All the examples in this worksheet and all the answers to questions are available as answer sheets or videos.


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1 BIRKBECK MATHS SUPPORT Numbers 3 In this section we will look at  improper fractions and mixed fractions  multiplying and dividing fractions  what decimals mean and exponents or indices can be used  how ratios and percentages work  exponents or indices with an introduction to algebra  some examples of calculations involving all numbers  some examples of word questions involving numbers Helping you practice At the end of the sheet there are some questions for you to practice. Don t worry if you can t do these but do try to think about them. This practice should help you improve. I find I often make mistakes the first few times I practice, but after a while I understand better. Videos All the examples in this worksheet and all the answers to questions are available as answer sheets or videos. Good luck and enjoy! Videos and more worksheets are available in other formats from
2 1. Improper and Mixed Fractions Here we will look at different ways that fractions can be written but mean the same thing. So far we have considered equivalent fractions, e.g. Now we will introduce Improper Fractions and Mixed Fractions. PROPER FRACTIONS Proper fractions are fractions where the numerator (number on the top) is always smaller than the denominator (number on the bottom), e.g. Any proper fraction is always less than one, and we can see that if we draw 2/3 as a diagram we don t quite have one whole, e.g. So all proper fractions are less than one. IMPROPER FRACTIONS Improper fractions are fractions where the numerator (number on the top) is always larger than the denominator (number at the bottom) e.g. Any improper fraction is always bigger than one whole and we can see this if we draw a diagram of four thirds.
3 So all improper fractions are bigger than the number one. MIXED FRACTIONS A mixed fraction is a fraction where we mix whole numbers and fractions. For example one and a third is a mixed fraction and we could write this as but by comparing the diagram above we can see that four thirds is the same as one whole and a third. We will now show this using maths Where we have used the fact that any number divided by itself is one, e.g. CONVERTING MIXED FRACTIONS AND IMPROPER FRACTIONS Here are some examples of converting between different types of fractions. To see videos explaining these calculations visit Example 1: is an improper fraction and can be written as a mixed fraction. First we need to know how many times the denominator of the fraction goes into the numerator of the fraction. Well 2 is the denominator and I can write the numerator 5 as So I can write So five out of two is the same as two and a half.
4 Example 2: One and a quarter is a mixed fraction and is written write it as an improper fraction in the following way. but we can Example 3: It also doesn t matter if the fraction is negative. If we have minus two and a fifth (a mixed fraction) we can change this to an improper fraction Notice that the whole expression is negative. This is because the whole of the mixed fraction is negative. 2. Multiplying and Dividing Fractions After the hard work on adding fractions and converting between improper and proper fractions you ll probably be pleased to know that multiplying and dividing fractions is much easy. We ll start with a simple example. MULTIPLYING FRACTIONS If I see a cake and eat a quarter of it and I do this three times in total, then I will have eaten three quarters of a cake. And another way to write this is as And here we can see that when we multiply fractions we just multiply all the numbers on the top (3 x 1) to get the total numerator (3), and all the numbers on the bottom (1 x 4) to get the total denominator (4).
