Smog City has a problem with air pollution. The concentration of pollution at the city centre is 0.4 ppm. For every additional five kilometres from the city centre, the concentration of pollutants in the air drops by 11 percent. For example, at 10 kilometres from the city centre, the concentration of air pollutants is 11 percent less than it is at five kilometres. Work out the concentration of air pollutants 25 kilometres away from the city centre. Give your answer to two significant figures.
We can answer this question by breaking it down into a few steps. First, we’ll begin by working out the concentration of pollution five kilometres away from the city centre. To do this, we’re going to need to reduce 0.4 ppm by 11 percent. And that’s because the concentration of pollution at the city centre is 0.4 ppm and we have moved five kilometres away.
There are two ways we could work this out. We could find 11 percent of 0.4 using a number of different methods. And then we could subtract that from 0.4. However, a much quicker way is to consider the decimal multiplier that represents an 11-percent reduction. The original amount is always 100 percent. So we can find an 11-percent reduction by subtracting that from 100. That’s 89 percent. So an 11-percent reduction to a number is the same as finding 89 percent of that same number.
Percent means out of 100. So we divide 89 percent by 100 to find the corresponding decimal multiplier. That’s 0.89. To find an 11-percent reduction, we multiply by 0.89. And that tells us that, five kilometres away from the city centre, the concentration of pollution is given by 0.4 multiplied by 0.89 ppm. We don’t actually need to work this out yet because we can use this exact sum in the next calculation.
Next, we want to find the concentration of pollution 10 kilometres away. To do this, we take the value at five kilometres — that’s 0.4 multiplied by 0.89 — and we multiply it once again by that decimal multiplier, by 0.89. In fact, we can say this is the same as 0.4 multiplied by 0.89 squared because 0.89 multiplied by itself is 0.89 squared.
Similarly, to find the concentration of pollution 15 kilometres away, we take the value at 10 kilometres and work out an 11-percent reduction. Once again, we use the decimal multiplier. And that’s 0.4 multiplied by 0.89 squared multiplied by 0.89. This time, that’s the same as 0.4 multiplied by 0.89 cubed.
You should be able to spot a pattern now. At 20 kilometres, we multiply the value from 15 kilometres by 0.89 again. That’s 0.4 multiplied by 0.89 to the power of four. And we multiply this by 0.89 once again to get the value of air pollution at 25 kilometres. It’s 0.4 multiplied by 0.89 to the power of five.
We can now type this sum into our calculator. And we get 0.223362. We’re being told though we need to give our answer correct to two significant figures. The first significant figure is the first nonzero digit. Here that’s two. That means the second significant figure in our number is the digit immediately to its right. It’s this two. The three is what we call the deciding digit.
And remember, if that deciding digit is less than five, we round our number down. And if it’s five or above, we round the number up. In this case, it’s less than five. So our number is closer to 0.22 than it is to 0.23. And we can say that the concentration of air pollutants 25 kilometres away from the city centre is 0.22 ppm.