## Percentages & surds

How can you turn a raw score in a test into a percentage? If you get 15/20 for a test, you have done as well as someone who got 75 out of 100!

See also Indices, logs, surds

- Percentages
- Surds

### Percentages

Percentages are another way to express a fraction. However, it is always expressed out of 100. This video will show you how to convert between fractions, decimals and percentages.

Hi, I’m Martin Lindsay from the Study and Learning Centre at RMIT University. This is a short movie on percentages.

A percentage means out of 100. For example, 25 percent means 25 out of 100 or 25 divided by a 100. Seven percent means seven divided by a 100 or seven out of 100. And 98 percent means 98 out of 100 or 98 divided by 100. Let’s start by changing percentages into fractions. So the first one, 75 percent is 75 divided by 100. What we need to do is to break that down into smaller numbers, in other words we’re simplifying the fraction, so let’s divide by 25, top and bottom, and we get three divided by four, so 75 percent, as a fraction is three-quarters, similarly 40 percent is 40 divided by 100. Notice here that there are zeroes top and bottom, cancel one on the top with one on the bottom, that leaves us with four over 10 which cancels even further into two over five, so 40 percent is two-fifths.

Let’s look at a large percentage and change this to a fraction, 175 percent is the same as 175 divided by 100, but now we turn this into a mixed fraction, which is one and 75 over 100. Notice again that 75 over 100 can be cancelled down, so we divided both top and bottom by 25 leaving us with an answer of one and three-quarters. So 175 percent is one and three-quarters.

Now let’s change a decimal percentage into a fraction. For instance we have the percentage 17.5 percent, which is 17.5 over 100. What we need to do here is to turn that 17.5 into a whole number so that we can then cancel the terms down. So we’re multiplying 17.5 by 10 and doing that to the top we must do the same thing to the bottom. In other words 17.5 by 10 is 175, 100 times 10 is 1,000, now we look to see if anything cancels and of course it does, we can divide both top and bottom by 25 leaving us with an answer of seven over 40, so 17.5 percent is seven over 40 as a fraction.

Now let’s look at a small decimal percentage. We have here 1.25 percent, which is the same as 1.25 divided by 100, again we need to turn the 1.25 into a whole number so this time we multiply it by 100 because 1.25 has got two decimal places and similarly we must multiply the denominator by 100 as well, so the top line becomes 125, the bottom line becomes 10,000, which cancels down into one over 80, so 1.25 percent is one over 80 as a fraction.

Now let’s change percentages into decimals. An example, 15 percent is the same as 15 over 100, change a percentage into a decimal we always move the decimal point two places to the left, so if you move the decimal point two places to the left we end up with 0.15. Similarly with a large percentage, 220 percent is the same as 220 divided by 100, so the quick way to do this is to move the decimal point two places to the left, so we end up with an answer of 2.2. So 220 percent is 2.2 as a decimal.

Finally 7.9 percent is 7.9 divided by 100 and dividing by 100 you simply move the decimal point two places to the left, so moving it two places to the left leaves us with an answer of 0.079, so 7.9 percent is 0.079.

Now let’s change fractions into percentages. So here we’re working in reverse to what we were doing in previous slides. Here’s an example, seven over 10. To change a fraction into a percentage we always multiply it by 100 and because the fraction is written as seven over 10 we can write the 100 as a fraction as well, 100 over one. Notice that one zero on the top will cancel with one zero on the bottom, in other words I’m dividing by 10, which leaves us with 70 over one, which gives us an answer of 70 percent. In other words seven-tenths is 70 percent as a percentage.

How about changing decimals into percentages, well we use the same idea as before. Here’s an example, 0.55, to change that decimal into a percentage we multiply it by 100 and when you multiply by 100 what you’re actually doing is you’re moving the decimal point two places to the right, so we start at the decimal point, move two places and we finish after the last digit, the five in this case. So we have an answer of 55, which is 55 percent, so 0.55 is 55 percent as a percentage.

What about small decimals, changing those into percentages. We use exactly the same rule as before. Here’s an example, 0.09, to change that into a percentage we multiply by 100, so again, as before, we’re moving the decimal point two places to the right and here the decimal point ends up on the far right hand side after the last digit, the nine, so our answer is nine percent, so 0.09 is nine percent as a percentage.

Now try some problems for yourself. The answers to these questions are on the next slide. Thanks for watching this short movie.