5 Here is another example But notice that we can simplify the answer, since the numerator and the denominator can be written as multiples of 2: Key Points: When multiplying fractions make sure all the fractions are in the form of proper or improper fractions. If any fractions are written as mixed fractions you will need to change them to improper fractions. When multiplying fractions just multiply the numerators together (numbers on top) and the denominators together (numbers on bottom). Remember you may be able to simplify your answer. Example with mixed fractions 1) Calculate Well we can t multiply these at the moment as the first fraction is a mixed fraction. So first we have to change the mixed fraction to an improper fraction Now we can rewrite the original equation and calculate the answer More videos of fractions are available at
6 DIVIDING FRACTIONS Dividing one fraction by another is not so difficult. To help us think about this, consider an example. If I have half of a cake and I divide this between 2 people what do I get? To see the answer we can draw a picture. Here is my half a cake: I now divide this between 2 people; and we can see that they each get a quarter of the original cake. Here is one of those pieces So this means one half divided by two is a quarter half divided by two equals a quarter Where each of these statements mean the same thing. Notice that we get the answer by turning the dividing fraction upside down, so To see examples of dividing fractions visit
7 4. Decimals Once we have started to understand fractions, then decimals are easier to get to grips with. We start with the simplest decimals Where 0.1 is pronounced as zero point one or nought point one. If we now consider some more general decimals So generally we can write the following eight fractions as decimals Any decimal that can be expressed exactly as a fraction is called a rational number. If a decimal can t be written exactly as a fraction it is called an irrational number. Decimals can be converted to fractions as follows: For more examples see the videos on
8 5. Percentages and Ratios If you can get confident with fractions then you will be much happier with ratios and percentages, as calculating ratios or percentages involve the same ideas. PERCENTAGES The word percentage means out of 100 and is written as %. So 50% means 50 out of 100, which we can write as a fraction then as a decimal But if we look at the fraction now we see that we can simplify the fraction So 50% is the same as 0.50 or the same as half. Here is another example. Seven percent is 7% which means 7 out of 100, and we can write it as a fraction and as a decimal We can t simplify this fraction though as no number goes into both 7 and 100. Here are some examples of questions that can be asked about percentage. We try to answer these questions in more than one way as this helps to check we have the right answer. Example 1: A dress is in the sale. The price says 25, but there is a 20% reduction. What is the actual price of the dress? Answer 1: first we need to find out what 20% of 25 is, and we can write this as 20% of 25 Therefore if the dress is reduced by 5, it means it is 255 = 20
9 Answer 2: Now we ll do the same question but a different way. If the dress is reduced by 20%, then 20% has been removed from the original price, since 100%  20% = 80%. The new price is 80% of 25. So the new price of the jumper is 20, which is the same as the answer above. Example 2: A roofer quotes 400 to replace a roof. He adds VAT of 20%. What is the full cost to replace the roof? Answer 1: The total cost is 400 plus the extra 20%. So first we find out what 20% of 400 is Then the total cost is 400 plus the extra 80 = 480 Answer 2: The total cost is 20% on top of the quote for 400. The 400 is 100% and then there is an extra 20%. This makes 120% in total. So the full cost is 120% of 400 So 120% of 400 gives the total amount of 480. RATIOS Ratios are a simple way to compare amounts, usually when mixing things together. For example to make orange juice you need 1 part orange juice to 3 parts water. This is a ratio of 1 part 3 parts, so the glass would look like this. This glass has 3 parts water and 1 part orange juice way. It is in the ratio 3 to 1. Written as 3:1
10 We can now see that because there is 1 part of juice and 3 parts water, there are four parts in total. So orange juice is one part out of four, and the juice is three parts out of the four. We find we are back to fractions and percentages. So if we want to make 6 litres in total, we need 1/4 (or 25%) orange juice and 3/4 (or 75%) water, which can be calculated as Or we can carry out the same calculation using percentages 6. Exponents, Indices or Powers Lots of jobs, such as nursing, business or engineering involve large and small numbers and decimals. A very useful way to write very small and very large numbers is to use exponents, indices or powers. For example we could write 10x10 as 10 to the power 2, which means multiply 10 by its self two times, so In the case of 10 3, the number 10 is the base and the 3 is the exponent, index or power. We can use the same idea for any number, so
11 This becomes useful though for writing very large numbers. For example This way of writing numbers is called Scientific Notation or Standard Form, and we will look at this more in the worksheet on Measurements and Units But before we go any further we need to think about indices a bit more. RULES OF INDICES To make calculations and simplifications with numbers written with powers or indices we need to learn and use the Rules of Indices. Below we show how they are true, give examples and the general rule. We have used a small bit of algebra in this section and if you feel lost try reading the next section on algebra or watching the video on indices. Notice that This means that Now notice how the indices on the lefthand side (which are 2 and 3) add up to give the index (this is singular for indices) of the answer (which is 5). We can write is as a general rule For example or we could also write using symbols, where x could be any number, for example in this case 2 and y is any other number, in this example 3
12 But x could be any number and y could any other number and the rule would still be true. Let us look at We can see that the rule still works, and further that if we change 4 for 6, So if we use the symbol a to mean any number we can write the First Rule of Indices is Where n, x and y represents any numbers. Notice now what happens if we write the following fraction Well we know that if something multiplies the top of a fraction and the bottom of a fraction then we can cancel, so this fraction can be simplified to But if we rewrite the top and bottom in terms of indices we have We can see that the index on the right hand side can be got from the index of the numerator minus the index of the denominator: 53 = 2. The second rule of indices is
13 We now consider what happens when we have 2x2x2 and multiply this by itself four times, well lets write it out (we have left extra spaces in between each group of 2x2x2) But we can see on the left hand side that the total number of 2s multiplying each other is twelve which is the same as 4x3, so we can write (If you are not completely comfortable with using brackets there are lots of examples explaining brackets in the first worksheet in the Calculations section.) Writing this as a general rule for any number we have The third rule of indices is Finally: we notice three other things about indices: Often it is only through practicing examples that things become clear. So try these examples then read the theory again and it will become clearer. Examples: simplifying using indices: 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) It is possible to have fractional indices and we will look at the meaning of these in the Calculations section.
