Interest, Profit, Loss & Discount

Overview

This module covers three interconnected topics that appear on both CAT and GMAT: Simple & Compound Interest, Profit & Loss, and Discount & Markup. Together they account for 2–4 questions per CAT exam and appear regularly in GMAT Problem Solving at the 500–700 difficulty range.

Simple Interest (SI)

SI = P × R × T / 100

Where: P = Principal, R = Rate % per annum, T = Time in years.

Amount (A) = P + SI = P(1 + RT/100)

Key deductions:

• SI is the same every year (linear growth)
• If two amounts are given at different time points, the difference = SI earned between those two periods
• From two data points (A₁ at T₁, A₂ at T₂): SI per year = (A₂ − A₁)/(T₂ − T₁), Principal = A₁ − SI×T₁

Rate = years trick: If R = T (rate equals number of years), then SI = P×R²/100. If SI = k×P, then R² = 100k.

Compound Interest (CI)

A = P(1 + r/n)^(nt)

Where: r = annual rate (as decimal), n = compounding frequency per year, t = years.

For annual compounding (n = 1): A = P(1 + R/100)^T

CI = A − P

CI vs SI difference formulas

For 2 years: CI − SI = P(R/100)²

For 3 years: CI − SI = P(R/100)²(3 + R/100)

These shortcut formulas avoid computing CI fully when asked only for the difference.

Example: P=5000, R=12%, T=2 years. CI−SI = 5000×(0.12)² = 5000×0.0144 = 72

Effective rate for multiple periods

For half-yearly compounding at R% annually: effective annual rate = (1 + R/200)² − 1

Profit and Loss

Key formulas:

• Gain% = (Gain/CP) × 100
• Loss% = (Loss/CP) × 100
• SP = CP × (100 + Gain%)/100
• SP = CP × (100 − Loss%)/100
• CP = SP × 100/(100 + Gain%) — to find CP when SP and Gain% are given
• CP = SP × 100/(100 − Loss%)

Example: Sold at 10% gain → SP = 1.10 × CP. Sold at 10% loss → SP = 0.90 × CP.

Successive profit/loss

If an item is marked up by a% and then discounted by b%, net change% = a − b − ab/100.

Two successive discounts of 10% each: net = 10 + 10 − 10×10/100 = 19% discount (not 20%).

Discount and Markup

• Marked Price (MP): price listed on the item
• Discount = reduction on MP. SP = MP × (100 − Discount%)/100
• Markup: increase on cost. MP = CP × (100 + Markup%)/100

Relationship: CP → Markup → MP → Discount → SP

Example: 40% discount on retail means CP = 0.6 × retail. Markup on CP = (retail − CP)/CP = 0.4/0.6 = 66.7%.

Loan Repayment with CI

When a loan is repaid in equal instalments at CI:

If loan P at rate r/year is repaid in 2 equal instalments x, one at end of each year:

P(1+r)² = x(1+r) + x → solve for P or x as needed.

Common Mistakes

• Profit/Loss% is ALWAYS calculated on CP, never on SP
• Discount% is calculated on MP (marked price), not CP
• In SI, "amounts to 704 in 2 years" means P + 2-year SI = 704 (not just SI = 704)
• CI−SI formulas are only for equal rate, compounded annually; don't use them for half-yearly CI
• Successive discounts: never add percentages — use (1−d₁/100)(1−d₂/100) approach

Exam Tips

• For SI problems with two unknown (P and R), extract the SI per year from two amount data points first
• When CI−SI difference is given for 2 years: P = (CI−SI)/(R/100)² — fastest path to principal
• For "sold at x% gain and y% loss on two items with total SP" problems: avoid equations, use multiplier approach
• Markup vs Discount: markup is on CP, discount is on MP — confusing these is the #1 error
• For CAT: when "effective discount" questions appear, use the formula (1−d₁)(1−d₂) and subtract from 1