Percentage, discount, profit and loss

Unit 5 Percentage, discount, profit and loss

Key unit competence: Solve problems that involve percentage, discount, commission, profit, loss, simple interest and taxes.

Key ideas

• A percentage is a fraction out of 100: $\text{percentage}=\frac{\text{part}}{\text{whole}}\times 100\%$. To find $p\%$ of a quantity, multiply it by $\frac{p}{100}$.
• Multipliers: an increase of $p\%$ means multiply by $1+\frac{p}{100}$; a decrease of $p\%$ means multiply by $1-\frac{p}{100}$. New amount = old amount × multiplier.
• Discount is a reduction on the marked price: selling price = marked price − discount.
• Commission is a payment to an agent, a percentage of the sales: commission = rate × sales.
• Profit = selling price − cost price; loss = cost price − selling price. Percentage profit or loss is always calculated on the cost price.
• Simple interest is calculated only on the principal: $I=\frac{PRT}{100}$, where $P$ is the principal, $R$ the rate per year (%), $T$ the time in years. Amount $A=P+I$.
• Taxes such as VAT (18% in Rwanda) are added to the price: total cost = price × $(1+\frac{18}{100})$.

Formulas to know

Common misconceptions

• "Percentage profit is profit divided by the selling price." No: divide by the cost price.
• "An increase of 20% followed by a decrease of 20% returns to the original price." No: $1.2\times 0.8=0.96$, a 4% decrease overall.
• "If 6 months, $T=6$." No: $T$ is in years, so $T=\frac{6}{12}=0.5$.

Worked examples

Practice questions

Easier and harder questions are mixed, just like in a real exam.

1. Express 45 minutes as a percentage of 2 hours.

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   $\frac{45}{120}\times 100\%=37.5\%$.

   Common mistake: Using different units (45 and 2).

2. A sum of money doubles in 8 years at simple interest. Find the rate.

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   $I=P$, so $P=\frac{P\times R\times 8}{100}$, $R=\frac{100}{8}=12.5\%$.

   Common mistake: Thinking the rate must be 50% because the money doubled.

3. A pair of shoes is marked 25 000 FRW. A 12% discount is given. Find the discount and the selling price.

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   Discount $=3000$ FRW; selling price $=22000$ FRW.

   Common mistake: Adding the discount instead of subtracting.

4. A school's enrolment fell from 1 250 to 1 100 students. Find the percentage decrease.

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   Decrease 150; $\frac{150}{1250}\times 100\%=12\%$.

   Common mistake: Dividing by the new value 1 100.

5. A farmer bought 50 chickens at 3 000 FRW each and sold them all at 3 600 FRW each. Find the total profit and the percentage profit.

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   Cost $150000$, sales $180000$, profit $30000$ FRW; $\frac{30000}{150000}\times 100=20\%$.

   Common mistake: Dividing the profit by the selling price (giving 16.7%).

6. A shopkeeper's price for a jacket is 15 000 FRW before VAT. Find the VAT (18%) and the price paid.

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   VAT $=2700$ FRW; price paid $=17700$ FRW.

   Common mistake: Subtracting VAT instead of adding it.

7. Mukamana saves 120 000 FRW at 6% p.a. simple interest. How much will she have after 4 years?

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   $I=\frac{120000\times 6\times 4}{100}=28800$; $A=148800$ FRW.

   Common mistake: Giving only the interest instead of the amount.

8. A house agent charges 3% commission. Find the commission on a house sold for 25 000 000 FRW.

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   $0.03\times 25000000=750000$ FRW.

   Common mistake: Writing 3% as 0.3.

Matching and drag-and-drop

1. Matching

   Match each item in column A with the correct answer in column B. Write the letter next to each number.

   Column A	Column B
   1. Find the simple interest on 150 000 FRW at 8% p.a. for 3 years.	a. 20%
   2. Find the simple interest on 20 000 FRW at 6% p.a. for 6 months.	b. 36 000 FRW
   3. A salesperson earns 5% commission on sales of 50 000 FRW. Find the commission.	c. 2 500 FRW
   4. A price rises from 2 500 FRW to 3 000 FRW. Find the percentage increase.	d. 600 FRW

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   1 → b (36 000 FRW) · 2 → d (600 FRW) · 3 → c (2 500 FRW) · 4 → a (20%)

   • $I=\frac{150000\times 8\times 3}{100}=36000$.
   • Step 1. $T=0.5$ year.
     Step 2. $I=\frac{20000\times 6\times 0.5}{100}=600$.
   • $0.05\times 50000=2500$.
   • Step 1. Increase $=3000-2500=500$ FRW.
     Step 2. $\frac{500}{2500}\times 100\%=20\%$ (always divide by the original value).

2. Drag and drop

   Choose the correct word or phrase from the box for each gap. One word in the box is not needed.

   15 000 FRW  ·  principal + interest  ·  218 000 FRW  ·  236 000 FRW
   1. The total amount after simple interest is .
   2. A trader buys a goat for 60 000 FRW and sells it for 75 000 FRW. Find the profit. →
   3. A TV costs 200 000 FRW before VAT of 18%. Find the total cost. →
   Show answer

   1: principal + interest · 2: 15 000 FRW · 3: 236 000 FRW

   • $A=P+I$.
   • Profit $=75000-60000=15000$ FRW.
   • $200000\times 1.18=236000$.

3. Matching

   Match each item in column A with the correct answer in column B. Write the letter next to each number.

   Column A	Column B
   1. Decrease 60 kg by 5%.	a. 10% p.a.
   2. Find 15% of 8 000 FRW.	b. 34 000 FRW
   3. 100 000 FRW grows to 130 000 FRW in 3 years at simple interest. Find the rate.	c. 57 kg
   4. A radio marked 40 000 FRW is sold at a 15% discount. Find the selling price.	d. 1 200 FRW

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   1 → c (57 kg) · 2 → d (1 200 FRW) · 3 → a (10% p.a.) · 4 → b (34 000 FRW)

   • $60\times 0.95=57$.
   • $\frac{15}{100}\times 8000=1200$.
   • Step 1. Interest $I=130000-100000=30000$ FRW.
     Step 2. $R=\frac{100I}{PT}=\frac{100\times 30000}{100000\times 3}=10\%$.
   • $40000\times 0.85=34000$.

4. Drag and drop

   Choose the correct word or phrase from the box for each gap. One word in the box is not needed.

   53 100 FRW  ·  5 600 FRW  ·  30 000 FRW  ·  5 012 FRW
   1. After a 20% discount, a bag costs 24 000 FRW. What was the marked price? →
   2. Increase 5 000 FRW by 12%. →
   3. An item marked 50 000 FRW gets a 10% discount, then 18% VAT is added. Find the final price. →
   Show answer

   1: 30 000 FRW · 2: 5 600 FRW · 3: 53 100 FRW

   • Step 1. $0.8\times MP=24000$.
     Step 2. So $MP=30000$.
   • $5000\times 1.12=5600$.
   • Step 1. After discount: $50000\times 0.9=45000$ FRW.
     Step 2. With VAT: $45000\times 1.18=53100$ FRW.

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