Percentage Questions

"Find % of something"

remember "of" means $*$. And then treat it as a fraction:

"Find 35% of $70": $\dfrac{35}{100 \div \color{red}{10}} * \dfrac{70 \div \color{red}{10}}{1} = \dfrac{245}{10} = 24.5$

"Find the new amount after a % increase or decrease"

For these questions, its a good idea to add or subtract the percentage from 100%, since the original value is 100%. For example:

"A GCSE textbook is reduced 60% from $100 work out the new amount:"

$(100\% - 60\%) * 100 = \dfrac{40}{100 \div 100} * \dfrac{100 \div 100}{1} = 40$

"Jeff Bezos' net worth was $90 billion until it increased by 20% after Amazon stock went up. Calculate the new amount:"

$(100\% + 20\%) * 90b = \dfrac{120}{100 \div 10} * \dfrac{90 \div 10}{1} = \dfrac{9 * 120}{10} = \dfrac{1080}{10} = 108bn$

Express one number as a percentage of another

Piece of cake. Make one the fraction of the other then multiply by 100!:

"Make 12 a percentage of 101":

$\dfrac{12}{100} * \dfrac{101}{1} = \dfrac{1212}{100} = 12.12\%$

Calculating Percentage Change

This question becomes simple once you learn that: $\text{Percentage Change} = \dfrac{\text{New value - Original Value}}{\text{Original value}} \times 100$

In general, change equals: $\text{New Value - Original Value}$

Whenver you have to use a formula, it might be a good idea to write it and then add in the values.

For example a question might ask:

"For every $10 worth of Iphones sold, Apple makes $13.5 on average. Calculate the percentage profit made":

$\begin{aligned} \text{Percentage Change} &= \text{Percentage Profit =} \dfrac{\text{New value - Original Value}}{\text{Original Value}} * 100 \\ \text{Percentage Profit} &= \dfrac{13.5 - 10}{10 \div 10} * \dfrac{100 \div 10}{1} = 3.5 * 10 = 35 \end{aligned}$

"Finding the original value after a percentage increase or decrease:"

All you need to do is make the new value equal to the total percentage change. So for example:

"Bill Gates is now worth $86bn however a year ago he was worth 20% more, work out how rich he was then:" $\begin{aligned} 86bn &= (100\% - 20\%) \\ \dfrac{86}{\color{red}{80}} &= \dfrac{80\%}{\color{red}{80\%}} = 1\% \\[15mu] \dfrac{86}{80 \div \color{red}{20}} * \dfrac{100 \div \color{red}{20}}{1} &= \dfrac{86 * 5}{4} = \dfrac{430 \div \color{red}{2}}{4 \div \color{red}{2}} = \dfrac{215}{2} = 107.5bn \end{aligned}$ Notice that because he was worth more (+) we have to subtract instead of add. We need to use the opposite sign.

Simple Interest

Simple interest is when the same amount of money is paid over a regular time period. Here is the formula to work it out:

$\begin{aligned} \text{Simple Interest Per Period} &= \text{Initial Amount} \times \text{Interest rate} \\ \text{Total Amount} &= \text{Initial Amount} + \text{Simple Interest Per Period * Number of Periods} \end{aligned}$

Sometimes they will ask you to work out the total amount, sometimes it will be only the profit from simple interest.

For example: "Mr Ex. invests $1069 into a bank account which will pay him 4% a year every year in simple interest. Calculate how much profit from interest he will have after 5 years:"

$\begin{aligned} \text{Simple Interest Per Period} &= 1069 * 4\% = 42.76 \\ \text{Profit from Interest} &= \text{Initial Amount}\color{red}{\text{(- Initial Amount)}}\text{ + Simple Interest Per Period * Number of Periods} \\ \text{Profit from Interest} &= 42.76 * 5 = 213.8 \end{aligned}$

Compound Interest

Compound Interest is when you reinvest interest back into something paying a certain percentage a year.

A super simple way to work this out is to work out the simple interest each year for every year. Such as:

For example "A company buys $1050 worth of shares which return, on average, 15% a year. They reinvest their earnings each year. After 3 years what is the total amount of money they have?"

$\begin{aligned} \text{Amount after 1 year} &= 1050 * (100\%+15\%) = 1207.5 \\ \text{Amount after 2 years} &= 1207.5 * 1.15 = 1388.625 \\ \text{Amount after 3 years} &= 1388.625 * 1.15 = 1596.91875 \end{aligned}$

This method is really slow however, and for those who want to save time you should learn this formula:

$\begin{aligned} A &= P(1 + \dfrac{r}{n})^{nt} \\ A &= \text{Amount}, P = \text{Initial Amount}, r = \text{interest rate as decimal} \\ n &= \text{number of times interest is compounded per year} \\ t &= \text{total time period in years} \end{aligned}$

We can apply the above formula to the last question to get the same answer:

$\begin{aligned} A = 1050 * (1 + \dfrac{0.15}{1})^{1 * 3} = 1050 * 1.15^{3} = 1596.91875 \end{aligned}$