Percentage, Profit and Loss: Formulas and Worked Examples

Percentage basics

A percentage is a fraction with denominator 100, so x% = x/100.

• Part = Percent × Base. Example: 15% of Rs 2,400 = 0.15 × 2,400 = Rs 360.
• Base = Part ÷ Percent. If 35% of a number is 280, the number is 280 ÷ 0.35 = 800.
• Percentage change = (New value - Old value) ÷ Old value × 100. The old value is always the base.
• If A is 25% more than B, then B is 25/125 = 20% less than A. The two percentages differ because the base changes.

Successive changes

Convert each change to a multiplying factor and multiply the factors.

• An increase of a% followed by an increase of b% gives a net change of a + b + ab/100 percent (use a minus sign for a decrease).
• Up 20% then down 20%: 1.20 × 0.80 = 0.96, a net decrease of 4%, not zero.
• Successive discounts of 20% and 10%: 0.80 × 0.90 = 0.72, equal to a single discount of 28%, not 30%.

Profit and loss

Cost price (CP) is what the seller paid; selling price (SP) is what the seller receives. Profit and loss percent are always calculated on CP unless the question says otherwise.

When two articles are sold at the same selling price, one at a gain of x% and the other at a loss of x%, the result is always an overall loss of x²/100 percent.

Marked price, discount and VAT

• Marked price (MP) is the list price. Discount is a percentage of MP, so SP = MP × (100 - d) ÷ 100.
• If goods are marked m% above cost and sold at a discount of d%, SP = CP × (1 + m/100) × (1 - d/100).
• VAT is added to the price after discount. Amount paid = MP × (1 - d/100) × (1 + v/100).
• In Nepal, section 7 of the Value Added Tax Act, 2052 sets a flat rate of 13%, with zero rate for the items in Schedule 2. Always use the rate printed in the question.

Worked examples

1. Percentage change. A price rises from Rs 800 to Rs 920. Change = 120 ÷ 800 × 100 = 15%.
2. Finding CP. An article sold for Rs 1,710 gives a loss of 5%. CP = 1,710 × 100 ÷ 95 = Rs 1,800. Check: 5% of 1,800 = 90 and 1,800 - 90 = 1,710.
3. Discount then VAT. MP = Rs 5,000, discount 10%, VAT 13%. Price after discount = Rs 4,500. VAT = 13% of 4,500 = Rs 585. Amount paid = Rs 5,085.
4. Working backwards. A customer pays Rs 9,040 after a 20% discount and 13% VAT. Price before VAT = 9,040 ÷ 1.13 = Rs 8,000. MP = 8,000 ÷ 0.80 = Rs 10,000.

Exam pointers

• Identify the base first: old value for percentage change, CP for profit and loss, MP for discount, discounted price for VAT.
• Never add or subtract successive percentages directly; multiply the factors.
• To move from one SP to another SP, find CP first. Do not apply the new percentage to the old SP.
• Marking up 40% and discounting 25% gives 1.40 × 0.75 = 1.05, a profit of 5%, not 15%.
• To remove 13% VAT from a total, divide by 1.13. Subtracting 13% of the total is wrong.
• Same SP with equal gain and loss percent always means a loss.