Applying Ratios to Problems
Ratios compare and scale quantities in practical scenarios like mixing solutions or converting units, and are 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.