Percentages
Neil Trivedi
Teacher
Contents
Percentages
Interest can be defined as the money earned or paid on an initial amount of money, also known as the principal amount and denoted by , where represents the amount and represents the time at which we are observing the amount of money we have which in this case means the beginning. The amount of interest depends on the interest rate, denoted by , and the the length of time, denoted by and usually in years, that the money is invested or borrowed for. This can also apply to other situations such as the value of an object increasing or decreasing over time.
Appreciation happens when an amount increases in value over time. Examples include interest being added to an account or population growth.
Depreciation happens when an amount decreases in value over time. Examples include a car losing its value as they are used or an electronic device losing value as newer models are released.
In GCSE Maths, we mainly work with two types of interest, which are simple interest and compound interest.
Simple Interest
Simple interest is when interest is calculated based only on the initial amount, . Any interest that is added does not earn further interest. This means that the amount of interest earned each year is fixed. When working with simple interest questions, we follow these steps:
1) Find the interest earned in year.
2) Multiply this interest by the number of years to obtain the total interest.
3) Add the total interest to the initial amount to obtain the final amount.
Example 1:
I put into into an account with simple interest per annum (per year). How much money will be in the account after years?
Since this is simple interest, the interest earned each year is the same.
Step 1: Work out how much interest is earned in year.
To work out the interest, we work out of , which is equivalent to calculating .
So, is earned each year.
Step 2: Work out the total interest earned.
Total interest earned after years:
Step 3: Work out the final amount by adding the total interest onto the initial amount.
Therefore, after years, there will be in the account.
Compound Interest
Compound interest is interest calculated not only on the initial amount, , but also on any interest that has been added in previous years. This means interest is calculated on the new amount each year, so the amount increases/decreases by the same percentage each year. In practice, simple interest is rarely used. So, unless it’s specified in the question, we must assume that we are dealing with compound interest.
To find a new amount with compound interest, we multiply the initial amount by a multiplier. The multiplier, written as a decimal, represents the new amount as a proportion of the initial amount, which we treat as .
An increase of (appreciation) means the new amount is of the initial amount.
Similarly, a decrease of (depriciation) means the new amount is of the initial amount.
To find the multiplier, we convert this percentage into a decimal by dividing by .
Let’s look at an example of how compound interest works and then derive a general formula.
Example 2:
I put into an account with interest per annum. How much money is there in the account after years?
Step 1: Find the multiplier as a decimal.
An interest rate means the initial amount increases by .
So, after year, the new amount in the account is of the initial amount. We convert this to a decimal to obtain the multiplier.
Step 2: Use this multiplier to find the amount in the account after years.
Each year, the new amount becomes of the amount at the end of the previous year, so we multiply by each time. We start with the initial amount, which is .
For the first year,
For the second year,
For the third year,
Therefore, after years, there is in the account.
Notice that we multiplied by three times. This can be summarised as multiplying by . The power of represents the number of years that has elapsed. So, step 2 can be shortened to one line of working:
Note: to get , we technically did which came from doing .
Compound Interest
For compound interest, the amount after years can be found using the following formula:
where:
is the final amount.
is the initial amount.
is the interest rate (or percentage increase/decrease).
is the number of years.
The term represents the multiplier, which comes from doing , and raising this to the power of indicates that we’re multiplying the initial amount, by the multiplier times to obtain the final amount, .
Example 3:
Abigail bought a car for . Each year, the value of the car depreciates by . Work out the value of the car at the end of years.
Step 1: Find the multiplier as a decimal.
A depreciation of means the value of the car decreases by each year.
We convert this to a decimal to obtain the multiplier.
Step 2: Work out the value of the car after years using this multiplier.
Each year, the value of the car becomes of what it was at the end of the previous year. Since we are finding the value after years, we multiply the initial value by four times, which can be written as .
Therefore, after years, Abigail’s car is worth .
Dealing with Unknowns
Sometimes, we are given the final amount and are asked to find one of the other values such as the initial amount, the percentage change or the time period.
Example 4:
The population of foxes decreased by each year for years, before dropping to foxes. How many foxes were there originally?
Step 1: Find the multiplier as a decimal.
The population of foxes is decreasing by each year.
We convert this to a decimal to obtain the multiplier.
Step 2: Set up an equation to illustrate this situation, using this multiplier, and rearrange it to find the initial population.
Each year, the population of foxes becomes of what it was a year ago. Since is the population after years, we multiply the initial population (let’s call it ) by six times, which can be written as . We, therefore, form the following equation.
Dividing both sides by ,
Since we are dealing with population, the initial amount must be a whole number. So,
Therefore, there were initially foxes.
Example 5:
I put into a bank account. After years, with compound interest, I have . What percentage interest did I get each year? Give your answer to decimal place.
When the percentage interest is unknown, we must use our general compound interest formula and rearrange for .
Step 1: Substitute the known values into the compound interest formula.
We are asked to find the percentage interest, . We are given the initial amount, , the final amount , and the number of years elapsed, . Substituting these values into the formula,
Step 2: Solve for by rearranging the equation.
To find , we first divide both sides by , and we get
Cube rooting both sides,
Subtracting from both sides,
Multiplying both sides by to isolate ,
Therefore, the percentage interest earned each year was approximately , to decimal place.
Reverse Percentages
In regular percentage change questions, we are usually asked to find a percentage of an amount or to work out the final amount after a percentage increase or decrease. The final amount is found using
In reverse percentage questions, we work backwards. We are given the final amount and are asked to find the initial amount. To do this, we rearrange the equation above by dividing both sides by the multiplier to give
Example 6:
The price of a laptop is reduced by . The sale price is . What is the normal price?
Step 1: Find the multiplier as a decimal.
The initial price is reduced by .
Therefore, the sale price is of the normal price. We convert this to a decimal to obtain the multiplier.
Step 2: Work out the initial price.
Let the initial price be . Since the initial price has been multiplied by to give the sale price, we can write
To find , we divide both sides by .
So, the initial price of the laptop is .
Example 7:
Amy works in an investment banking company. She started her career as an investment banking analyst. A few years later, she was promoted to investment banking associate, and her salary increased by . Later, she was promoted to vice president, and her salary increased by a further .
As a vice president, Amy’s salary is .
What was Amy’s salary as an investment banking analyst?
Step 1: Find the multipliers as decimals.
At first, Amy’s salary increases by .
Therefore, Amy’s salary as an investment banking associate is of her salary as an investment banking analyst. We convert this to a decimal to obtain the multiplier.
Amy’s salary then increases by a further .
Therefore, Amy’s salary as a vice president is of her salary as an investment banking associate. We convert this to a decimal to obtain the multiplier.
Step 2: Work backwards to find the initial salary.
Let Amy’s salary as an investment banking analyst be . Her salary was first multiplied by , and then by , to give the final salary of . So, we can write
To find , we divide both sides by .
Therefore, Amy’s salary as an investment banking analyst was .
Challenging Question