What is the average annual increase in the pollination rate?

Question

This is an SAT practice-test question:

The graph above shows the pollination rate of plants in a forested area every two years. Based on the trend line, what is the average annual increase in the pollination rate?

a) $5\%$
b) $3.8\%$
c) $2.5\%$
d) $1.5\%$

I first calculated the total percentage increase from $2000$ to $2014$ $$90/53 = 1.698113...$$

Since I am looking for the average annual increase $A,$ I then find the total increase during $14$ years $$A^{14} = 1.698113.$$

Thus option (b) $A = 1.038547$ is my answer, but the answer key says that the answer is option (c). Is the answer key wrong, or have I made a mistake?

Answer 1

The dotted line that is shown is the line of best fit, the slope of which better demonstrates the average rate of change than first minus last. So rather than taking the $90$ and $53$ in that equation, look at the dotted line and estimate where it is in $2014$ and in $2000$.

THEN also, you shouldn't be thinking $A^{14}=\text{final/initial}$ since the $y$-axis is already measuring percentages. Instead, we want $14A=\text{final} - \text{initial}$, which is just a rearrangement of the regular equation for slope:

$$ A=\frac{\text{final} - \text{initial}}{2014-2000}. $$

Answer 2

I first calculated the total percentage increase: $90/53 = 1.698113...$

To be clear: from 2000 to 2014,

• the pollination rate (itself a percentage figure) increased from $53\%$ to $90\%;$

• its percentage increase was $69.8\%$

  (it increased by $69.8\%$ of its original value);

• its (total) increase was $37\%$

  (it increased by $37$ percentage points).

Because the quantity under discussion is a percentage figure, the statement $$\text{“the pollination rate increased from $10\%$ by $8\%$”}$$ is ambiguous: it's unclear whether the final pollination rate is $10.8\%$ or $18\%.$

(Typically, the quantity being discussed isn't a percentage figure, and “the length increased from $10$cm by $8\%$” and “the length increased from $10$cm by $8$cm” unambiguously mean that the new length is $10.8$cm and $18$cm, respectively.)

what is the average annual increase in the pollination rate?

By definition, the average annual increase in the pollination rate is $$\frac{\text{the total increase in pollination rate}}{\text{the number of years in the entire duration}}=\frac{37\%}{14}=2.6\%.$$ In other words: every year, on average, the pollination rate increased by $2.6$ percentage points.

Since I am looking for the average annual increase $A, A^{14} = 1.698113.$ Thus $A = 1.038547$

No, $3.9\%$ is the average annual percentage increase in the pollination rate.

In other words: every year, on average, the pollination rate increased by $3.9\%$ of its previous year's value.