Percentages In this presentation we will cover the skills required to deal with problems invoving percentages Most problems involve finding the percentage of a value but there are other situations where we will need to apply different processes to find the solution we require

1

Percentages If you require to find the percentage value of a number then the process is straightforward Example

find 5% of 60

STEP 1

multiply the values provided (5 and 60)

STEP 2

divide this answer by 100

STEP 1

5 ร 60 = 300

STEP 2

300 รท 100 = 3 2

Percentages Let’s apply this process to a problem where you want to find the interest earned on an account Example

you invest £1000 at 2.5 % per annum. How much will you have after 1 year?

STEP 1

multiply the values provided (2.5 and 1000)

STEP 2

divide this answer by 100

STEP 1

2.5 × 1000 = 2500

STEP 2

2500 ÷ 100 = 25 Answer: Interest earned = £25 total in bank = £1000 + £25 = £1025

3

Percentages Try the following problems 1. Find 12% of 550 2. Find the interest earned in one year if you invest £5000 at an interest rate of 3% 3. A house bought for £120 000 rises in value by 15% in 3 years. What is the house worth now? 4. A car bought new is said to lose 20% of its value as soon as you drive it away. If you paid £12 000 for a new car, what is it worth 5 minutes later? 5. Find the VAT at 17.5% on an item advertised at £24.50 ExVAT

4

Percentages Answers 1. Find 12% of 550

12 ÷ 100 X 550 =

2. Find the interest earned in one year if you invest £5000 at an interest rate of 3% 3. A house bought for £120 000 rises in value by 15% in 3 years. What is the house worth now? 4. A car bought new is said

5

Percentages You should now have no problem finding the percentage of a value. We will look at further applications of this. Problem

You invest £1000 for two years at an interest rate of 3%. How much do you have after 2 years?

Answer Year 1 1000 × 3 = 3000 ÷ 100 = 30

Total = £1030

Year 2 1030 × 3 = 3090 ÷ 100 = 30.90

Total = £1060.90

Note

total interest earned = £1060.90 - £1000 = £60.90 6

Percentages Try the following problems 1. You invest £5000 at an interest rate of 2% per annum for 2 years. How much will you have after 2 years and what will be the total interest earned 2. You invest £2500 at an interest rate of 2.5% per annum for 2 years 3. You invest £1000 at 4% interest per annum for 3 years

7

Percentages Try the following problems 1. You invest £5000 at an interest rate of 2% per annum for 2 years. How much will you have after 2 years and what will be the total interest earned 2. You invest £2500 at an interest rate of 2.5% per annum for 2 years 3. You invest £1000 at 4% interest per annum for 3 years

8

Percentages What happens if something drops in value by a stated percentage. For example a car Example

A car costing £10 000 drops in value by 20%. What is it worth now?

Method 1 find 20% of £10 000 and subtract it 20 × 10 000 ÷ 100= 2 000 £10 000 - £2 000 = £8 000 Method 2

Loses 20% so still worth 80%. Find 80% 80 × 10 000 ÷ 100 = 8 000 answer = £8 000

You can use either method, stick to the one you are happiest with. You may use either method depending on the problem. 9

Percentages Try the following problems 1. An item with a value of £1500 drops in value by 10% 2. A car bought for £8000 loses 12% of its value in the first year and then a further 10% in the following year 3. Shares that were worth £8.40 each have fallen by 60% during the recession. What are they worth now? 4. Someone’s salary is £24 600 per annum. If the total tax they pay on this is 21% how much do they have left?

10

Percentages Try the following problems 1. An item with a value of £1500 drops in value by 10% 2. A car bought for £8000 loses 12% of its value in the first year and then a further 10% in the following year 3. Shares that were worth £8.40 each have fallen by 60% during the recession. What are they worth now? 4. Someone’s salary is £24 600 per annum. If the total tax they pay on this is 21% how much do they have left?

11

Percentages Finding the percentage value has been straightforward. Using it to complete the problem is called â€˜problem solvingâ€™. When you have to problem solve always take it spep by step. In the previous problems we stuck to finding the percentage value then returned to the question to decide how we use it to solve the problem. We will now move on to problems that are different to the ones we have been solving so far

12

Percentages Comparing values and expressing one as a percentage of another A good example of this is one where we look at exam results and turn it into a percentage result Example Darren sat a class test and achieved a mark of 8 out of 40. We want to express this as a percentage so we can state the grade he achieved

13

Percentages Darren sat a class test and achieved a mark of 8 out of 40. We want to express this as a percentage so we can state the grade he achieved How would you normally write down a test result? 8 40 This written like a fraction. A fraction tells you to multiply by the top value but divide by the bottom value

14

Percentages When we convert a fraction or decimal to a percentage we simply multiply by 100 For a fraction the top number multiplies with 100 and the bottom number divides. Let’s look at Darren’s result

8 40

×

÷

= 20% Don’t forget to put the % sign in as you now have a percentage 15

Percentages Examples 1. Joyce tried her hand at archery and hit the target 16 times from 25 attempts. What is the percentage success rate? 16 25

