Interest, Profit, Loss & Discount

Interest, Profit & Loss

Two families of percentage problems. Interest grows a principal over time; profit & loss compares selling price to cost. Both reward the percentage fluency from the Arithmetic overview.

Interest formulas

For principal $P$, rate $r\%$ per annum, time $t$ years:

$\text{Simple Interest}=\frac{P\,r\,t}{100},\qquad \text{Amount}_{\text{CI}}=P\left(1+\frac{r}{100}\right)^{t}.$

A useful fact: for 2 years, the CI–SI difference is exactly $P\left(\dfrac{r}{100}\right)^2$ — it is the "interest on the first year's interest."

Profit & loss formulas

• $\text{Profit}\%=\dfrac{\text{SP}-\text{CP}}{\text{CP}}\times100$ — always on cost price.
• Marked price with discount $d\%$: $\ \text{SP}=\text{MP}\left(1-\dfrac{d}{100}\right)$.
• Successive markup $a\%$ then discount $b\%$: net $=a-b-\dfrac{ab}{100}\,\%$.

A worked example

Find the difference between CI and SI on ₹10,000 at 10% p.a. for 2 years.

$\text{SI}=\frac{10000\cdot10\cdot2}{100}=2000.$ $\text{CI}=10000\left(1.1^2-1\right)=10000(0.21)=2100.$

Difference $=2100-2000=\mathbf{₹100}$. The shortcut confirms it: $P\left(\dfrac{r}{100}\right)^2=10000(0.01)=100.$

Common traps

• Profit % on SP instead of CP. The base is cost price unless stated otherwise.
• Discount on cost. Discount is always on the marked price, not cost.
• CI compounding period. Half-yearly at $r\%$ p.a. means rate $r/2$ over $2t$ periods.

Checklist

• Anchor profit %/loss % on cost price
• Use $P(1+r/100)^t$ for CI; the 2-year gap is $P(r/100)^2$
• Apply discount to the marked price
• Adjust rate & periods for non-annual compounding