Part A: Mathematics
Ratios, rates and proportions
Key takeaways
- A ratio compares two or more quantities of the same kind, written a:b; a rate compares quantities measured in different units as a single value, such as km per hour or KES per US dollar.
- A proportion states that two ratios are equal, a:b = c:d, so a missing value can be found once the other three are known.
- The same value can always be written as a fraction, a decimal or a percentage: 3/4 = 0.75 = 75%.
- Carry extra precision through a multi-step calculation and round only the final answer, because rounding early compounds error through every later step.
- Where a ratio meets a total, turn the ratio into fractions of the whole first: a 2:3 ratio splits a total into 2/5 and 3/5.
This opening chapter covers the simplest tools business mathematics uses to compare numbers to each other: ratios, rates and proportions, and the three interchangeable ways a single quantity can be written, as a fraction, a decimal or a percentage.
Key definitions
Ratio versus proportion
A ratio compares two quantities directly to each other; a proportion is an equation stating that two ratios are equal. In practice, a proportion is what lets a known ratio be scaled up or down to solve for a missing figure.
Solving a problem involving rates and proportions
A recipe uses flour and sugar in a 5:2 ratio. How much sugar is needed for 15 kg of flour?
5/2 = 15/x, so x = (15 × 2) / 5 = 6 kg of sugar
Fractions, decimals and percentages
The same value can always be written in any of the three forms.
Rounding
A figure is rounded to a stated number of decimal places or significant figures by looking at the digit immediately after the cut-off point: 5 or above rounds the preceding digit up, below 5 leaves it unchanged. Rounding too early in a multi-step calculation compounds error through every later step, so a full computation should carry extra precision throughout and round only the final answer.
Percentages in business problems
A typical business percentage problem
A shop's revenue was KES 450,000 last month and rose by 12% this month.
Increase = 450,000 × 0.12 = KES 54,000
This month's revenue = 450,000 + 54,000 = KES 504,000
Exam tip
Whenever a question mixes a ratio and a total figure, convert the ratio to fractions of the whole first (a 2:3 ratio splits a total into 2/5 and 3/5), rather than trying to work with the ratio numbers directly.
Self-check
(a) A recipe uses flour and sugar in a 7:3 ratio. How much sugar is needed for 21 kg of flour? (3 marks)
(b) A shop's revenue was KES 380,000 last month and rose by 8% this month. Compute this month's revenue. (3 marks)
(c) Distinguish between a ratio and a rate, giving one business example of each. (4 marks)
(d) Convert 0.625 to a fraction in its lowest terms, and convert 5/8 to a percentage. (4 marks)
(e) Explain why rounding too early in a multi-step calculation is a risk, and state the correct practice to avoid it. (4 marks)
Show worked answer
(a) 7/3 = 21/x, so x = (21 x 3) / 7 = 9 kg of sugar. (3 marks)
(b) Increase = 380,000 x 0.08 = KES 30,400. This month's revenue = 380,000 + 30,400 = KES 410,400. (3 marks)
(c) A ratio compares two or more quantities of the same kind, for example mixing cement to sand in a 2:3 ratio. A rate compares two quantities measured in different units, expressed as a single value, for example a delivery speed of 50 km per hour or an exchange rate of KES per US dollar. (4 marks)
(d) 0.625 = 625/1000, which simplifies to 5/8 in lowest terms. 5/8 as a percentage = 5/8 x 100 = 62.5%. (4 marks)
(e) Rounding too early compounds error through every later step of a multi-step calculation, since each subsequent step then works from an already-imprecise figure rather than the true value, and the accumulated error can meaningfully distort the final answer. The correct practice is to carry extra precision (unrounded, or several extra decimal places) throughout the calculation and round only the final answer, once no further arithmetic will be performed on it. (4 marks)