vms Year 9 Mathematics Percentages

Learning Intentions Pupils should be able to: express one quantity as a percentage of another find a percentage of a quantity increase or decrease a quantity by a given percentage calculate simple Interest calculate the new amount with interest added calculate the principal, time or interest rate

Percentages Percentage means per cent or out of 100 If 60% of people own a pet, then, on average, 60 people in every 100 own a pet Percentage are often used in shops to indicate a sale - for example, 20% off In banks to indicate interest rates – loans at 7%

Writing Fractions Before changing numbers into percentages, it is important to be able to write one number as a fraction of another. For example, write: 16 marks out of 20 20cm out of 1.5m 15min as a fraction of 3 hours 5.6 cm as a fraction of 12 cm It is important to make sure that the two quantities are in the same units!

Making it a percentage To change a fraction into a percentage, we simply multiply by 100%. For example, change the following into percentages: 16 marks out of 20 20cm out of 1.5m 15min out of 3 hours 5.6 cm out of 12 cm

… and Decimals To change a decimal into a percentage we also multiply by 100% For example, change into a percentage Do these too…

Changing back We need to be able to change percentages back to fractions Remember % means out of 100, so 4% means 4 out of 100 We can write this as a fraction = Change these into fractions 28% 114% 5.5% 7½%

… and to Decimals To change a percentage to a decimal we also divide by 100 For example 4% = Try converting these into decimals too 42% 75.5% 106 2/3% You may have to round some decimals!

To find the percentage of a quantity, we simply change the percentage into a fraction and then multiply Remember % means out of 100 (divided by 100) For example, find 20% of £150 More examples, find: 24% of % of % of £ % of 7kg Finding the percentage Yes – you can use your calculator! - But not your phone!

Percentage Increase Now that we can find a percentage of a number we can add that value to find a percentage increase. For example, increase £140 by 15% =£ % of 140 =140 + =£

Percentage Decrease Now that we can find a percentage of a number we can subtract that value to find a percentage decrease. For example, decrease £120 by 20% =£120 – 20% of 120 =120 – =£

Another Way? We can also work out percentage increase or decrease by calculating the new percentage and then working out the new percentage. For example, Increase £140 by 15% Since the original amount represented 100%, if we increase it by 15% we now have 115%. =115% of £140 =

Examples Increase £3000 by 16% Decrease £400 by 20%

It’s Sale Time! Calculate the new prices £700 £350 £105 £10.00 £54 £70 25% 15% 10% 15% 8% 5% ££££££

Interest If you put money in a bank, they will use it for other purposes and pay you for that use –they pay you interest! If you borrow someone else's money, you must pay a fee for this use – you pay them interest! Interest is normally given as a percentage of the amount borrowed. An interest rate from the bank might be 5%. That means that for every year you keep your money in the bank they will pay you 5%.

Calculating Interest If £400 is invested at a 5% interest rate how much interest is earned each year? A one year loan of £650 is to be paid back at 8% interest. How much has to be repaid?

Simple Interest Simple Interest (I) is calculated only on the initial money invested. This money is called the Principal (P) and it is invested at Rate (R) over a period of Time (T). Simple interest calculations can be useful in giving you a rough idea of how much interest you earn. We can use the following formula to calculate the simple interest:

Calculating Find the simple interest earned when £2000 is invested for 5 years at an annual interest rate of 4½%.

How much do I have? So I put £500 into the bank for 3 years at 5% interest pa. We can now calculate the interest the bank will add on, but how much do I now have? To find the Amount (A) we need to add the Interest (I) to the Principal we started with. A = P + I

So how much? I put £500 into the bank for 3 years at 5% interest pa, how much will I have? I = = A =P + I A=

Calculations! Find the simple Interest on each of the following and the amount at the end of the given period of time 1. £100 invested for 2 years at 2% interest per annum. 2. £150 invested for 2 years at 12% interest per annum. 3. £500 invested for 3 years at 9% interest per annum. 4. £1000 invested for 4 years at 10% interest per annum. 5. £1500 invested for 3 years at 7% interest per annum. 6. £2000 invested for 3 years at 4% interest per annum.

Swap Shop! We sometimes need to be able to calculate the Principal (P), the interest Rate (R) or the Time (T) using our formula. For example, find the principal that will earn £400 simple interest in 2 years at 5% pa.

More examples What is the interest Rate (R) if the cost of borrowing £300 for 4 years is £180?