Profit, loss, discount, and tax

When you buy or sell something, three numbers matter: cost price (CP) , what the seller paid for it; selling price (SP) , what the seller charges; and the profit or loss between them. Add to this the marked price (MP) , the printed sticker price , and the discount, tax, and you have the full machinery of shop-floor maths.

Concept

Profit and loss.

• If $SP > CP$: profit $= SP - CP$.
• If $SP < CP$: loss $= CP - SP$.

Profit/Loss percentage is always taken with respect to cost price:

$\text{Profit}\% = \frac{\text{Profit}}{\text{CP}} \times 100, \quad \text{Loss}\% = \frac{\text{Loss}}{\text{CP}} \times 100.$

Example: bought for $\mathbb{?}\,500$, sold for $\mathbb{?}\,600$. Profit $= \mathbb{?}\,100$. Profit% $= \dfrac{100}{500} \times 100 = 20\%$.

Finding SP from CP and profit/loss %.

• $SP = CP \cdot (1 + p/100)$ where $p$ is profit %.
• $SP = CP \cdot (1 - l/100)$ where $l$ is loss %.

Example: a phone bought at $\mathbb{?}\,8000$ is sold at $15\%$ profit. $SP = 8000 \cdot 1.15 = \mathbb{?}\,9200$.

Marked price and discount. Shopkeepers often mark a higher price (MP) and then offer a discount.

• Discount $= MP - SP$.
• Discount% $= \dfrac{MP - SP}{MP} \times 100$.

Example: MP $= \mathbb{?}\,2000$, discount $20\%$. Discount $= \mathbb{?}\,400$. SP $= \mathbb{?}\,1600$.

Tax (GST, sales tax). A tax is added on top of the selling price.

• Final price $= SP \cdot (1 + t/100)$ where $t$ is the tax rate.

Example: a TV with SP $\mathbb{?}\,20{,}000$ and GST $18\%$. Final price $= 20{,}000 \cdot 1.18 = \mathbb{?}\,23{,}600$.

Beware: "$20\%$ off then $10\%$ off" is not "$30\%$ off". Discounts compound multiplicatively.

Example: MP $= \mathbb{?}\,1000$.

• After $20\%$ off: $\mathbb{?}\,800$.
• Then $10\%$ off the $\mathbb{?}\,800$: $\mathbb{?}\,720$.
• Effective discount: $28\%$, not $30\%$.

This is the same kind of mistake students make with percentage increases. Be careful.

Worked examples

Example 1. Cost price $= \mathbb{?}\,750$, selling price $= \mathbb{?}\,900$. Find the profit and profit%.

• Profit $= 900 - 750 = \mathbb{?}\,150$.
• Profit% $= 150 / 750 \times 100 = 20\%$.

Example 2. A trader buys a watch for $\mathbb{?}\,2400$ and sells it at a loss of $5\%$. Find the SP.

• $SP = 2400 \cdot (1 - 5/100) = 2400 \cdot 0.95 = \mathbb{?}\,2280$.

Example 3. A shirt has MP $\mathbb{?}\,1800$ and is on $25\%$ discount. After GST of $5\%$, what does a customer pay?

• After discount: $1800 \cdot 0.75 = \mathbb{?}\,1350$.
• After GST: $1350 \cdot 1.05 = \mathbb{?}\,1417.50$.

Example 4. A book is sold for $\mathbb{?}\,150$ at a profit of $20\%$. Find the cost price.

• $SP = CP \cdot 1.20$, so $CP = 150 / 1.20 = \mathbb{?}\,125$.

Try it yourself

1. CP $= \mathbb{?}\,300$, SP $= \mathbb{?}\,330$. Find profit%.
2. A trader sells a chair for $\mathbb{?}\,1080$ at a loss of $10\%$. Find the cost price.
3. A jacket marked $\mathbb{?}\,2500$ is sold at a $15\%$ discount. Find SP.
4. Find the GST amount on an item of SP $\mathbb{?}\,800$ at $12\%$ GST.
5. A laptop bought at $\mathbb{?}\,40{,}000$ is sold at $25\%$ profit. Find SP.
6. A toy is marked $\mathbb{?}\,400$. After a $20\%$ discount and $5\%$ GST, what does a customer pay?
7. Successive discounts of $10\%$ and $5\%$ are offered on a $\mathbb{?}\,500$ item. Find the effective discount %.
8. A trader marks up his cost by $40\%$ and then offers a $20\%$ discount. Find his actual profit %.

Activity / Insight

Discount detective. Look at the back of a popular toothpaste tube and find the MRP. Now find the actual selling price online or at a nearby shop. Compute the effective discount % yourself. Many "promotions" turn out to be just $5$-$10\%$ off the MRP , which is the discount the shop is required to offer anyway. Becoming numerate about percentages is real-world consumer power.