Unitary method
So far we've considered finding a certain percentage of an amount, now we will look at how to find the total amount after being given a percentage of it.
For example, if $40%$40% of a dog's weight is $10$10 kg, how heavy is the dog? We can use what is called the unitary method here, which means finding out what one unit of something is first. In our case we're going to find what one percent is first. And then multiplying to find the required amount, in our case $100%$100%.
Let's see how this works in the following diagram:
There are three stages in this method:
- Start with the amount that we know and its related percentage.
- Convert both numbers (percentage and amount) to what $1%$1% would be.
- Multiply by $100$100 to get $100%$100%, which is the whole.
Exploration
Sometimes questions will involve starting amounts over $100%$100%.
For example, say we knew that a bank account was worth $\$770$$770 after $10%$10% interest was paid. To find the original $100%$100% we would first need to figure out what the starting percentage is. If $\$770$$770 is the amount after interest then it equals $100%+10%=110%$100%+10%=110%.
Then to find the total amount we would follow step $2$2 above and divide everything by $110$110 to get $1%$1%, and then finally multiply by $100$100 to get the whole amount.
So we can find the original amount as follows:
| Amount and interest: | $110%$110% | $=$= | $\$770$$770 |
| $\div110$÷110 |  |
$\div110$÷110 | |
| Unitary step: | $1%$1% | $=$= | $\$7$$7 |
| $\times100$×100 | |
$\times100$×100 | |
| Original amount: | $100%$100% | $=$= | $\$700$$700 |
Practice questions
Question 1
$12%$12% of a quantity is $24$24.
What is $1%$1% of the quantity?
Hence find the total quantity.
Question 2
Find the number if $10%$10% of the number is $12$12.
Question 3
$6%$6% of a number is $36$36.
Find $1%$1% of the number.
Hence find $36%$36% of the number.
QUESTION 4
$870%$870% of a number is $696$696. What is the number? Write your answer in simplest form.
Algebraic method
Exploration
Algebra is an alternate strategy which can be used to find the total amount. Consider the following example. An item in a shop is priced for sale at $\$50$$50. It is known that all items in the shop are marked-up by $25%$25% from the cost price. Find the cost price, $c$c.
If we knew what $c$c was, we would multiply it by $125%$125% in order to increase it by $25%$25% and we would get the selling price as the answer. So as an equation that looks like this:
$c\times1.25=50$c×1.25=50
We can then solve this equation to find $c$c.
| $c\times1.25$c×1.25 | $=$= | $50$50 |
| $c$c | $=$= | $\frac{50}{1.25}$501.25 |
| $=$= | $40$40 |
Hence, the cost price was $\$40$$40.
QUESTION 5
Peter and Luigi are building a new supercomputer. When finished, the supercomputer will cost $\$195300$$195300. The price of the supercomputer is $2%$2% higher than the price when the original plans were made.
Let $p$p represent the original price of the supercomputer in dollars, before the plans were changed.
By first converting the percentage to a decimal, write an algebraic expression in terms of $p$p for the amount of cost added.
Using your answer from part (a), write and solve an equation to find $p$p, the price of the supercomputer before the plans were changed.
Write your answer to two decimal places.
Hence, the price of the supercomputer before the change of plans was $\editable{}$ dollars.
Multiple percentage changes
The process of finding the final(net) amount after a series of percentage increases/decreases can be reversed through division to find the original amount.
Worked example
Solve: A computer is on sale for $\$957$$957 after being discounted by $25%$25% and then a further $12%$12%, what was the original selling price of the computer?
Think: To obtain the sale(net) price the original price would need to be multiplied by $0.75$0.75 to discount by $25%$25% and also multiplied by $0.88$0.88 to discount by $12%$12%. Set this up as an equation and then undo the multiplications by division.
Do:
| $\text{Net price}$Net price | $=$= | $\text{original price}\times0.75\times0.88$original price×0.75×0.88 | |
| $\$957$$957 | $=$= | $\text{original price}\times0.75\times0.88$original price×0.75×0.88 | Substitute in sale price |
| $\therefore\text{original price}$∴original price | $=$= | $\$957\div0.75\div0.88$$957÷0.75÷0.88 | Find the original price by dividing both sides by the discount multipliers |
| $=$= | $\$1450$$1450 |
The original selling price of the computer was $\$1450$$1450.
Practice question
Question 6
A guitar is on sale for $\$403.69$$403.69 after a discount of $21%$21% and an additional discount of $27%$27%.
Fill in the gaps below:
Sale price$=$=Original price$\times$×$\editable{}\times\editable{}$×
What overall percentage change is this equivalent to?
Find the original selling price of the guitar.
Round your answer to the nearest dollar.
Question 7
The population of a town is $64609$64609 after two years of increasing at $2%$2% per year. Find the population of the town two years ago.
Find the population of the town two years ago.
Round up to the nearest person.