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Daily Maths

Daily Maths

Profit, loss, discount and other money problems

Percentages- Formula Core + Extended 9 min read

On this page
  1. In this note
  2. Formula bank
  3. Profit and loss
  4. Discounts and sales
  5. Tax, instalments and earnings
  6. Ways they test it
  7. Mistakes to avoid
  8. Summary
  9. Practice
  10. Can I do this?
  11. Quiz

In this note

Syllabus 1.13: money with percentages
  • 1.13Find a profit or loss, and the percentage profit or loss (of the cost price).
  • 1.13Find a sale price and a percentage discount; compare offers.
  • 1.13Work with tax, deposits and instalments, commission and income tax.

Before you start: percentage increase and decrease and reverse percentages. Every question here is one of those with money words on it.

Formula bank

Profit / lossLearn itprofit \(=SP-CP\); loss \(=CP-SP\)CP = cost price (what the seller paid), SP = selling price.Use it when buying and selling.
Percentage profit or lossLearn it\(\dfrac{\text{profit or loss}}{CP}\times100\)always out of the COST price.Use it when "find the percentage profit".
Selling priceLearn itprofit \(p\%\): \(SP=CP\times\left(1+\dfrac p{100}\right)\); loss: \(\times\left(1-\dfrac p{100}\right)\)Use it when a target profit or a given loss.
Sale priceLearn it\(\text{marked}\times\left(1-\dfrac d{100}\right)\)d% discount off the marked (label) price.Use it when a sale.
Percentage discountLearn it\(\dfrac{\text{marked}-\text{sale}}{\text{marked}}\times100\)out of the MARKED price.Use it when "find the percentage discount".
InstalmentsLearn ittotal = deposit + number × paymentextra = total − cash price.Use it when "pay monthly" plans.

Profit and loss

The cost price (CP) is what the seller paid; the selling price (SP) is what they sold it for. Profit and loss percentages are always percentages of the COST price: that is the 100%.

General methodHow to find any percentage profit or loss

  1. Profit = SP − CP (or loss = CP − SP).
  2. Divide by the COST price, × 100.
  3. Bought in bulk and sold one by one? Compare TOTALS (or compare per item) - never a total with one price.

Example A shop buys a fan for $40 and sells it for $52. Find the percentage profit.

Profit ÷ cost price × 100.

  1. profit \(=52-40=12\)
  2. \(\tfrac{12}{40}\times100=30\)

Answer 30% profit

Try it yourself

1
Asma bought a bike for $250 and sold it for $210. Percentage loss?
Show answer

Answer 16%

2
A watch costs $80. The trader wants a 45% profit. Selling price?
Hint
× 1.45.
Show answer

Answer $116

3
24 mangoes cost $18; each is sold for $1.05. Percentage profit?
Hint
Total selling price = 24 × 1.05.
Show answer

Answer 40%

Discounts and sales

General methodHow to work with any discount

  1. Sale price: marked price × (1 − discount as a decimal). 20% off → × 0.8.
  2. Percentage discount: (marked − sale) ÷ marked × 100.
  3. Comparing offers: work out the actual price at each shop, then compare the prices (not the percentages).

Example A game is marked $48 and sold for $39.60. Find the percentage discount.

Discount ÷ marked price × 100.

  1. discount \(=48-39.60=8.40\)
  2. \(\tfrac{8.40}{48}\times100=17.5\)

Answer 17.5%

Try it yourself

1
A dress is marked $75 and sold at 20% off. Sale price?
Show answer

Answer $60

2
"Buy one, get the second half price": what percentage do you save on two?
Hint
You pay 1.5 items' worth out of 2.
Show answer

Answer 25%

Tax, instalments and earnings

General methodHow to handle tax, plans and pay

  1. Tax / VAT / service charge: an increase. × (1 + rate).
  2. Deposit + instalments: total = deposit + (number of payments × each payment); extra = total − cash price.
  3. Commission: pay = basic + rate × sales.
  4. Income tax with a tax-free allowance: tax only the part ABOVE the allowance.

Example A TV costs $1200 cash. On a plan you pay a 15% deposit then 18 payments of $65. How much more is the plan, as a percentage of the cash price?

Total paid, minus the cash price, out of the cash price.

  1. deposit \(=0.15\times1200=180\)
  2. payments \(=18\times65=1170\)
  3. total \(=1350\); extra \(=150\)
  4. \(\tfrac{150}{1200}\times100=12.5\)

Answer 12.5% more

Try it yourself

1
A meal costs $45 before a 12% service charge. Total?
Show answer

Answer $50.40

2
Nadia earns $1800 plus 3% commission on $84 000 of sales. Total pay?
Show answer

Answer $4320

3
Tariq earns $38 000. The first $12 000 is tax free; 20% tax on the rest. How much tax?
Hint
20% of 26 000.
Show answer

Answer $5200

Ways they test it

Type 1 Percentage profit / loss

"Bought for $45, sold for $63. Find the percentage profit." (2 marks)

Spot it "bought", "sold", "profit", "loss".

  1. Profit or loss.
  2. ÷ cost price × 100.

Type 2 Bulk buying

"A seller buys 50 pens for $2.40 each... sells 90% of them at $3.20..." (3 marks)

Spot it a box or pack, sold one by one.

