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

Transcription

1 Numbers 2 BIRKBECK MATHS SUPPORT In this section we will look at - the meaning of dividing - an introduction to fractions - when fractions are equivalent - adding and subtracting fractions - practicing calculations with fractions - practicing word questions with fractions 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 lots of mistakes the first few times, 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. What does Dividing mean? If we have eight pounds ( 8) shown below as eight circles We can divide these between 2 people who then get four ( 4) pounds each We can write this in the language of maths in several ways: Eight divided into two gives four each Eight divided by two is four 8 divided by 2 is = 4 8 / 2 = 4 So divided by can be written as or as / And gives or is can be written as an equal sign = Another way to write this division is as a fraction with the eight on top and the two at the bottom and as before this still gives us four as the answer.

3 2. Fractions A fraction is written with a number on the top (this is called the numerator ) and a number on the bottom (called the denominator ) For example: And this means One divided into three pieces One divided by three / 3 We can also say one third or we can draw a diagram to show what we mean. Suppose we have one piece of chocolate cake One whole = 1 = We can divide this one piece into three pieces And one of these smaller pieces is a fraction of the original piece. So of the whole cake is this much

4 3. More about fractions Let s look at some more fractions Example 1: One quarter is written as One divided by four / 4 Or as a diagram we can take any shape and divide it in to four pieces. So if one whole, written as 1 = Then each of the four smaller pieces is one out of all the four. So one quarter = = Example 2: If we now consider two quarters this means two of each quarter. We can write this as 2 times 1/4 which gives us two quarters which is 2/4 and we can draw this as two quarters = =

5 But if we now consider two whole things and divide them amongst four people, Then each of the four people gets one quarter of the two whole things, which is equal to one half of a whole Writing this mathematically we say this in the following different ways two divided between four gives one half each two divided by four is one half 2 4 = 1/2 From this last line we can say something important about fractions: we can say the two fractions 2/4 and 1/2 are equivalent. We often use cancelling or simplifying to show that fractions are equivalent in this way. The next section deals with cancelling and simplification in detail.

6 4. Simplifying fractions and equivalent fractions First we start by saying something very simple. We can write any integer (whole number) as a fraction, for example four is the same as four divided by one. four divided by one is four 4 1 = 4 This may seem strange but it is very useful because we can now write eight divided by two is four as But there is another way we can see this. Because we know eight is the same as 4 x 2 and two is the same as 2 x 1 (and when we multiply numbers it does not matter which number goes first) so we can write is the same as And here on the right hand side we can see we have 2 multiplying everything on the top and 2 multiplying everything on the bottom. When this happens we can cancel these numbers (shown by writing 2) to give us Here is another example showing simplifying fractions by cancelling: And here is another example with two cancellations in: If you want to see videos of simplifying and cancelling fractions visit

7 5. Adding and subtracting fractions How do we add fractions? Well let s consider this example. Here we have half of a circle and next to that we have a quarter of a circle. If we combine these two fractions, the result looks like this: We can see we have three out of a total of four pieces or three quarters. So by just looking at the diagrams, we can see that But now we use the fact that one half is the same as two quarters. So By comparing the two lines above it is now easier to see how to add fractions together. When the fractions have the same denominators the fractions are easy to add, so two quarters plus one quarter gives three quarters.

8 Another Example: If we have half of something and we add one third. The result looks like this And we can write this as a fraction by looking at the way we can divide it up And now we see that we have five out of the six possible pieces. This means And we can see that this is true by re-writing the first two fractions so they have the same denominator (6) so we get

9 KEY POINT: The key point about adding and subtracting fractions is that we can only add or subtract fractions if the denominators are the same. (Remember denominators are the numbers at the bottom of the fraction). If the denominators are the same the fractions have common denominators Examples: 1. We can add these fractions straight away as the denominators are the same: 2. We can also subtract these fractions straight away as they have the same denominators 3. We can add these fractions too 4. These fractions are OK to add too 5. and these ones 6. But we cannot add these fractions straight away as they have different denominators So we need to find a way to write them so they have common denominators

10 FINDING COMMON DENOMINATORS To do this we use the ideas from page 5: Simplifying and Equivalent fractions. Well we can multiply the denominator of any fraction by any number as long as we do the same to the top of the fraction. and So now we can write Example 1: We can t add these straight away as they have different denominators. So we need to give them the same denominators. To choose a common denominator we need to find a number that can be exactly divided by both 5 and 2. One answer is 10, since 2 x 5 = 10 So now we re-write both fractions so they have the denominator 10. To do this for the first fraction we have to multiply the top and bottom by 2 And for the second fraction we have to multiply the top and bottom by 5 So we can re-write the original question and complete the addition Example 2:

11 6. 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 The first one in each case is an example. Answers available as videos at and A) Try to find the simplest equivalent fractions. The first one is an example Example: 1) 6) 2) 3) = 7) 8) 4) 9) 5) 10) B) Add and subtract these fractions. Hint: do they have the same denominator? As extra practice see if you can simplify your answer. Example: 1) 2) 3) 4) 5) 6) 7) 8) 9) 10)

12 C) Add and subtract the following fractions. Hint: how can you find common denominator, the first one is an example. Also check to see if you can write your answer in the simplest possible fraction Example: 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) D) Combining all types in word form. The first one is an example Example: Eight out of ten people at the cinema bought an ice cream or popcorn. Write this number as a fraction, then simplify it into its simplest form. 1) At a wedding the bride gives half the cake to her mum and a quarter of the cake to her sister. How much cake has she given away in total? 2) If the major shareholder owns four tenths of a company and the only other investor owns half of the amount that the major shareholder owns, how much of the company is not owned by these two combined? 3) If it takes Phil 3 days to build a brick wall and it takes Vasil 2 days to build a wall, how much of the wall do they manage to finish in one day if they both work together? 4) Orange juice is sold in 600ml cartons, for a special promotion they offer 1/3 extra in each carton. What is the total amount in the promotional cartons?