Preparing for KS3- Probability, Formulae and Equations, Ratio and Proportion, Percentages of Quantities and Fractions of Quantities.
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The document explains various concepts of probability, ratios, and proportions, highlighting methods to express probability as fractions, decimals, or percentages. It provides examples of calculating probabilities and understanding ratios, including direct proportion and simplifying ratios. Additionally, it discusses practical applications like cooking and sharing amounts based on these mathematical principles.
Preparing for KS3- Probability, Formulae and Equations, Ratio and Proportion, Percentages of Quantities and Fractions of Quantities.
- 1. ProbabilityHow to express Probability.
- 2. How to Express it...There are two main methods of expressing probability:Probabilities are most commonly shown as fractions:The probability of getting 'tails' when you toss a coin is a 1 in 2 chance, or1/2The probability of getting a 3 when you roll a dice is a 1 in 6 chance, or 1/6Probabilities can also be shown as decimals or percentages:A probability of 1/2 can also be shown as 0.5 or 50%A probability of 3/4 can also be shown as 0.75 or 75%The results of an experiment can give different probabilities to those you expect. If you throw a dice 6 times, you would expect to get one six as the probability of throwing a six is 1 in 6. But in fact you might get more than one six or no sixes at all!
- 3. Scales...Probability scalesProbabilities can be shown on a scale between 0 (impossible) and 1 (certain).For example, imagine a shopping bag contains three bananas and nothing else:The probability of reaching into the bag and pulling out a banana is 1 (certain), as there is nothing else in the bag.The probability of reaching into the bag and pulling out an apple is 0 (impossible), as there are no apples in the bag.
- 4. Working it all out...Working out probabilityWork out the probability of throwing an even number on a dice.Count up the total number of possible results.When throwing a dice, for example, there are 6 possible numbers the dice can land on (1, 2 ,3 ,4, 5, 6).Then count up the number of results you are interested in.In this example, you are only interested in throwing a 2, 4 or 6 (all the even numbers on a dice). So you are interested in 3 numbers.The probability of getting an even number on a dice is 3 chances out of 6 chances which you write as 3/6.3/6 is the same as 1/2 which can also be expressed as 'half', 0.5 or 50%.
- 5. Formulae and equations.Function Machines and More.
- 6. Function Machines...Function machines have three parts. You may have questions which ask you to find the numbers coming in and out of the machine, or even the missing function. If you are asked which number goes into the machine, you will need to know the inverse function:The opposite of adding is subtracting, The opposite of multiplying is dividing et cetera.Some functions may have more than one operation. Just put a number in and carry out the operations in order: For example: (6x2)-3=9.Top Tip!
- 7. Formulae and Equations...You need to work out what the symbol or letter stands for every time. Use numbers to help you, and say it as a sentence: “What added to 2 makes 6?” etc.If you are finding it difficult to work out the value of a letters write it out again using a box instead of a letter. You can them try different numbers un the box to see if the equation works.Eg. 3t-1=14Add the one, (do the opposite.) Divide the number by three, to get what one ‘t’ equals.A number before a letter or a symbol indicates how many of them are there. If just one is there, no numbers will appear.Remember!
- 8. Tricky Equations...If it is a complicated equation, you may need to work it out step my step.4y+3=15Do the opposite! Minus the three from the fifteen. You now have twelve, what do you do now? Divide this by 4 to find one y- SIMPLE!Simples!
- 9. RATIO AND PROPORTION.Simplifying, Doubling- fantastic!1:27= 1/27.
- 10. Direct Proportion...Understanding direct proportionTwo quantities are in direct proportion when they increase or decrease in the same ratio. For example you could increase something by doubling it or decrease it by halving.If we look at the example of mixing paint the ratio is 3 pots blue to 1 pot white, or 3:1.But this amount of paint will only decorate two walls of a room. What if you wanted to decorate the whole room, four walls? You have to double the amount of paint and increase it in the same ratio.If we double the amount of blue paint we need 6 pots.If we double the amount of white paint we need 2 pots.Look at this graph:Two quantities which are in direct proportion will always produce a graph where all the points can be joined to form a straight line.
- 11. Using it...Using direct proportionUnderstanding proportion can help in making all kinds of calculations. It helps you work out the value or amount of quantities either bigger or smaller than the one about which you have information. Here are some examples:Example 1: If you know the cost of 3 packets of batteries is £6.00, can you work out the cost of 5 packets?To solve this problem we need to know the cost of 1 packet.If three packets cost £6.00, then you divide £6.00 by 3 to find the price of 1 packet. (6 ÷ 3 = 2)Now you know that they cost £2.00 each, to work out the cost of 5 packets you multiply £2.00 by 5. (2 x 5 = 10)So, 5 packets of batteries cost £10.00 Example 2: You've invited friends round for a pizza supper. You already have the toppings, so just need to make the pizza base. Looking in the recipe book you notice that the quantities given in the recipe are for 2 people and you need to cook for 5!Pizza base - to serve 2 people: 100 g flour 60 ml water 4 g yeast 20 ml milk pinch of saltThe trick here is to divide all the amounts by 2 to give you the quantities for 1 serving. Then multiply the amounts by the number stated in the question, 5.For 1 serving, divide by 2: 100 g ÷ 2 = 50 g 60 ml ÷ 2 = 30 ml 4 g ÷ 2 = 2 g 20 ml ÷ 2 = 10 mlFor 5 servings, multiply by 5: 50 g x 5 = 250 g 30 ml x 5 = 150 ml 2 g x 5 = 10 g 10 ml x 5 = 50 ml
- 12. Simplifying Ratios...Simplifying ratiosWe can often make the numbers in ratios smaller so that they are easier to compare. You do this by dividing each side of the ratio by the same number, the highest common factor. This is called simplifying.Example: In a club the ratio of female to male members is 12:18Both 12 and 18 can be divided by 2. 12 ÷ 2 = 6 18 ÷ 2 = 9So a simpler way of saying 12:18 is 6:9.To make the ratio simpler again, we can divide both 6 and 9 by 3 6 ÷ 3 = 2 9 ÷ 3 = 3So a simplest way of saying 12:18 is 2:3.These are all equivalent ratios, they are in the same proportion. All these ratios mean that for every 2 female members in the club there are 3 males: 12:18 6:9 2:32:3 is easier to understand than 12:18!
