Ratio, Proportion, Partnership & Ages
🔒 Log in to trackhigh importance~2 Q in Tier 122 formulas⚡ 15 shortcuts5 subtopics
Every formula in this topic, grouped by subtopic. Print it and pin it above your desk.
Ratio basics & dividing amounts
Share of the total
\text{share} = \frac{\text{ratio term}}{\text{sum of terms}} \times N
Combining ratios
A:B = m:n,\ B:C = p:q \Rightarrow A:B:C = mp : np : nq
Cross-multiplication test
a:b > c:d \iff ad > bc
Multiplier method
\text{shares } ax, bx,\ (b-a)x = \text{given difference}
Duplicate / sub-duplicate
a:b \Rightarrow a^2:b^2 \text{ (duplicate)},\ \sqrt{a}:\sqrt{b} \text{ (sub-duplicate)}
Proportion & proportional division of terms
Basic proportion
\frac{a}{b} = \frac{c}{d} \iff ad = bc
Fourth proportional
d = \frac{bc}{a}
Third proportional
c = \frac{b^2}{a}
Mean proportional
\text{mean} = \sqrt{ab}
Componendo & dividendo
\frac{a}{b} = \frac{c}{d} \Rightarrow \frac{a+b}{a-b} = \frac{c+d}{c-d}
Invertendo / alternando
\frac{b}{a} = \frac{d}{c},\quad \frac{a}{c} = \frac{b}{d}
Partnership
Simple partnership
P_1 : P_2 = C_1 : C_2 \quad (\text{same time})
Compound partnership
P_1 : P_2 : P_3 = C_1T_1 : C_2T_2 : C_3T_3
Working partner
\text{profit} = \text{manager's cut} + \text{residual split by } C_iT_i
Capital change mid-year
\text{effective capital} = C_a t_1 + C_b t_2 + \cdots
Problems on ages
Present = a x, b x
\frac{ax + n}{bx + n} = \frac{p}{q} \quad (\text{ratio after } n \text{ years})
Ratio n years ago
\frac{ax - n}{bx - n} = \frac{p}{q}
Constant difference
A - B \text{ is the same at every time}
Multiple of age
F = kS \Rightarrow \frac{F - n}{S - n} = \frac{p}{q}
Money ratios: income–expenditure, coins & mixed amounts
Savings equations
ax - py = s_1,\quad bx - qy = s_2
Coin value
\text{total value} = k \sum_i (n_i \times d_i)
Difference of shares
\text{given excess} = (b - a)x