🔀 Split Sums & Rate Schemes

Split-sum problems. Call one part $x$; the other part is then automatically $(\text{Total}-x)$. Write the interest on each part with $\dfrac{P\times R\times T}{100}$, add the two, and set the sum equal to the total interest you are given. That is just one equation in one unknown.

For example, ₹10,000 split with a part at 8% and the rest at 12% for 2 years giving ₹1,920 total interest becomes $\dfrac{16x}{100}+\dfrac{24(10000-x)}{100}=1920$, which solves to $x=6000$ — so ₹6,000 at 8% and ₹4,000 at 12%.

Alligation shortcut. If you only need the ratio of the parts, work with the effective rates (rate $\times$ time). The parts come out in the inverse ratio of how far each effective rate sits from the overall rate; and when the overall rate is the exact average of the two, the sum splits a clean $1:1$.

Rate-change and scheme problems. Here the secret is to work with the change, not the whole sum:

• A rate that rises by $d\%$ for $T$ years adds extra interest of just $\dfrac{P\times d\times T}{100}$ — nothing else moves.

• Two equal-time schemes differ by $\dfrac{P\times \Delta R\times T}{100}$, so never compute both interests and subtract.

• When a fee or tax is deducted, find the gross interest, subtract the deduction — it comes out of the interest, never the principal — then take the leftover as a percentage of the original principal to get the effective yield.