Basic Numeracy — Quick Revision

Every shortcut from Percentages, Ratio and Proportion and Averages on one page, for the last week before Prelims Paper II.

Percentages

Fraction–percentage table

Fraction	%	Fraction	%	Fraction	%
\( \frac{1}{2} \)	50	\( \frac{1}{7} \)	\( 14\frac{2}{7} \)	\( \frac{1}{12} \)	\( 8\frac{1}{3} \)
\( \frac{1}{3} \)	\( 33\frac{1}{3} \)	\( \frac{1}{8} \)	12.5	\( \frac{1}{15} \)	\( 6\frac{2}{3} \)
\( \frac{1}{4} \)	25	\( \frac{1}{9} \)	\( 11\frac{1}{9} \)	\( \frac{1}{16} \)	6.25
\( \frac{1}{5} \)	20	\( \frac{1}{10} \)	10	\( \frac{1}{20} \)	5
\( \frac{1}{6} \)	\( 16\frac{2}{3} \)	\( \frac{1}{11} \)	\( 9\frac{1}{11} \)	\( \frac{1}{25} \)	4

Multiples: \( \frac{3}{8} = 37.5\% \), \( \frac{5}{8} = 62.5\% \), \( \frac{7}{8} = 87.5\% \), \( \frac{2}{3} = 66\frac{2}{3}\% \), \( \frac{5}{6} = 83\frac{1}{3}\% \).

Core formulas

$$\boxed{x\% \text{ of } N = \frac{x}{100}\times N} \qquad \boxed{x\% \text{ of } y = y\% \text{ of } x}$$$$\boxed{\text{Percentage change} = \frac{\text{New} - \text{Old}}{\text{Old}}\times 100}$$$$\boxed{\text{Successive change of } a\% \text{ and } b\% = a + b + \frac{ab}{100}\ \%}$$$$\boxed{\text{Rise of } x\% \text{ then fall of } x\% = \text{net fall of } \frac{x^2}{100}\ \%}$$$$\boxed{\text{A is } r\% \text{ more than B} \Rightarrow \text{B is } \frac{r}{100+r}\times 100\ \% \text{ less than A}}$$$$\boxed{\text{A is } r\% \text{ less than B} \Rightarrow \text{B is } \frac{r}{100-r}\times 100\ \% \text{ more than A}}$$$$\boxed{\text{Price up } r\%,\ \text{same spend} \Rightarrow \text{cut consumption by } \frac{r}{100+r}\times 100\ \%}$$$$\boxed{P_n = P\left(1 + \frac{r}{100}\right)^n} \qquad \boxed{P_n = P\left(1 - \frac{r}{100}\right)^n}$$

Ready-made successive changes

Changes	Net	Changes	Net
+10, +10	+21%	+20, −20	−4%
+10, +20	+32%	+10, −10	−1%
+20, +20	+44%	−10, −10	−19%
+25, −20	0%	+50, \( -33\frac{1}{3} \)	0%

Ratio and Proportion

$$\boxed{a : b = \frac{a}{b}} \qquad \boxed{a : b :: c : d \iff ad = bc}$$$$\boxed{\text{Share of } a \text{ in } a : b : c = \frac{a}{a+b+c}\times N}$$$$\boxed{\text{A : B} = p : q,\ \text{B : C} = r : s \Rightarrow \text{A : B : C} = pr : qr : qs}$$

Term	Formula	Example
Duplicate / triplicate ratio	\( a^2 : b^2 \) / \( a^3 : b^3 \)	2 : 3 → 4 : 9 / 8 : 27
Sub-duplicate ratio	\( \sqrt{a} : \sqrt{b} \)	4 : 9 → 2 : 3
Inverse ratio	\( b : a \)	5 : 7 → 7 : 5
Compound ratio	\( ac : bd \)	2 : 3 and 4 : 5 → 8 : 15
Fourth proportional to \( a, b, c \)	\( \frac{bc}{a} \)	2, 3, 4 → 6
Third proportional to \( a, b \)	\( \frac{b^2}{a} \)	4, 6 → 9
Mean proportional of \( a, b \)	\( \sqrt{ab} \)	9, 16 → 12

$$\boxed{\text{Partnership: profit share} \propto \text{capital} \times \text{time}}$$$$\boxed{\text{Inverse proportion: } x_1 y_1 = x_2 y_2 \quad (\text{workers} \times \text{days} = \text{constant})}$$

Averages

$$\boxed{\text{Average} = \frac{\text{Sum}}{\text{Number}}} \qquad \boxed{\text{Sum} = \text{Average}\times\text{Number}}$$$$\boxed{\text{Average of an AP} = \frac{\text{First} + \text{Last}}{2}}$$$$\boxed{\text{Weighted average} = \frac{n_1 a_1 + n_2 a_2}{n_1 + n_2}}$$$$\boxed{\text{Correct average} = \text{Old average} + \frac{\text{Correct} - \text{Wrong}}{n}}$$$$\boxed{\text{Average speed (equal distances)} = \frac{2xy}{x+y}}$$

Set / situation	Result
First \( n \) natural numbers	\( \frac{n+1}{2} \)
First \( n \) even numbers	\( n + 1 \)
First \( n \) odd numbers	\( n \)
Replacement, average of \( n \) rises by \( d \)	New member \( = \) old member \( + nd \)
Member joins, average of \( n \) rises by \( d \)	New member \( = \) new average \( + nd \)
Overlapping groups (shared member)	Group sum \( + \) group sum \( - \) overall sum
Equal times at speeds \( x, y \)	\( \frac{x+y}{2} \)

Don’t Confuse

This	vs	That
Percentage point (simple difference)		Percent change (difference ÷ old value)
A 25% more than B		B 20% less than A
Third proportional \( \frac{b^2}{a} \)		Mean proportional \( \sqrt{ab} \)
Duplicate ratio \( a^2 : b^2 \)		“Double” of \( a : b \) (unchanged)
Equal distances → \( \frac{2xy}{x+y} \)		Equal times → \( \frac{x+y}{2} \)
Replacement: count stays \( n \)		Joining: count becomes \( n + 1 \)