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The value of investment Q increased by q percent from the beginning of a particular year through June 30 of that year, and then experienced no net change from June 30 to the end of the year. The value of investment P increased by p percent from the beginning of that year to June 30, and the new value increased again by p percent from June 30 to the end of the year. If the percent increase in value from the beginning to the end of the year was the same for both investments, which of the following expressions gives the value of p in terms of q?
We can pick our own smart number or use an algebraic approach; both methods are shown below.
Let the initial value of the investment be $100 (a value that works nicely with percent changes). Note that both investments can start off at $100; nothing in the problem prevents this and choosing the same starting point for the two investments will make the math easier.
It’s better to select a value for p and use that to calculate q, because investment P undergoes two consecutive percent changes while investment Q undergoes just one. Say p = 20.
In that case, investment P increases by 20% during the first half of the year. Twenty percent of the initial value of $100 is $20, so investment P goes from $100 to $120. Then, the new value of investment P increases by a further 20% during the remainder of the year. This time, 20% of $120 is $24, so investment P goes from $120 to $144.
Now, figure out investment Q, which also starts at $100. If the percent increase over the year for investment Q is the same as for investment P, then investment Q will also be worth $144 at the end of the year. That’s a percent increase of 44%, so q = 44.
Plug q = 44 into the answer choices, and look for the desired result of p = 20:
The correct answer is A.
If the overall percent increase was the same for both investments, then the initial investments must have been multiplied by the same overall factor in the end. (For instance, if both investments went up by a net total of 30 percent, then both initial values would be multiplied by 1.30 to produce the final values.)
Investment P undergoes two consecutive changes of p percent, so the initial value of that investment is multiplied by a factor of . Investment Q increases by q percent and then undergoes no further changes, so the initial value of that investment is multiplied by a factor of . Since both factors must be the same in the end, set them equal to each other and solve for p:
Multiply by 100 on the left and (the same as 100) on the right:
The correct answer is A.
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