Calculations with Fractions and Percentages
ESAT problems frequently present numerical information in a mix of fractions, decimals, and percentages. This topic covers the essential skill of converting fluently between these forms to choose the most efficient calculation method for a given problem.
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
- To convert a fraction to a decimal, divide the numerator by the denominator.e.g., 3/4 = 0.75
- To convert a decimal to a percentage, multiply by 100.e.g., 0.75 = 75%
- To convert a percentage to a fraction, place the number over 100 and simplify.e.g., 75% = 75/100 = 3/4
- Equivalent fractions represent the same value. Create them by multiplying or dividing both the numerator and denominator by the same non-zero number.e.g., 2/3 = 4/6 = 10/15
- Choose the best format for the operation: fractions are often best for multiplication and division due to cancellation, while decimals can be easier for addition and subtraction.
- Memorising common conversions is a significant time-saver.e.g., 1/8 = 0.125, 1/3 ≈ 0.333, 1/5 = 0.2
Formulae
Percentage = Decimal × 100 To convert a number from its decimal form to its percentage form.
Fraction = Percentage / 100 To convert a percentage into a fraction, which can then be simplified.
a/b = (a × n) / (b × n) To create an equivalent fraction, for example when finding a common denominator for addition or subtraction.
Definitions
- Equivalent Fractions
- Fractions that have different numerators and denominators but represent the same value or proportion of a whole. For example, 1/2 and 5/10 are equivalent.
Worked example
An engineering alloy is 3/8 aluminium, 45% copper, and 0.15 tin by mass. The rest of the alloy is zinc. If a component made from this alloy contains 40 grams of zinc, what is the mass of the copper in the component?
- 1
Convert all proportions to a single, consistent format.
Let's use decimals.
- 2
Aluminium:
3/8.
To convert, we can do 3 divided by 8.
(1/8 = 0.125, so 3/8 = 3 × 0.125 = 0.375) - 3
Copper:
45% = 0.45 - 4
Tin:
0.15.
- 5
Add the known proportions:
0.375 + 0.45 + 0.15 = 0.975.
- 6
The total proportion is 1.
The proportion of zinc is 1 - 0.975 = 0.025.
- 7
We are told that this proportion (0.025) corresponds to 40 g of zinc.
Let T be the total mass.
- 8
So, 0.025 × T = 40.
Note that 0.025 is 1/40.
So (1/40) × T = 40.
- 9
Therefore, the total mass T = 40 × 40 = 1600 g.
- 10
The question asks for the mass of copper, which is 45% of the total mass.
- 11
Mass of copper = 0.45 × 1600 g.
This is (45/100) × 1600 = 45 × 16 = 720 g.
Answer: 720 g
Common mistakes
- ×Decimal placement errors during conversion, for instance writing 6% as 0.6 instead of the correct 0.06.
- ×When adding or subtracting, failing to convert all numbers to the same format first (e.g. attempting to calculate 3/4 + 0.1 directly).
- ×Incorrectly simplifying fractions, or failing to simplify them, which makes subsequent calculations more difficult.
No-calculator tips
- ✓When multiplying mixed formats, such as 0.25 × 36%, converting both to fractions is usually fastest: (1/4) × (36/100) = 9/100.
- ✓To calculate a percentage of a number, use building blocks. For 45%, find 50% (half) and subtract 5% (a tenth of the 50% value).
- ✓To divide by a decimal, convert it to a fraction. For example, 20 / 0.1 is the same as 20 / (1/10), which is 20 × 10 = 200.