### Surds

How do you find a square root of any number that isn’t already a square? The square root of 16 is 4 and the square root of 25 is 5 (because 4 x 4 is 16 and 5 x 5 is 25), and so on. But can you find the square root of a number between these …for example √20? Watch this video and see how to deal with this group of numbers called surds.

I’m Martin Lindsay from the Study and Learning Centre at RMIT University. This is a short movie on surds. A surd is a number that cannot be written as a fraction.

For instance numbers like root two and root seven have got a series of decimal places after the decimal point if you type those into the calculator so you can’t write those as fractions. Numbers that are not surds are written below. You can see that two can be written as a fraction two over one, one point three can be written as 13 over 10 and root four, which is two, can be written as two over one. In other words they can all be expressed as fractions.

The trick in simplifying surds is to look for square factors. Here are a number of square factors, four, which is two squared, nine, which is three squared and so on up to 100, which is 10 squared. Look at this example, we have the square root of 200 and what we do is we express it so that there’s a square factor under the square root sign which in this case is 100, shown in red, 100 times two under the square root sign is the same as the square root of 100 times the square root of two. Notice the square root of 100 is 10 so our answer is 10 root two. Notice also the square root sign is sometimes called the radical. Similarly three root 48, 48 is not a square factor but 16 times three is, which gives us three the square root of 16 times three, the root 16 is four and root three is left on its own, in other words we now have two whole numbers, three and four, which gives us 12 root three, so our answer is 12 root three.

When it comes to adding and subtracting surds one has to be mindful of the fact that only like surds can be added or subtracted. For instance, here’s a question eight root five minus three root five. The root fives are the same in both terms, in other words we can subtract these two terms, eight root five minus three root five is five root five. However, if we have this question, where we have two root three plus five root seven minus 10 root three, notice there are two different surds here, a root three and a root seven, so we have to keep those separate in our answer, so that gives us five root seven minus eight root three.

Here’s the rule for multiplying surds and it’s probably easier to illustrate this with an example. For instance root three times root two we can place both the root three and the root two under one square root sign, in other words that becomes the square root of three times two, three times two is six so the answer is root six.

A second example, we have three root five times four root seven. Three and four are the whole numbers so you multiply those two separately, whereas the root five and the root seven can go under one square root sign, the square root of five times seven, so that gives us an answer of 12 root 35.

Finally here’s another one. We start with two root 10 times seven root six, which gives us initially 14 times the square root of 10 root six, which is root 60, but when you get a large square root sign in particular see if it will simplify into something a bit smaller, in other words look for a square factor which in this case is four, in other words root 60 is the square root of four times 15. Four times root 15, the square root of, gives us two times root 15, so we now have four times two which is 28 root 15.

In dividing surds we use a similar idea and again let’s look at the example here, root 18 divided by root six is the same as the square root of 18 divided by six, which is three. In other words our answer is root three. Here’s a number four divided by a surd, root two, so here we have four, which can be expressed as root 16 divided by root two. Notice I’ve expressed both numerator and denominator as surds because we can now use the rule, which is the square root of 16 divided by two, which is root eight, and again notice that this simplifies because we can use a square factor here of four, so root four times two is two root two.

Rationalising a surd is a way of simplifying an expression where the surd is on the denominator. For instance if we have two over root five notice the surd is in the denominator here and to simplify that we can multiply two over root five by root five over root five, which is effectively one. In other words two over root five times root five over root five is two root five multiplying the top line, root five times root five is root 25 which is five. Notice the answer to root five over five has got no surd in the denominator.

Similarly here we have surds top and bottom but the purpose here is to try and get rid of the root two in the denominator, so we do as before by multiplying both top and bottom by root two over root two, which gives us root 10 on the top, root two times root two on the bottom is two times three which gives us six on the bottom. Again notice in both examples here that there are no surds in the denominator, that’s rationalising surds.

Conjugate surds are just a slight extension of the rationalising surds problem we were doing in the previous slide. Here’s an example; root three over five plus root two. What we do is we look at the denominator and then we write the conjugate of that, meaning we write the same terms five and root two but with a negative sign between them. In other words we’re multiplying by five minus root two over five minus root two. Now we multiply out the problem, this gives us root three brackets five minus root two over five plus root two multiplied by five minus root two. If we multiply out the bottom line we’ll get four terms, but notice in particular that when we multiply out this bottom line it’ll be the surds that will cancel out. In other words we’ll have no surd left in the denominator, so our answer is five root three minus root six divided by 23. Always make sure that the answer does not have a surd.

Now try some problems for yourself. The answers to these questions are on the next slide. Thanks for watching this short movie.