- 13. Quick Tips!Tips for ratio and proportion sumsRatio can be used to solve many different problems, for example recipes, scale drawing and map work.Changing a ratio A common test question will ask you to change a ratio - the reverse of cancelling down.Example: A map scale is 1 : 25 000. On the map the distance between two shopping centres is 4 cm. What is the actual distance between the shopping centres? Give your answer in km.A scale of 1 : 25 000 means that everything in real life is 25 000 times bigger than on the map.So 4 cm on the map is the same as 4 x 25 000 = 100 000 cm in real life.(Reminder 1 m = 100 cm and 1 km = 1 000 m)Now change the real life distance of 100 000 cm to metres100 000 ÷ 100 = 1 000 mAnd 1 000 m is the same as 1 km.So the shopping centres are 1 km apart.
- 14. Quick Tips!Keeping things in order When working with ratios keep both the words and the numbers in the same order as they are given in the question.Example: Share a prize of £20.00 between Dave and Adam in the ratio 3:2.The trick with this type of question is to add together the numbers in the ratio to find how many parts there are, divide by the number of parts to find the value of 1 part, then multiply by the number of parts you want to calculate.First add together the number of parts in the ratio: 3 + 2 = 5Divide to find out how much 1 part will be: £20.00 ÷ 5 = £4.00To find Dave's share multiply £4.00 x 3 = £12.00Adam's share is £4.00 x 2 = £8.00Dave's £12.00 is of £20.00 (3 of 5 parts). Adam's £8.00 is of £20.00 (2 of 5 parts).You can check that you have worked out the ratio correctly by adding the shares together. In this sum Dave's and Adam's shares should equal £20.00Let's check: £12.00 + £8.00= £20.00 Correct!
- 15. Quick Tips!Use the same units Always check that the things you are comparing are measured in the same units. Example: Jenna has 75 pence. Hayley has £1.50 What is the ratio of Jenna's money to Hayley's?.In this problem one amount is in pence, the other in pounds. Before you calculate the ratio you have to make sure they are the same units. We have to convert Hayley's amount into pence first. There are 100 pence to a pound Hayley's £1.50 = 150 pence So the ratio is 75 : 150You can simplify this ratio as both numbers are divisible by 75. The ratio is 1:2.
- 16. Film time!Watch a short video on percentages.
- 17. Fractions of quantitiesLearn how to find complex fractions of a number.
- 18. Finding Fraction Parts...What if four people want to share a box of 24 chocolates? A practical way would be for each person to take it in turns to choose a chocolate, until all the sweets have gone. They would have six chocolates each. You could also find the answer by working out a quarter of the total number of chocolates. of 24 is easy to do, just divide 24 by 4. 24 ÷ 4 = 6.So the answer is six chocolates.Similarly if three people wanted to share 24 chocolates you could find the answer by working out a third. In this case you would divide by 3. 24 ÷ 3 = 8So the answer is eight chocolates.Example £50 has been won in a pub quiz. Five people were in the team. How much does each person get? Find a of £50 by dividing by 5. 50 ÷ 5 = 10So each person gets £10.
- 19. Fraction Problems...You can use unit fractions to help you solve harder problems:Find of a box of 24 chocolates.It is easy to do this if you think of the sum as 3 lots of of the chocolates. First find of 24: 24 ÷ 4 = 6 Then multiply the answer by 3: 3 x 6 = 18So the answer is 18 chocolates. Find of a box of 24 chocolates.Think of this as 2 lots of the chocolates. First find of 24: 24 ÷ 3 = 8 Then multiply the answer by 2: 2 x 8 = 16So the answer is 16 chocolates.These questions look easy, as long as you can see what numbers to divide and multiply by. You do this by looking at the fraction you want. The bottom number (denominator) is the dividing number and the top number (numerator) is the multiplying number.
- 20. Percentages of quantitiesLearn how to find complex percentages of a number.
- 21. You can use the fraction method!Remember that percentages are always out of one hundred. So 50& is the same as 50/100, and 27% is the same as 27/100- they’re all the same! If you wanted to find out 27% of 250, you’d do this:Divide 250 by 100, to find one percent, or 1/100. (2.5)Multiply this by 27, to find the percentage you want. (67.5)It’s that easy!
- 22. GAMESTest your knowledge by playing these challenging games.Probability.Ratio and Proportion.Percentages.Fractions.