Profit, Loss and Interest

Profit and loss

• Bought for ₹400 and sold for ₹500: profit 100/400 = 25%.
• Successive discounts combine like successive percentage changes: 20% and then 10% off ₹1000 gives 1000 × 0.8 × 0.9 = ₹720, a single discount of 28% (not 30%).
• If selling at a 10% profit instead of a 10% loss earns ₹50 more, the 20-point difference is ₹50, so CP = ₹250.
• A dishonest seller who sells at cost price but gives 900 g for every kilogram gains 100/900 = 11.11%.

Simple interest

SI = P × r × t / 100 for principal P, rate r% per year and t years. The amount is A = P + SI. ₹5000 at 8% for 3 years earns 5000 × 8 × 3 / 100 = ₹1200.

Compound interest

Interest is added to the principal each period, so it earns interest itself:

A = P (1 + r/100)ᵗ, CI = A − P.

• ₹10,000 at 10% for 2 years: A = 10000 × 1.1² = 12,100, so CI = ₹2100, while SI would be ₹2000.
• Compounded k times a year, use rate r/k per period and kt periods: ₹8000 at 10% per year compounded half-yearly for one year gives 8000 × 1.05² = 8820, interest ₹820.
• CI − SI for 2 years = P (r/100)²: on ₹10,000 at 10% that is ₹100. For 3 years it is P (r/100)² (3 + r/100).
• Doubling time at r% compound interest is about 72/r years (the "rule of 72"): at 8%, log 2 / log 1.08 ≈ 9.01 years.

import math

def compound(P, rate, years, times_per_year=1):
    return P * (1 + rate / 100 / times_per_year) ** (years * times_per_year)

print("profit %:", (500 - 400) / 400 * 100)
print("successive discounts 20% + 10% =", round(100 - 100 * 0.8 * 0.9, 2), "% in one go")
print("SI on 5000 at 8% for 3 years:", 5000 * 8 * 3 / 100)
print("CI on 10000 at 10% for 2 years:", round(compound(10000, 10, 2) - 10000, 2))
print("half-yearly CI on 8000 at 10% for 1 year:", round(compound(8000, 10, 1, 2) - 8000, 2))
print("doubling time at 8%:", round(math.log(2) / math.log(1.08), 2), "years (rule of 72 gives 9)")

profit %: 25.0
successive discounts 20% + 10% = 28.0 % in one go
SI on 5000 at 8% for 3 years: 1200.0
CI on 10000 at 10% for 2 years: 2100.0
half-yearly CI on 8000 at 10% for 1 year: 820.0
doubling time at 8%: 9.01 years (rule of 72 gives 9)

Practice questions