Business Mathematics: Ratio, Proportion and Logarithms
Domain 1 examines foundational algebra used in commercial calculations: ratio and proportion, indices, logarithms and their applications. Questions usually require simplifying expressions, solving proportional equations or using log laws without a calculator.
Core rules
- A ratio compares like quantities in the same unit; clearing units prevents inverted answers.
- If a:b = c:d, then ad = bc. In continued proportion a:b = b:c, b² = ac.
- For positive bases, a^m × a^n = a^(m+n), a^m ÷ a^n = a^(m-n), and (a^m)^n = a^(mn).
- log_a(xy) = log_a x + log_a y; log_a(x/y) = log_a x − log_a y; log_a(x^k) = k log_a x. Logarithms are defined only for positive arguments, with base >0 and ≠1.
Worked example
If x:y = 3:4 and y:z = 6:7, make the y-terms equal. The first ratio becomes 9:12 and the second 12:14, so x:y:z = 9:12:14 and x:z = 9:14. For a log check, log10 8 = log10(2³) = 3 log10 2 = 3 × 0.3010 = 0.9030.
Traps
Do not write log(a+b) = log a + log b. Do not cancel terms inside a ratio as if it were a fraction. When converting a ratio to a fraction of the total, use the sum of all parts. Check that a logarithm's argument is positive.