and the fraction that is B is B/(A+B).
Formulae
Value of one part = Total Quantity / Sum of ratio parts Use this when you have a total amount shared in a known ratio and you need to find the specific amounts of the components.
Definitions
- Ratio
- A way to compare the relative sizes of two or more quantities. It is written with a colon, for example, 3:2.
- Proportion
- A statement that two ratios are equal. Proportions are used to solve problems where quantities scale up or down at the same rate.
Worked example
A paint mixture is created by combining two paints, X and Y. Paint X contains pigment and linseed oil in a ratio of 2:3. Paint Y contains pigment and linseed oil in a ratio of 1:5. If 100 ml of paint X is mixed with 300 ml of paint Y, what is the final ratio of pigment to linseed oil in the new mixture?
- 1
Step 1:
Calculate the amount of pigment and oil in the 100 ml of paint X.
The ratio is 2:3, so there are 2+3=5 parts.
One part is 100 ml / 5 = 20 ml.
Pigment = 2 × 20 = 40 mlOil = 3 × 20 = 60 ml - 2
Step 2:
Calculate the amount of pigment and oil in the 300 ml of paint Y.
The ratio is 1:5, so there are 1+5=6 parts.
One part is 300 ml / 6 = 50 ml.
Pigment = 1 × 50 = 50 mlOil = 5 × 50 = 250 ml - 3
Step 3:
Sum the total amounts of pigment and oil from both mixtures.
Total Pigment = 40 ml (from X) + 50 ml (from Y) = 90 ml.
Total Oil = 60 ml (from X) + 250 ml (from Y) = 310 ml - 4
Step 4:
Express the total amounts as a ratio and simplify.
The ratio of pigment to oil is 90:310.
Divide both sides by 10 to simplify.
- 5
Final Answer:
The ratio is 9:31.
Answer: 9:31
Common mistakes
- ×Mistaking the ratio for the fraction. In a 2:3 ratio of A to B, the amount of A is 2/5 of the total, not 2/3. This 'off by factor' error is common.
- ×Making arithmetic errors when scaling up ratios. If you scale one part of a ratio, you must scale all other parts by the exact same factor.
- ×Incorrectly converting units in scaling problems. For example, in a 1:1000 scale, converting 5 cm on a map to 50 m instead of 5000 cm (which is 50 m) is a frequent slip-up.
No-calculator tips
- ✓Always simplify ratios to their smallest integer values as your first step. It's much easier to work with 3:4 than 27:36.
- ✓When dividing a total quantity by the sum of ratio parts, look for ways to break down the division. To calculate 240 / 8, you could do 240 / 2 = 120, 120 / 2 = 60, 60 / 2 = 30.
- ✓When combining ratios like A:B and B:C, choose the lowest common multiple for B to keep the numbers small and manageable for mental arithmetic.