⚖️ Difference & Equal SI

The interest a sum earns each year is fixed at $\dfrac{P\times R}{100}$, so over $T$ years it simply piles up to $\dfrac{P\times R\times T}{100}$. Everything below is just this one fact, used twice.

1. Difference of two SIs (same principal). Subtract the two formulas and the principal stays out front:

$\text{Difference}=\dfrac{P\times(R_1T_1-R_2T_2)}{100}$

So you never need to compute both interests — just find the combined gap $R_1T_1-R_2T_2$ in the rate$\times$time percent, then take that slice of the sum. Two handy special cases:

• Only the time changes (same $P,R$): the gap is just the interest for the extra years, $\dfrac{P\times R\times(T_2-T_1)}{100}$.

• Only the rate changes (same $P,T$): the gap is the interest at the rate-difference, $\dfrac{P\times(R_1-R_2)\times T}{100}$.

Only when the principal, rate and time are all different is there no shortcut — there you genuinely compute both SIs and subtract.

2. Equal interest (set the two SIs equal). If the two interests are the same, the whole $\dfrac{P\times R\times T}{100}$ matches on both sides and the $100$ cancels, leaving the clean product rule:

$P_1\,R_1\,T_1=P_2\,R_2\,T_2$

Plug in and divide for a missing principal or time. For the ratio of the two sums use the inverse rule $P_1:P_2=(R_2T_2):(R_1T_1)$ — the bigger a sum's rate$\times$time, the smaller the principal it needs to earn the same money. And if you are also told the difference of the two sums, turn the equality into a ratio first, then split the difference into equal parts.