14 6. Writing big and small numbers with indices We already mentioned that it is possible to write large numbers using index notation. For example. But we will see now that it is also possible to write small numbers in index form: Consider the following decimals: Continuing to follow this pattern we see that we could write So for any small number, for example we can write If your job is writing small numbers regularly then it is easy to miss out the zeros and so using indices is a much safer and easier way to write the numbers and the index tells you how many zeros, including the one before the decimal point. Here we will look at more examples: 1) 2) 3) 5) 6) 7) 8) 9)
15 7. So what is Algebra? Algebra uses the rules of numbers, such as addition, division, but instead of using numbers it use variables. For example if I buy two arm chairs and a sofa, then I can use the variable A to represent the armchairs and the variable S to represent the sofa, then I have If I now decide to buy another arm chair I have But this is the same as three armchairs and a sofa, so we can simplify SUBSTITUTION Often the letters x and y are used to represent variables. For example we can let x represent one burger and y represent one portion of fries. So if the first customer orders two burgers and three portions of fries I can write this as But now the second customer orders five burgers and two fries we write Suppose we now find out that one place sells burgers for 1.50 and fries for 1.00, we can substitute these in the following way But if the burgers are 1.00 and the fries are 1.20, substitution gives So we can see that it is cheaper for the first customer to go the first cafe but it is cheaper for the second customer to go to the second cafe.
16 8. Now your turn Generally the more maths you practice the easier it gets. If you make mistakes don t worry. I generally find that if I make lots of mistakes I understand the subject better when I have finished. If you want to see videos explaining these ideas and showing the answers visit A) Convert the following improper factions into mixed fractions 1) 2) 3) 4) 5) 6) B) Multiplying and Dividing Fractions, try also to simplify your answer 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) C) Decimals, Percentages and Ratios 1) 2) 3) 4) 5) 6) 7) Water is mixed with glue in the ratio 3:1. If I use 1 litre of glue, how much water do I need? 8) Orange juice is mixed in the ratio 1 parts juice to 4 parts water. How much water is in 100ml of mixed juice? 9) The ratio of milk to dark chocolates is 3:4. If there are 140 chocolates in a bag, how many are milk chocolates?
17 D) Working with indices, simplify the following 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) E) Word Questions 1) In a restaurant each slice of cake sold is one eight of a total cake. If there are three and a half cakes left how can you show mathematically how many slices are left for sale? 2) A suit has a price tag 100 and a shirt has a price tag 40. But there is a special offer and everything is reduced by 15%. How much would the suit and shirt cost with the reduction? 3) Humans produce 2 million red blood cells each second. If each blood cell has mass of. Show using indices the mass of blood cells produced each second. 4) Two brothers decide to share the cost of a car. One of them pays 800 and the other one 200. Write these amounts as the simplest ratio possible. 5) Jeans are sold for 30 each. The selling price is 50% more than the cost price. What is the cost price of each pair of jeans? 6) The ratio of men to women working in a factory is 5:4. There are a total of 20 women. How many people work at the factory in total? 7) A survey reports that one half of families living in a particular block of flats have a pet. Of these people half had a fish, a quarter had a hamster and a quarter had a dog. What fraction of families had a fish? 8) A television has a mark up on the cost price of 40% and is then sold in the sale for 10% discount. If the TV originally cost 100, how much was it sold for?
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