Finding $10%$10%, $1%$1%, $5%$5% without the use of a calculator
Being able to find some percentages quickly by hand can come in handy in every day life. Calculating a tip, a discount at a sale or the tax to be added to an item.
- To find $10%$10% simply move the decimal one place to the left.
- To find $1%$1% simply move the decimal two places to the left.
- To find $5%$5%, find $10%$10% and then divide it by two.
We can also use combinations of $10%$10%, $5%$5% and $1%$1% to find other percentages.
Practice questions
Question 1
A salesperson earns a $13%$13% commission on their sales each week. In one week, their sales amounted to $\$640$$640.
First, find $10%$10% of their total sales.
Now find $3%$3% of their total sales. Leave your answer to two decimal places.
Hence, find the total commission they made this week. Write your answer to two decimal places.
Question 2
We want to find $45%$45% of $5$5 hours.
How many minutes are there in $5$5 hours?
What is $10%$10% of $300$300 minutes?
What is $5%$5% of $300$300 minutes?
Hence find $45%$45% of $300$300 minutes.
Goods and services tax (GST)
The Howard Government introduced Goods and Services Tax, or GST, in 2000. It is a value added tax, usually of $10%$10%, on most goods and services transactions. In other words, the prices businesses charge you include an additional $10%$10% of the original price as a GST amount. For example, if the original price of an item was $\$10$$10, the GST on this item is $\$1$$1 since $10%$10% of $\$10$$10 is $\$1$$1. The price you would be charged would be $\$11$$11 since $10+1=11$10+1=11. For tax reasons, businesses in particular need to keep track of how much GST they pay and receive, so it is important to be able to calculate prices before and after GST, as well as the amount and rate of GST.
To calculate the price of an item after GST is added we multiply by $1.1$1.1, since we are finding $110%$110%.
To calculate the original price of an item before GST if we know the final price including GST we divide by $1.1$1.1.
Practice questions
Question 3
Calculate the GST-included sales price for an item with an original price of $\$20$$20.
Question 4
The sales price of an item, including GST, is $\$40$$40. Calculate the price of the item without GST.
Round your answer to the nearest cent.
Appreciation
When an object or investment is said to appreciate, this means it increases in value by a given percentage.
For example, property prices in your suburb might have appreciated by $0.8%$0.8% over the last year. We can use the average annual appreciation rate to predict the value at the end of each year by repeated percentage increases, that is repeatedly multiplying by $(1+0.008)$(1+0.008). For a house valued at $\$580000$$580000, we could calculate the estimated value after $t$t years as follows:
| Value after $1$1 year | $=580000\times(1+0.008)$=580000×(1+0.008) |
| Value after $2$2 years | $=580000\times(1+0.008)\times(1+0.008)=580000\times(1+0.008)^2$=580000×(1+0.008)×(1+0.008)=580000×(1+0.008)2 |
| Value after $3$3 years | $=580000\times(1+0.008)\times(1+0.008)\times(1+0.008)=580000\times(1+0.008)^3$=580000×(1+0.008)×(1+0.008)×(1+0.008)=580000×(1+0.008)3 |
Observing the pattern above we can create a formula which will encounter again when we look at compound interest.
$A=P\left(1+r\right)^t$A=P(1+r)t
where:
$A$A is the future value of the item (the initial value plus the amount the item or investment appreciated by)
$P$P is the principal (the initial value of the item or investment)
$r$r is the appreciation rate per time period expressed as a decimal
$t$t is the number of time periods
Worked example
Example 1
A collectable coin valued at $\$50$$50 appreciates in value at an average of $4%$4% per year. Estimate its value in $3$3 years time.
| Value after three years: | $A$A | $=$= | $A\times\left(1+r\right)^t$A×(1+r)t |
| $=$= | $\$50\times\left(1+0.04\right)^3$$50×(1+0.04)3 | ||
| $=$= | $\$56.24$$56.24 |
Practice question
Question 5
A house was valued $6$6 years ago to be worth $\$548000$$548000. Its value appreciated at $5.2%$5.2% p.a. What is its appreciated value? Give your answer correct to the nearest dollar.
Depreciation
If something depreciates in value over time, this means it is reducing in value by a given average percentage at regular time periods. We can use the average annual depreciation rate to predict the value at the end of each year by repeated percentage decreases, that is repeatedly multiplying by $1-r$1−r, where $r$r is the depreciation rate.
Thus, we could form a similar formula to the appreciation formula given above:
$A=P\left(1-r\right)^t$A=P(1−r)t
where:
$A$A is the future value of the item (the initial value plus the amount the item or investment appreciated by)
$P$P is the principal (the initial value of the item or investment)
$r$r is the depreciation rate per time period expressed as a decimal
$t$t is the number of time periods
Worked example
Example 2
An item of jewellery is purchased for $\$1025$$1025 and it has an average depreciation in value by $5%$5% each year. How much would it be worth two years after purchase?
| Value after two years: | $A$A | $=$= | $P\left(1-r\right)^t$P(1−r)t |
| $=$= | $\$1025\times\left(1-0.05\right)^2$$1025×(1−0.05)2 | ||
| $=$= | $\$925.06$$925.06 |
Practice question
Question 6
The value of my car depreciated $5%$5% this year.
If its value was $\$4000$$4000:
By how much did it depreciate?
What is the current value of the car?
Inflation
Inflation is a very similar concept to appreciation, but instead of looking at the increase in value of an investment, we instead examine the increase in the prices of goods and services in an economy over time.
We refer to inflation as the rate of inflation and it is expressed as a percentage. Often, the rate of inflation in a particular country is reported as the average annual inflation rate. And we can use the appreciation (compound interest) formula once again to predict future prices.
If we know the average rate of inflation over a period of time and we know the current value of something, we can also use the formula and rearrange to predict what its value would have been in the past. See the second example below.
Practice questions
Question 7
The price of pens and pencils in 1996 was $\$5$$5. If the value inflated at an average rate of $\frac{16}{5}%$165% per annum, what would the price have been in 2005?
Question 8
An item costs $\$15000$$15000 today. Solve for its price, $P$P, $4$4 years ago if the inflation rate was an average of $3.5%$3.5% per annum. Give your answer to the nearest dollar.
Enter each line of working as an equation.