Fractions, decimals and percentages all deal with parts of things, but they are all subtly different.
Fractions are, in fact, incredibly precise, but they are often not thought of that way, because they are also the most naturalistic way of dealing with parts of things. When we share a pizza out five ways we naturally think of dividing into fifths (even though the pizza shop cuts it into sixths). Doing calculations with fractions can cause some confusion, and is hard to do on a calculator, which thinks in binary.
Decimals are basically the calculator way out of fractions, although they are used in a number of technical fields and have been for some time - logs and slide rules work more easily with decimals too. They are regarded as precise, although some simple fractions (1/3 for example) can't be expressed exactly this way - it is a product of the fact they're a computed answer and we tend to trust answers from computers and calculators.
Percentages are used slightly differently to the other two, they nearly always refer to parts of a group or changes. You can use decimals, or more usually fractions, to represent this data just as well, but percentages have the benefit of being easier to compare than fractions. They have a double-edged benefit and drawback of suppressing information about scale. If you want to compare the uptake of women into Higher Education over the years that is a good thing, the percentage of women has increased since the 1960's, but the number of students has increased a lot too. Suppressing the scale information here is useful. Knowing that 100% of children born with SCID (a very rare immunodeficiency disease) die by the age of 1 suppresses the fact that (in the figures I have to hand) only 1 such child has ever reached term and been born alive.
Arithmetic with fractions is not hard, but does require a couple of tricks.
We will start with multiplication (which is the easiest of them all). Consider 2/3 X 5/7. When multiplying you multiply the top lines (numerators) together, and the bottom lines (denominators) together and put the answers in the same place in the answer. 2 x 5 / 3 X 7 = 10/21. Dividing uses the same idea, but you reverse the second number, then multiply (you can remember this if you think of dividing by two, or halving being synonymous, and indeed equivalent operations... ÷2 is the same as X ½ 2/3 ÷ 5/7 = 2/3 X 7/5 = 14/15.
Adding and subtracting works with the trick you need to remember. This trick is related to BODMAS which you can see in the calculator page. 2/3 can be read as two-thirds or as 2 divided by 3. Division is a more important operation than addition and subtraction, so we need to 'cheat' to get around the fact there are division sums in any fraction. What we do is make the bottom lines the same so we can (implicitly at least) factorise, add and then reconvert to a fraction. That sounds far scarier than it actually is. You may remember, however vaguely, something about lowest common multiples. This is the way it is commonly taught at school, and correct, but not as fast as bending that rule (which often gives you an extra step at the end). All you need to do is multiply each fraction top and bottom, but the denominator of the other fraction. This will make the denominators the same. Then you add (or subtract) the numerators. Finally you cancel the fraction. Let's see 2/3 + 5/7. The denominators are 3 and 7. We need to multiply 2/3 by 7/7 (7/7 =1 so we're not actually changing the sum, just changing how we write it). Similarly we need to multiply 5/7 by 3/3. So, 2/3 X 7/7 + 5/7 X 3/3 becomes 14/21 + 15/21. Because the bottom lines are the same we can add the top lines, to get 29/21.
Canceling fractions comes, usually in two parts. You can do them in either order.
- You look for numbers that will go into both the numerator and denominator (common factors) and divide through. You keep doing this until you can find no more. This is also known as simplifying fractions. 29/21 doesn't actually factorise. 64/256 does. What is the biggest number you can think of that will divide into both? Try 4... 64/4 = 16, 256/4 = 64, so 64/256 = 16/64. If you're quick you'll realise that 16 goes into both of these, but let's say you recognise 8 goes into both. 16/8 = 2, 64/8 = 8, so 16/64 = 2/8. Finally two goes into both of these, so 64/256 = 1/4 when simplified.
- "Top heavy" fractions (such as 29/21) can also be written as an integer plus a fraction. Division is covered below, but you work out how many times the denominator goes into the numerator (once in our case) and the remainder is the numerator of the fractional part. 29/21 = 1 and 8/21.
Finally, if you are working with a mixture of whole numbers and fractions you have a couple of choices: one is to treat the whole numbers separately and remember the fraction part also represents division. 9 X 2/3 is 9 x 2 ÷ 3, or 6. The second is to rewrite the integer as a fraction - 9 becomes 9/1 and apply the rules above. Either works well.
Working with decimals also requires a couple of tricks. When adding decimals, or subtracting them, the thing that matters is lining up the decimal points. If you remember your columns for numbers: Hundreds, tens, units - this extends to tenths, hundredths etc. When you add integers lining up to the right is short hand for lining up the units columns, the tens columns and so on. Lining up the decimal points is the same thing.
When multiplying, as discussed in a previous class, the trick is to forget the decimal points and multiply the numbers as if integers. At the end count the number of significant digits (other than trailing zeros) after the decimal point in the numbers you've multiplied together and bring the decimal point in that number of places from the right hand end of the number. 3.1 X 2.45 = 7.295. If you do the sum without decimal points you get 7295. There are three digits after the decimal point between the two numbers, so we come in three places... to give 7.295 (your estimating skills can be useful here too: 3 X 2 = 6, so an answer of 7.295 is probably right, whilst 0.7295 or 72.95 would both be wrong).
There is an extra trick for dividing with decimals. Consider 7.295/2.45: it looks very scary. However you can multiply top and bottom of this by 100 to get rid of the decimal point on the bottom, to 729.5/245. Now, my 245 times table is not that good, but I feel more comfortable about the division at this point, and will eventually get to 3.1 (see below).
Working with percentages and arithmetic is unusual. If you ever need to do it I strongly suggest you convert to decimals (divide by 100), work out the sums, and convert back to a percentage. The two sums you will be expected to do with decimals are sums of the form of "what is x% of y?" and, more usually, what is the percentage change (or sometimes percentage increase) in these values (usually between two times).
Questions of the form, what is x% of y can be handled easily as a fraction. Consider the sum, what is 25% of 24? Well 25% is 25/100 (per cent meaning per 100 or divided by 100 after all), so the question is what is 25/100 X 24. You can work this in anyway you choose: for this example the fact that 25/100 = 1/4 (25 divides into both), and 1/4 X 24 = 6. If the question becomes what is 3% of 17 say... we have 3/100 X 17. 3 x17 =51, 51/100 = 0.51.
Percentage changes are calculated by the formula (new - old)/old * 100%. If you are good at comparing fractions you can save a little time and avoid the * 100% at the end and work with the fractions directly.
Finally we come to division. There are two methods that can be used: I suggest you read about them on the skillswise website, where it deals with whole number divisions. Long division is here and repeated subtraction here. Repeated subtraction is a good, although slower, alternative method.
The skillwise website does not go in to answers with remainders. One of the easiest ways out to give the answer as a fraction. What the remainder is, over the divisor, cancelled is a correct answer. 5/2 becomes 2 r 1 or 2 ½ for example. Often for tests you will not need to go this far, but you may. If you want or need an answer in decimals there are two tricks depending on your preferred method of dividing.
- If using long division, you simply put a decimal point, as many zeros after it as required and keep on dividing - the decimal point in the answer being over the decimal point in the number inside the sum. If the question already contains a decimal point you work past it as normal, copying it up as you pass it's column inside the sum.
- If you are using the repeated subtraction method, the trick is to take your remainder, multiply it by some suitably large number (100 if you want 2 decimal places for example, or 10,000 if you want 4 decimal places), then repeat the operation on this new number, writing the answer after the decimal point.
There is a third method of dividing which more or less combines these. It is only available in class: if you make it, you'll see why!
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