theperfectgentleman wrote:

At the end of the year 1998, Shepard bought nine dozen goats, Henceforth, every year he added p% of the goats at the beginning of the year and sold q% of the goats at the end of the year where p > 0 and q > O. If Shepard had nine dozen goats at the end of year 2002, after making the sales for that year, which of the following is true?

(1) p = q

(2) P < q

(3) p> q

(4) P = q/2

(5) P = q/4

Good question.

1. When you increase by a percent, then decrease by that same percent, you do not end up where you began. That's a trap. Eliminate A.

2. In fact, if the original ends up at the same value, p% increase is inversely related to q% decrease. Forget 9 dozen. We just need to watch any quantity increase and decrease by percentages and return to its original value. Use 100.

3. If that 100 increases by p% = 25 at the beginning of the year, there are 100 * 1.25 = 125 goats that, um, Shepard :wink: , must herd. Or feed. Or whatever.

By what percentage q must that 125 decrease in order to return to 100?

By 1 - (fractional inverse of the increase).The increase, from decimal to fraction form, is 1.25 = 1\(\frac{25}{100}\) = 1\(\frac{1}{4}\) = \(\frac{5}{4}\)

Because percent increase and percent decrease are inversely proportional when the original quantity is the start and end value, flip the percent increase fraction from \(\frac{5}{4}\) to \(\frac{4}{5}\) to get the percent decrease multiplier. ------> 125 *\(\frac{4}{5}\) = 100

So 1 - \(\frac{4}{5}\)= \(\frac{1}{5}\) or a 20% decrease. Shepard sells 20% of the goats at the end of the year.

Alternatively, \(\frac{4}{5}\) = .8 = 80%, which is a 20% decrease.

4. p% increase = 25, q% decrease = 20.

p > q p > q. It doesn't matter that there are four years involved.

It wouldn't matter if there were 40 years involved.

Every year, the number of goats starts at the same value and returns to that same value;

the +25% and -20% just keep getting repeated.

p > q Answer C

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In the depths of winter, I finally learned

that within me there lay an invincible summer.

-- Albert Camus, "Return to Tipasa"