When an object is sold for more money than it was bought for,
the difference is called profit.
John bought a CD for £12 and sold it for £15.
What was his profit?
When an object is sold for less money than it was bought for,
the difference is called loss.
This is really a negative profit.
John bought a CD for £12 and sold it for £10.
Did he make a profit?
This is the profit or loss expressed as a percentage of the original price.
John bought a CD for £12 and sold it for £15.
What was his profit as a percentage of the price he paid?
Step 1: Find the profit
\[ \text{Profit} = 15 - 12 = 3 \]Step 2: Use the percentage profit formula
\[ \text{Percentage profit} = \frac{\text{Profit}}{\text{Cost price}} \times 100 \] \[ = \frac{3}{12} \times 100 \] \[ = 25\% \]John made a 25% profit.
Value Added Tax (VAT) is added to most goods and services.
On 4th January 2011, the rate of VAT changed from 17.5% to 20%.
A scooter is advertised at £350.
VAT is an additional 17.5%.
What is the total cost including VAT?
Step 1: Find the VAT amount
\[ \text{VAT} = 17.5\% \times 350 \] \[ = 0.175 \times 350 = 61.25 \]Step 2: Add VAT to the original price
\[ \text{Total cost} = 350 + 61.25 = 411.25 \]The scooter costs £411.25 including VAT.
A washing machine is advertised at £350, ex VAT.
VAT is an additional 20%.
What is the total cost including VAT?
Step 1: 20% of £350
\[ 20\% = \frac{20}{100} = \frac{1}{5} \] \[ \frac{1}{5} \times 350 = 70 \]Step 2: Add VAT to the original price
\[ \text{Total cost} = 350 + 70 = 420 \]The washing machine costs £420 including VAT.
VATWhen an object appreciates, it increases in value.
Glenfox Lodge was valued at £165,000 on 30th April 2001.
If appreciation is 4% per annum, what is the value of Glenfox Lodge
on 30th April 2003?
Step 1: Appreciation for one year
\[ \text{Increase} = 4\% \times 165000 \] \[ = 0.04 \times 165000 = 6600 \]Step 2: New value after one year
\[ 165000 + 6600 = 171600 \]Step 3: Appreciation for the second year
\[ \text{Increase} = 4\% \times 171600 \] \[ = 0.04 \times 171600 = 6864 \]Step 4: New value after two years
\[ 171600 + 6864 = 178464 \]
The value of Glenfox Lodge after 2 years is
£178,464.
The CRy method is a shortcut for repeated appreciation or depreciation. Instead of increasing or decreasing the value year by year, we multiply the original amount by a single compound factor.
Use + for appreciation and − for depreciation.
This gives the final value in one step, even over many years.
Using the CRy method
Appreciation rate: 4% per year
\[ \text{New value after } y \text{ years} = \text{Original value} \times (1 + r)^y \] \[ = 165000 \times (1 + 0.04)^2 \] \[ = 165000 \times 1.04^2 \] \[ = 165000 \times 1.0816 \] \[ = 178464 \]
The value of Glenfox Lodge after 2 years is
£178,464.
When an object depreciates, it decreases in value.
A car was bought for £10,000 in 1998.
Each year, it depreciated in value by 20%.
What was the car worth 4 years later?
Depreciation rate: 20% per year
\[ \text{Value after 1 year} = 10000 \times 0.8 = 8000 \] \[ \text{Value after 2 years} = 8000 \times 0.8 = 6400 \] \[ \text{Value after 3 years} = 6400 \times 0.8 = 5120 \] \[ \text{Value after 4 years} = 5120 \times 0.8 = 4096 \]After 4 years, the car is worth £4096.
Since the rate of depreciation remains constant at 20%, this can also be done using CRy
Using the CRy method
\[ \text{Value after } y \text{ years} = \text{Original value} \times (1 - r)^y \] \[ = 10000 \times (1 - 0.20)^4 \] \[ = 10000 \times 0.8^4 \] \[ = 10000 \times 0.4096 = 4096 \]Same result: £4096.
Here, the original cost needs to be found.
A television set is sold for £150, including VAT at 17.5%.
Find the cost of the TV without VAT.
Step 1: Understand what £150 represents
\[ 150 = 117.5\% \text{ of the original price} \]Step 2: Convert 117.5% to a decimal
\[ 117.5\% = 1.175 \]Step 3: Divide to find the original price
\[ \text{Original price} = \frac{150}{1.175} \] \[ = 127.66\ldots \]
The cost of the TV before VAT was approximately
£127.66.
Alternatively
Using proportional reasoning
\[ 117.5\% \rightarrow 150 \] \[ 1\% = \frac{150}{117.5} \] \[ 100\% = 100 \times \frac{150}{117.5} \] \[ = 127.66\ldots \]Same result: £127.66.