I covered a maths class last week for an absent colleague and some of the work that had been set included questions such as “Increase 9 in the ratio 11:3”. The pupils had some examples of how this was done and I was able to help them. However, one of the pupils asked me the point of the exercise and I couldn’t tell him! Can you enlighten me?
I’m sure you will have had problems that are similar to: “If it takes a man three days to dig a hole, how many days will it take six men to dig four holes?” It’s often quoted as an example of what people hate about maths.
Doesn’t it depend on the men and the ground they are digging? Lots of hidden assumptions are made. It’s a horrible problem and it’s not very realistic.
Before I tackle how the increase and decrease in a certain ratio is applied I will explain how you would tackle the calculation: “Increase 9 in the ratio 11:3”. The ratio given, 11:3, is written as a fraction to provide a scale factor of enlargement. For “increase” the larger number is written on the top: 11Z3 (which is greater than one). For “decrease” the smaller number is written on the top: 3Z11 (which is less than one). To increase 9 in the ratio 11:3, the 9 is multiplied by the “increase” scale factor (ie, 11Z3) giving 9 x 11Z3 = 33.
So how does this relate to a real problem, such as one about similar triangles (these not drawn to scale)?
The ratio of the lengths is written DB:DA
3:11
From the diagram we have the lengths of corresponding sides, which is how we obtained the ratio and then the scale factor of enlargement. We would expect an increase calculation from triangle B to triangle A, so we use the increase scale factor as shown in the calculation above.
Now consider this problem (this method could be used in questions involving conversions between currencies): the exchange rate for the euro is about e1.4 to pound;1. We can write this as a ratio and simplify it e:pound;
1.4:1
14:10 (x 10 on both sides to eliminate the decimal)
7:5 ( V 2 to calculate simplest equivalent ratio)
Looking at the numerical size of the numbers, as opposed to their value as euro or as pounds, we can set up a mnemonic to assist in calculating the correct conversion. I have presented this as the following diagram:The diagram below offers an explanation of why this works: The fraction provides the units’ equivalence. For pound;1 you receive e7Z5 (1.40) and for e1 you receive pound;5Z7. For example, to convert e294 to pounds sterling, the calculation would be a decrease: 294 x 5Z7 = pound;210.
Note that I have used an approximate exchange rate for euro and not one that reflects the current rate. You could use the new rate to extend this work.
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