Ratios and Percentages

A dimensionless ratio compares like quantities by division after converting their units; a percentage is a ratio whose second term is fixed at 100. The standard numerical tasks reduce to one of a handful of ratio moves: scale a ratio, combine two ratios that share a term, split a total in a ratio, change a quantity by a percentage, or undo such a change. Rates such as speed, work done per day and price per kilogram are ratios too, as are mixture and average calculations that look different on the surface.

The tools are the decision procedure for each move, the multiplier view of percentage change that makes successive and reverse changes mechanical, and the traps that separate a correct answer from a nearby wrong option. The same moves apply to numbers read from a chart.

Same-unit convention. For a dimensionless comparison of like quantities, convert to the same unit first: 45 minutes to 2 hours is $45:120=3:8$, not $45:2$. A rate can compare different dimensions, such as distance divided by time, and carries the quotient unit, such as km/h.

Fractions worth knowing by heart. $1/8 = 12.5\%$, $1/6 = 16.\overline{6}\%$, $1/7 \approx 14.29\%$, $1/9 = 11.\overline{1}\%$, $1/11 = 9.\overline{09}\%$, $1/12 = 8.\overline{3}\%$, $3/8 = 37.5\%$, $5/8 = 62.5\%$. Recognising them turns a long division into a lookup.

Given $A : B = 2 : 3$ and $B : C = 4 : 5$, the two values of $B$ (3 and 4) must be made equal. Scale each ratio so that $B$ becomes the least common multiple of 3 and 4, which is 12: $A : B = 8 : 12$ and $B : C = 12 : 15$. Hence $A : B : C = 8 : 12 : 15$.

To split a total $T$ in the ratio $a : b : c$, one part is worth $T / (a + b + c)$ and the shares are $a$, $b$ and $c$ parts.

A change of $r\%$ multiplies a quantity by $m = 1 + r/100$ (with $r$ negative for a decrease). Then:

All four are percentages of a stated base, and the base is the whole question:

On principal $P$ at $r\%$ per year for $n$ years, simple interest is $P r n / 100$; the amount under annual compounding is $P(1 + r/100)^n$. Compound interest earns interest on interest, so it exceeds simple interest from the second year on.

Successive changes. For multipliers $m_1 = 1 + a/100$ and $m_2 = 1 + b/100$,

so the net change is $a + b + \frac{ab}{100}$ percent. With $a = 20$ and $b = -20$ this is $-4\%$: equal rises and falls never cancel.

Undoing a rise. If $x$ rises by $r\%$ to $x(1 + r/100)$, the fall $d\%$ that returns it to $x$ satisfies $(1 + r/100)(1 - d/100) = 1$, so $d = \frac{100r}{100 + r}$.

Alligation. Mixing quantities $u$ at value $p$ and $w$ at value $q$ gives mean $m = \frac{up + wq}{u + w}$. Solving, $u(m - p) = w(q - m)$, hence $u : w = (q - m) : (m - p)$. The rule is the weighted mean read backwards.

Average speed over equal distances. For distance $d$ each way at speeds $v_1$ and $v_2$, total time is $d/v_1 + d/v_2$, so the average speed is

the harmonic mean of the two speeds, which is never more than their arithmetic mean.

Compound minus simple interest over two years. $P(1 + r/100)^2 - P - 2Pr/100 = P(r/100)^2$: the extra is interest on the first year's interest.

Example 1 (combining and dividing). $A : B = 2 : 3$ and $B : C = 4 : 5$. Divide 1,050 among A, B and C in this ratio.

Example 2 (successive change). A price of 2,500 rises by 20% and then falls by 20%.

Example 3 (reverse percentage). After a 25% increase a quantity is 1,500. Find the original.

Example 4 (base switching). A earns 25% more than B. By what percent is B's pay less than A's?

Example 5 (percentage points). An interest rate goes from 8% to 10%. It rose by 2 percentage points, which is a relative increase of $2 / 8 = 25\%$.

Example 6 (mark-up, discount and profit). Cost price 800, marked up by 40%, then sold at a 15% discount.

Example 7 (interest). 10,000 at 10% per year for 2 years.

