Solve ratio and proportion problems such as sharing out
The whole grisly subject of RATIOS gets a whole lot easier when you do this:
TURN RATIOS INTO FRACTIONS
What the fraction form of the ratio actually means
- Suppose in a class there are girls and boys in the ratio 3:4. This means that there are 3/4 as many girls as boys.
- So if there were 20 biys, there would be 3/4 x 20 = 15 girls
You have got to be careful though - it doesn't mean 3/4 of the people in the class are girls. In fact, 3/7 of the class are girls (15/35).
Reducing Ratios to their simplest form
You reduce the ratios just like you'd reduce fractions to their simplest form. For the ratio 15:18, both numbers have a factor of 3, so divide them by 3 - that gives 5:6. We can't reduce this any fuirther. So the simplest form of 15:18 is 5:6.
Treat them just like fractions - use your calculator if you can
Now this is really sneaky. If you stick in a fraction using your calculators button, your calculator automatically cancels it down when you press .
So for the ratio 8:12, just press 8 12 , an you'll get the reduced fraction 2/3. Now you just change it back to a ratio form i.e. 2:3. Pretty nifty hey?!
The More Awkward Cases
1) The button will only accept whole numbers
So if the ratio is something like 2.4:3.6 then you must....
MULTIPLY BOTH SIDES BY THE SAME NUMBER UNTIL THEY ARE BOTH WHOLE NUMBERS
E.g. for 2.4:3.6, multiplying by 5 gives 12:18, then you can cancel down further to 2:3.
2) If the ratio is MIXED UNITS
CONVERT BOTH SIDES INTO THE SMALLER UNITS USING THE RELEVANT CONVERSION FACTOR
E.g. 24mm : 7.2cm (x 7.2cm by 10 to get it to mm) = 24mm : 72mm = 1:3 (using )
3) To reduce a ratio to the form 1:n or n:1
SIMPLY DIVIDE BOTH SIDES BY THE SMALLEST SIDE
E.g. take 3 : 56 - dividing both sides by 3 gives 1 : 18.7
This is often the most useful, since it shows the ratio very clearly.
Worked Through Example
Say you work with someone else, chopping wood over the weekend for a timber company. There are 500 logs for you both to chop, and between you you’ll be paid £100. By the end of the Saturday, you’ve chopped 300 logs, and your colleague chopped the remaining 200. “Excellent,” says the company owner, “here’s £50 each.” Your colleague thanks the owner, and heads off on her way. But hang on… are you happy with £50?
You chopped more than half of the logs, you did more than half of the work, why are you only being paid half of the money? Surely that’s not really your fair share... So, how could the £100 be shared out more fairly?
To answer this question, we can use something called a RATIO. We look at the ‘ratio’ of work you did, compared to your colleague.
- You chopped 300 logs, she chopped 200, so the ratio of work is 300:200 (read that as “300 to 200”, or “300 parts to 200 parts” – you’ll see this language on the back of squash bottles: “Dilute 1 part squash to 7 parts water”)
- We can simplify ratios the same way we simplify fractions, so 300:200 = 3:2 (“Three to two” – “Three parts to two parts”)
This means that for every £3 you get, your colleague should get only £2, rather than an even amount.
- To work out how much of the £100 you each get by the end, first add up the number of parts in your ratio: 3 + 2 = 5
- Then divide the money by the total number of parts: £100 / 5 = £20
- So each part is worth £20. Now return to your ratio, to see how many parts each person should get.
You get 3 parts , and your colleague gets 2 parts, so you get:
3 x £20 = £60
And your colleague gets the remaining:
£40. Now it’s fair.
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