# Interesting Numerical Reasoning Question

Q: Which period saw the greatest decrease in water prices?

How do I calculate the percentage decrease between period $y_0$ and $y_1$ for example? My initial instinct was to simply calculate: $\frac {y_0-y_1}{y_0}\times 100 = \frac{100-95}{100} \times100= 5\%$. However this is not correct. Any ideas how it's supposed to be calculated?

This is the proposed solution

• y0-y1 1-(100/95)=-5.3%
• y1-y2 1-(95/90)=-5.6%
• y2-y3 1-(90/85)=-5.8%

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Presumably, this question is asking about finding the greatest percentage decrease in water usage (seeing as your answer is a percent). If so, try computing the percentage decrease from year 1 to year 2 and from year 2 to year 3, and see what happens. If the question is asking about the greatest total decrease, then the greatest total decrease is $5$ points in the Inflation index. Finally, how do you know what is the right answer? – JavaMan Dec 19 '11 at 16:00
@JavaMan I have clarified it better in my question, I don't understand how the percentage change from y0 to y1 is calculated and so on. I have the solutions available, they say that's how you do it. I can't understand the rationale behind this method. – Catherine Dec 19 '11 at 16:12

Each of the years $y_1$, $y_2$, $y_3$ saw a decrease of 5 from the previous year; so there is no year that saw the "greatest decrease".

I suspect, they want percentage decrease, as JavaMan surmises. Then you compute $${5\over100}, {5\over95}, {5\over 90}$$ for the decimal representation of the percentage decrease for, respectively, the years $y_1$, $y_2$, and $y_3$. Then select the largest.

To find the percentage decrease, $D$, from one year, $y_{old}$ to the next year, $y_{new}$, solve $$y_{old}-\underbrace{\textstyle{D\over 100}y_{old}}_{\text{change in }\atop{\text{value}}}=y_{new}$$ for $D$. This gives $$D=100\cdot{y_{old}-y_{new}\over y_{old}}$$ (as you calculated).

In the "solution", it seems that they are giving the percentage decrease from one year to the previous year. This will allow you to select the year with the greatest percentage decrease; but it's a strange way to do it in my opinion.

I suspect the solution is in error, and they meant for the fractions therein to be "flipped". This would give the formula above (the percentage decrease from one year to the next year).

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In other words, Catherine and David Mitra used (y0-y1)/y0, but the solution used (y1-y0)/y1. This is just a matter of perspective, but I tend to think Catherine and David Mitra have the better perspective and that y0 changed into y1. – Jack Schmidt Dec 19 '11 at 17:16