  1. Total cost.
  2. Total takings (count how many are sold).
  3. Profit ÷ total cost × 100.

For example 50 pens at $2.40; 90% sold at $3.20

  1. cost \(=120\)
  2. takings \(=45\times3.20=144\)
  3. \(\tfrac{24}{120}\times100=20\)

Answer 20% profit

Type 3 Which is the better buy?

"Shop A: $90 with ⅓ off. Shop B: $84 with 30% off. Which is cheaper?"

Spot it two offers.

  1. Work out each actual price.
  2. Compare the prices, state the difference.

For example A: $90, ⅓ off; B: $84, 30% off

  1. A: \(90-30=60\)
  2. B: \(84\times0.7=58.80\)

Answer Shop B, by $1.20

Type 4 Cash or monthly plan

"Deposit 25% of $640, then 12 payments of $48. What percentage more than cash?"

Spot it deposit, instalments.

  1. Total paid.
  2. Extra ÷ cash × 100.

Type 5 Cost price from SP and % profit

"A toy sells for $96 at a 20% profit. Find the cost price." Extended

Spot it asks for the cost price.

  1. SP = 120% of CP.
  2. ÷ 1.2 (reverse percentage).

For example $96 at 20% profit

  1. \(96\div1.2=80\)

Answer $80

Mistakes to avoid

  • ✗ % profit = profit ÷ selling price✓ profit ÷ COST priceThe cost price is the 100%.
  • ✗ Comparing offers by their percentages✓ compare the final pricesThe starting prices differ.
  • ✗ "Second half price" = 50% off✓ 25% off two itemsOnly the second item is halved.
  • ✗ Income tax on the whole salary✓ tax only the part above the allowanceThe allowance is tax free.

Summary

  • Profit and loss percentages are out of the COST price.
  • Discount percentages are out of the MARKED price.
  • Tax is an increase; a discount or loss is a decrease: use multipliers.
  • Plans: total = deposit + all instalments.
  • Cost price from a selling price: a reverse percentage (divide).

If you see …

  • "percentage profit"profit ÷ cost × 100.
  • "percentage discount"discount ÷ marked × 100.
  • "which is cheaper"work out each price.

Practice

Practice: easy, medium, hard

Easy

1
Bought for $45, sold for $63. Percentage profit?
Show working
  1. \(\tfrac{18}{45}\times100=40\)

Answer 40%

2
Marked $120, sold for $96. Percentage discount?
Show working
  1. \(\tfrac{24}{120}\times100=20\)

Answer 20%

3
A trader buys a radio for $350 and wants a 16% profit. Selling price?
Show working
  1. \(350\times1.16=406\)

Answer $406

Medium

1
Which is cheaper: $80 with 25% off, or $72 with 15% off? Give the cheaper price.
Guide me

What is it asking? the cheaper price

What do we know? two offers

Plan work out both prices

Show working
  1. \(80\times0.75=60\)
  2. \(72\times0.85=61.20\)

Answer $60 (the $80 one)

2
Pay is $1500 plus 4% commission on $20 000 of sales. Total pay?
Guide me

What is it asking? total pay

What do we know? basic + 4% of sales

Plan 1500 + 800

Show working
  1. \(0.04\times20\,000=800\)
  2. \(1500+800=2300\)

Answer $2300

3
A $50 bill has 10% tax added, then a 10% discount. Final amount?
Guide me

What is it asking? the final amount

What do we know? +10% then −10%

Plan multiply the multipliers

Show working
  1. \(50\times1.1\times0.9=49.50\)

Answer $49.50

Hard

1
A laptop costs $640 cash, or a 25% deposit and 12 payments of $48. What percentage more is the plan?
Guide me

What is it asking? % extra

What do we know? deposit 25% of 640; 12 × 48

Plan total − cash, ÷ cash × 100

Show working
  1. deposit \(=160\)
  2. payments \(=576\)
  3. total \(=736\); extra \(=96\)
  4. \(\tfrac{96}{640}\times100=15\)

Answer 15%

PartAnswer / workingMarksGuidance
\(160+12\times48\) [= 736]M1
\(\tfrac{736-640}{640}\times100\)M1
15A1
Total: 3 marks
2
A seller buys 50 pens at $2.40 each. She sells 90% of them for $3.20 each. Percentage profit?
Hint
How many pens does she sell?
Guide me

What is it asking? % profit

What do we know? 50 pens bought, 45 sold

Plan total cost, total takings, then profit ÷ cost

Show working
  1. cost \(=50\times2.40=120\)
  2. takings \(=45\times3.20=144\)
  3. \(\tfrac{24}{120}\times100=20\)

Answer 20%

PartAnswer / workingMarksGuidance
120 and 144M1M1
20A1
Total: 3 marks
3
A toy sells for $96, making a 20% profit on the cost price. Find the cost price.
Guide me

What is it asking? the cost price

What do we know? $96 = 120% of CP

Plan ÷ 1.2

Show working
  1. \(96\div1.2=80\)

Answer $80

PartAnswer / workingMarksGuidance
\(96\div1.2\)M1
80A1
Total: 2 marks

Can I do this?

Anything unticked: revise that section first. All ticked: take the quiz.

Quiz

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