Example 8 (alligation). Rice at 60 per kg is mixed with rice at 90 per kg to give a mixture worth 72 per kg. Ratio of cheap to dear: $(90 - 72) : (72 - 60) = 18 : 12 = 3 : 2$. Check: 3 kg at 60 plus 2 kg at 90 costs $180 + 180 = 360$ for 5 kg, which is 72 per kg.

Example 9 (work). A finishes a job in 12 days and B in 18 days. Working together they do $1/12 + 1/18 = 3/36 + 2/36 = 5/36$ of the job per day, so they need $36/5 = 7.2$ days.

Example 10 (average speed). A car goes from P to Q at 40 km/h and returns at 60 km/h. Average speed $= \frac{2 \times 40 \times 60}{40 + 60} = 48$ km/h. Check with a distance of 120 km each way: $3 + 2 = 5$ hours for 240 km, which is 48 km/h. The tempting 50 km/h is the arithmetic mean and is wrong.

Example 11 (estimation). Estimate 49.8% of 3,012. Round to 50% of 3,000, which is 1,500. The exact value is 1,499.976, so the estimate is within 0.01%. Options that differ by more than a few percent can be eliminated without exact work.

Example 12 (three successive changes). A price rises by 10%, then by 20%, then falls by 25%. Multiplier: $1.10 \times 1.20 \times 0.75 = 0.99$. Net change: a 1% decrease. Adding the percentages ($10 + 20 - 25 = 5$) gives the wrong answer, a 5% rise.

Example 13 (simple versus compound interest over three years). 8,000 at 5% per year for 3 years. Simple interest: $8000 \times 0.05 \times 3 = 1200$. Compound amount: $8000 \times 1.05^3 = 9261$, so compound interest is 1,261. Difference: 61. For two years the difference is $P(r/100)^2 = 8000 \times 0.05^2 = 20$: it is the interest earned on the first year's interest.

Example 14 (removal and replacement). A can holds 40 litres of milk. 8 litres are removed and replaced with water, and this is done once more. Mix thoroughly before each withdrawal. Each round keeps $\frac{40 - 8}{40} = \frac{4}{5}$ of the milk, so the milk left is $40 \times \left(\frac{4}{5}\right)^2 = 25.6$ litres. The ratio of milk to water is $25.6 : 14.4 = 16 : 9$. Subtracting 8 litres of milk twice (leaving 24) ignores that the second removal takes out some water.

Example 15 (alligation with concentrations). Using percentages by mass, a 20% salt solution is mixed with a 50% salt solution to obtain a 30% solution. Ratio of the 20% solution to the 50% solution: $(50 - 30) : (30 - 20) = 2 : 1$. Check: 2 kg at 20% and 1 kg at 50% hold $0.4 + 0.5 = 0.9$ kg of salt in 3 kg, which is 30%.

Example 16 (work with a person leaving). A can finish a job in 10 days and B in 15 days. They work together for 4 days, then A leaves. Together they do $\frac{1}{10} + \frac{1}{15} = \frac{1}{6}$ per day, so in 4 days they finish $\frac{4}{6} = \frac{2}{3}$. B alone finishes the remaining $\frac{1}{3}$ in $\frac{1}{3} \times 15 = 5$ days. Total time: 9 days.

Example 17 (pipe and leak). A pipe fills a tank in 6 hours; a leak alone would empty the full tank in 9 hours. Net rate: $\frac{1}{6} - \frac{1}{9} = \frac{1}{18}$ per hour, so the tank fills in 18 hours. Treat emptying as negative work.

Example 18 (relative speed: trains). A train 240 m long runs at 54 km/h, which is $54 \times \frac{5}{18} = 15$ m/s.

Example 19 (relative speed: boat and stream). A boat goes 15 km/h downstream and 9 km/h upstream. Boat speed in still water: $\frac{15 + 9}{2} = 12$ km/h; stream speed: $\frac{15 - 9}{2} = 3$ km/h. A 36 km round trip (18 km each way) takes $\frac{18}{15} + \frac{18}{9} = 1.2 + 2 = 3.2$ hours.