🧩 Chains, Algebra & Changes

• Chained clues (“A is 7 times C, B is twice A”): anchor on the smallest partner. Let C $= 1$, then A $= 7$, B $= 14$, giving $7 : 14 : 1$ in one pass. For fractions of the total, take the total as the LCM of the denominators so each part is a whole number.

• Equal-gap capitals (each ₹$d$ more than the last): set them as $x,\ x+d,\ x+2d$, add to the total, solve for $x$. With equal gaps the middle partner always gets exactly $\dfrac{1}{3}$ of the profit.

• Capital from time and profit: capital $\propto \dfrac{\text{profit}}{\text{time}}$, so divide each profit by its time and clear fractions with the LCM. Multiplying the two ratios is the classic wrong turn.

• Percentage changes: apply each change to that partner's OWN capital — new $= \text{old} \times (1 \pm \text{rate})$ — then read off the new ratio. Never add the percent to the ratio number.