16 × 100 ÷ 25

= 64%

2. As part of a job interview Anwar sat a test and scored 42 from a possible 60. What is this expressed as a percentage? 42 60

42 × 100 ÷ 60

= 70%

There will be problems where the answer has a decimal part. You must decide how accurate the solution needs to be 16

Percentages Try the following problems 1. In a vote to be class rep John polled 5 votes from a class size of 20. What percentage of the vote did John get? 2. Janine and Amy are trying to raise £500 for charity. So far they have reached a total of £310. What percentage of the £500 have they still to achieve 3. Stephanie has an annual salary of £15 500. If she is awarded a rise £200 what is this expressed as a percentage rise? 4. A street seller started with 120 bags of sweets and manages to sell 96. What percentage of bags is the street seller left with? 17

Answers - Percentages Answers 1. In a vote to be class rep John polled 5 votes from a class size of 20. What percentage of the vote did John get? 5 x 100 ÷20 = 20% 2. Janine and Amy are trying to raise £500 for charity. So far they have reached a total of £310. What percentage of the £500 have they still to achieve 310 x 100 ÷ 500 = 62% - STILL TO ACHIEVE 38% 3. Stephanie has an annual salary of £15 500. If she is awarded a rise £200 what is this expressed as a percentage rise? 200 x 100 ÷ 1500 = 1.29% 4. A street seller started with 120 bags of sweets and manages to sell 96. What percentage of bags is the street seller left with? 96 x 100 ÷ 120 = 80% 18

Percentages There are some cases where we have the final value after adding or subtracting the calculated percentage value, but do not know the original value For example, a television cost ÂŁ423 including VAT. Can we find the cost of the TV before VAT at 17.5% was added? Remember the process you carried out when you were finding 17.5% and adding to the price of an item You multiplied the item value by the percentage value and divided by 100. In the case of VAT you added this amount onto the cost of the item 19

Percentages For example, a television costs £423 including VAT. Can we find the cost of the TV before VAT at 17.5% was added? To find the cost before VAT was added we must reverse the process. This is not quite as straightforward as you think. Many people would find 17.5% of £423 and subtract it. THIS DOES NOT WORK! Method:

multiply the value by 100. Now divide this by 100 + the percentage value

Original cost = £423 × 100 ÷ 117.5 = £360 excluding VAT Not sure! Then find cost by adding 17.5% VAT to £360 20

Percentages Summary

to find original value before the percentage was added we: Multiply the value by 100 Divide by 100 added to the percentage rise

Another example A house increased in value by 12.5% and is now worth £202 500. What value did it rise from? Original value = 202 500 × 100 ÷ 112.5

= £180 000

21

Percentages Try solving the following problems 1. A primary school role has increased by 5% to 189 pupils. How many did it have the year before? 2. The attendance at a sporting event has increased by 15% from the previous year to 7176. How many attended in the previous year 3. The annual rainfall for a specific region increased by 4 % on the previous year (probably Scotland) to 83.2 inches. What was the total rainfall in the previous year?

22

Percentages Answers 1. A primary school role has increased by 5% to 189 pupils. How many did it have the year before? 2. The attendance at a sporting event has increased by 15% from the previous year to 7176. How many attended in the previous year 3. The annual rainfall for a specific region increased by 4 % on the previous year (probably Scotland) to 83.2 inches. What was the total rainfall in the previous year?

23

Percentages You must be careful about whether the value given is the result of a percentage rise or drop. If the new value you have is the result of a drop in the original value then we change the process slightly. Example

A car has dropped in value by 20% in one year to a value of £6400. What value did it drop from?

Instead of dividing by the percentage added to 100 we divide by THE PERCENTAGE SUBTRACTED FROM 100 Original value = 6400 × 100 ÷ 80

= £8000 24

Percentages Try solving the following problems 1. A primary school role has decreased by 5% to 190 pupils. How many did it have the year before? 2. The attendance at a sporting event has decreased by 15% from the previous year to 4641. How many attended in the previous year 3. The annual rainfall for a specific region decreased by 8.5% on the previous year (probably not Scotland) to 54.9 inches. What was the total rainfall in the previous year?

25

Percentages Answers 1. A primary school role has decreased by 5% to 190 pupils. How many did it have the year before? 190 X 100 รท 95 = 200 PUPILS 2. The attendance at a sporting event has decreased by 15% from the previous year to 4641. How many attended in the previous year 4641 x 100 รท 85 = 5460 3. The annual rainfall for a specific region decreased by 8.5% on the previous year (probably not Scotland) to 54.9 inches. What was the total rainfall in the previous year? 54.9 x 100 รท 91.5 = 60 26

Percentages You now have the fundamental skills to understand and manipulate percentage problems. It should not take you long to refresh your memory on this when you need to apply these skills in the future. If you wish to complete further practice on dealing with percentages then ask your tutor for access to more problem solving questions

27