FRACTIONS, DECIMALS, PERCENTAGES how do they relate?

Transcription

1 CONNECT: Fractions FRACTIONS 4 FRACTIONS, DECIMALS, PERCENTAGES how do they relate? If you are unsure about what a fraction is, please refer to CONNECT: Fractions Manipulating fractions. To relate Fractions and Decimals, let s first think about our Decimal System of Numbers (or Place Value). This system gives us a very efficient way to write any number we please, just by using the digits 0,,,, 4,, 6, 7, 8, 9 in columns, where each column represents a different size. Here are the headings for the columns, which you are familiar with: millions hundred ten hundreds tens units Unlike English language, we read numbers from right to left. Each time we move from a column on the right to one directly on its left, we have multiplied by 0 (so that for example, x 0 = 0, or 8 x 0 = 80) we attach a 0 on the end because it tells us we have moved across a column. In the same way, if we divide by 0, we are moving across a column from left to right so, for example, = 49 we ve picked up our number and moved it across a column to the right. x 0 0 millions hundred ten hundreds tens units All the columns listed so far refer to whole numbers. But what happens if we have a whole number where the last digit is not 0 and we want to divide it by 0? How can we pick the number up and move across a column? We need to be able to show that we are no longer dealing with a whole number and will only have a fraction as our answer. This is why we have a

2 decimal point. The decimal point separates whole numbers from parts of a whole, and parts if a whole are fractions. So: millions hundred ten hundreds tens units. Let s begin by dividing by 0. We can write this as 0. You probably realise we can also write this as 0 or. Using the fact that when we 0 divide by 0 we move our number across a column from left to right, we would have: 0 units.?. So what do we call our new column? 0., (which is the same as.), must be the same as and so we call the new column tenths. (The th on the end 0 tells us that we are not dealing with a whole number any more, but part of a number, that is, a fraction.) We now know that 0. (meaning 0 units and tenth) is exactly the same as 0 and in maths we would write 0. =. 0 Our columns continue on the right. For example, if we divide each tenth into 0 bits (that is if we calculate 0) we would get 00 bits and so we call 0 the next column across hundredths. So, 0 =, or 0. 0 =

3 This is what we have if we put it all together in our decimal system: millions hundred ten hundreds tens units. tenths hundredths thousandths So 0 = 0., 00 = 0.0, 000 = Also, = 0., = 0.0, = 0., 94 4 = 0.0, = 0.94, = Remember, we can only put one digit in each column of our decimal system. The number of 0s in the denominator tells us which column the last digit of the numerator goes into. Here are some for you to try. Match the decimals and fractions in the following table (one is done for you). You can check with the table at the end of this resource. Fraction Decimal

4 We have just looked at the relationship between fractions and decimals, as long as the denominator of the fraction is 0, 00, 000. But what about fractions such as ½, ⅔, ¾, and so on? In these cases where the denominator is not a power of 0, we need to consider a different method to work out their decimal equivalents. For this, we go back to the definition of a fraction, where the top number is divided by the bottom number. For ½, we need to calculate. Set up a short division: ) We see that will not divide into, but we know that is the same as.0, so our division becomes ).0 There are no s in, so we put a 0 above the, followed by a decimal point above the decimal point in the ).0 Now ask how many s in 0? There are, so we put above the line. 0. ).0 There is no remainder. So, ½ = 0. For ⅔ we calculate, ) will not divide into so put a 0 above the, and attach a decimal point and a 0 after the. 0. ). 0 4

5 Now ask how many s in 0? There are 6, so: but there is left over, so 0.6 ) ). 0 0 Again ask how many s in 0? 6 again ). 0 0 There is a remainder of again. We can keep attaching 0 at the end and obtain for as long as we wish to keep writing 6s! So, ⅔ is the same as (the means continues the same ). You can also write this as This method works for any fraction but for a denominator of 0 or 00 or 000 (and so on) it is always easier to use the place value method from above. Here are some for you to try. One has been done for you. You can check with the solution at the end. Fraction Decimal (or 0.6 ) (or 0. ) (or 0.46 ) 0.7

6 We have found ways in which to convert fractions into their decimal form. Now, to work back the other way. (For people who do not like division, this is much easier!) Let s say we want to know what the decimal 0.9 is as a fraction. From our decimal system (or place value) above, we know that this number consists of tenths and 9 hundredths and we can write it as 9. We cannot simplify this 00 fraction any further so 0.9 = Now, what about 0.? We know that this is the same as and 000 can be divided by, so we get = and that both Here are some for you to try. Change these decimals to fractions and write your answer as the simplest fraction possible. (You can check your answers at the end.) () 0.7 () 0.89 () 0. (4) 0. () 0. (6) 0.0 (7) 0. (8) 0. (9) 0. (0).4 6

7 PERCENTAGES Percentages relate to fractions and decimals because per cent means out of 00. So, for example, 80% means 80 out of 00, or We can simplify this both as a fraction and as a decimal. Firstly as a fraction: 80 = (Try to divide by the highest number that will go into both numerator and denominator here.) So = 4 As a decimal: By place value, we know 80 is the same as 0.80, which is the same as So the relationship is 80% = 4 = 0.8. Notice that all of these indicate the same relatively high proportion. If you got 80%, or 4 out of, or 0.8 for an assessment task, you should be pretty happy! We have changed a percentage into a fraction and a decimal, now let s go the other way. Example: Change into a percentage. We ll do this as you probably already know the answer. But we can then use the same method for all other cases. To change fractions or decimals into percentages, we just multiply by 00 and call the answer %. So, = 00(%) = 00 (%) = 00 (%) = 0% 7

8 Let s try a fraction we do not know: change into a percentage. 8 8 = 8 00(%) = 8 00 (%) = 00 8 Here we can go two ways, either divide 8 into 00 with a short division, or divide both numerator and denominator by = or 8) = 7 = 7 Either way, we end up with 7.%. This seems right, especially because 7.% is halfway between % and 0%. Now let s try a decimal which is actually easier than a fraction! Change 0. to a percentage: just times by 00, so 0. x 00(%) gives 0%. What about changing. to a percentage?. =. x 00% = 0%. Does this make sense? It does if we remember that 00% is the same as whole! Here are some for you to try. Fill in the blanks in the table. Fraction Decimal Percent 0. 7% 0.0 0% 8

9 If you need help with any of the Maths covered in this resource (or any other Maths topics), you can make an appointment with Learning Development through Reception: phone (0) 4 977, or Level (top floor), Building, or through your campus. Solutions Solution to table exercise (from page ): Fraction Decimal Solution to table exercise (from page ): Fraction Decimal (or 0.6 ) (or 0. ) (or 0.46 ) 0.7 9

10 Solutions to questions from page 6 () 0.7 = 7 (cannot be simplified further) 0 () 0.89 = 89 (cannot be simplified further) 00 () 0. = (cannot be simplified further) 000 (4) 0. = 0 = () 0. = (cannot be simplified further) (6) 0.0 = = (Note that 0. and 0.0 are the same, and they are also the same as , etc!) (7) 0. = 0 = (8) 0. = (9) 0. = = = 00 = 8 (You could also divide numerator and denominator both by at the first step, or you could divide both numerator and denominator by and by again instead of by at the second step and you would obtain the same result.) (0).4 = 4 0 = 7. (You can also make 7 into.) 0

11 Solutions to Table (page 8) Fraction Decimal Percent % % 7 00 = % 0 00 = % % % (or 66 %)