I would like to address in this article an issue that I have seen crop up many times in the UKCAT Numerical Analysis questions.
I have written other articles about order of magnitude issues but this one addresses a particular problem.
Let’s look at an example:
Gross Domestic Product (€bn)^{*} |
|||
Country |
Population in 2005 (thousands) |
2005 |
2010 |
Belgium |
10418 |
256.1 |
269.2 |
*1 billion = 1 thousand million
Question: The GDP per capita in Belgium rose by €1208 per person between 2005 and 2010. What happened to the population over this period?
A. - Up 20000; B. – Up 10000; C. – No change; D. – Down 10000; E. – Down 20000.
The table is trimmed for brevity but all the required information is there.
So what do we do first?
We need to formulate our method of solution. We need to calculate the GDP per capita in 2005 which is the GDP divided by the population; we can then add in the increase stated in the question text; next we calculate the population in 2010 by dividing the GDP by the new GDP per person; and finally we subtract the 2005 population to ascertain the change.
So, let’s make a start…
Step 1: calculate the GDP per capita in 2005. This is (256.1/10418)=0.024582.
Step 2: add in the increase per person over the stated period…
I’m sure you can see the problem now. Normally we can leave issues of order of magnitude until the end of the question and deal with it in one step. This is fine when we are multiplying and dividing because all interim answers will only differ from the actual value by some order of magnitude. The actual digits of the answer will always be correct; it’s just the position of the decimal point that needs to be adjusted.
However...
When we are adding and subtracting values we must ensure that the orders of magnitude of both elements of the addition are the same. As you can see in the example above, we have a GDP per person of 0.024582 and the question text tells us that this increases by €1208 per person. These do not look compatible, and indeed they are not. We need to adjust the order of magnitude of one or the other so that they both have the same units.
In brief:
In step 1 we used the GDP in billions of euro and divided it by the population which was in thousands of people. This means that the answer is in €bn/1000 people or more simply, millions of euro per person. If we multiply it by 1 million it will then be in euros per person which is the same as the increase we are given in the text.
Step 1a: convert to euro per person. 0.024582*1m=24582.
Step 2: add in the increase per person… 24582+1208=25790.
Step 3: Calculate the population in 2010… 269.2 / 25790 = 0.010438.
Step 4: subtract the 2005 population…
Once again we have an incompatibility of the data to be able to do a straightforward subtraction. The GDP we calculated in step 2 and used in step 3 is in billions of euro and so the population we have now calculated is also in billions. We need to multiply it by 1 million to get it into thousands like the original data.
Step 3a: convert it to thousands… 0.010438 * 1m = 10438
Step 4: subtract the original population of 2005… 10438 – 10418 = 20.
The population data is in thousands and so the correct answer is A, increased by 20000.
Magnitudes are vital!
You must constantly be aware of the data magnitudes you are working with and ensure they are the same when adding and subtracting. If you are multiplying and dividing they you can leave the order of magnitude adjustment until the end.
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