In the country of Celebria, the Q-score of a politician is computed fr

From the question stem, we know that

--> Q=\(\frac{(41ab^2c^3)}{(d^2)}\)

--> Q(Mayor) = 250% * Q(Councillor) --------------------- (1)

Let the Q(Councillor) be measured with values a,b,c & d.

==> Q(Councillor) = \(\frac{(41ab^2c^3)}{(d^2)}\)

Let Q(Mayor) be measured with values A,B,C & D.

==> Q(Mayor) = \(\frac{(41AB^2C^3)}{(D^2)}\)

Let's rewrite the values A,B, & C of Q(Mayor) in terms of a,b,& c (from question stem)

A= 160% * a = \(\frac{160}{100}\) * a = \(\frac{8}{5}\) * a

B = 140% * b = \(\frac{140}{100}\) * b = \(\frac{7}{5}\) * b

C = 80% * c = \(\frac{80}{100}\) * c = \(\frac{4}{5}\) * c

Substituting the values in the equation (1),

Q(Mayor) = 250% * Q(Councillor)

\(\frac{(41AB^2C^3)}{(D^2)}\) = \(\frac{250}{100}\) (\(\frac{(41ab^2c^3)}{(d^2)}\))

\((41* \frac{8}{5} * a * \frac{7}{5}^2 * b^2* \frac{4}{5}^3 * c^3)/(D^2)\) = \(\frac{5}{2}\) (\(\frac{(41ab^2c^3)}{(d^2)}\))

Cancelling on both sides and cross- multiplying, we get

\(D^2\) = \((\frac{2}{5} * \frac{8}{5} * \frac{7}{5}^2 * \frac{4}{5}^3)\) * \(d^2\)

\(D^2\) = \((\frac{16}{25} * \frac{7}{5}^2 * \frac{4}{5}^3)\) * \(d^2\)

Taking square root on both sides,

D = \(\sqrt{(\frac{16}{25} * \frac{7}{5}^2 * \frac{4}{5}^3)}\) * \(\sqrt{d^2}\)

D= \(\frac{4}{5} * \frac{7}{5} * \sqrt{\frac{16}{25} * \frac{4}{5}}\) * d

D= \(\frac{4}{5} * \frac{7}{5} * \frac{4}{5} * \sqrt{\frac{4}{5}}\) * d

D= \(\frac{4}{5} * \frac{7}{5} * \frac{4}{5} * 1\) * d [ P.s Square root of 4 = 2 and Square root of 5 = 2.4 => Cancel out (Approx.)]

D = \(\frac{112}{125}\) * d =~ 0.8 * d = 80% * d

Therefore , D is 20% lower than d

Correct Answer is (D)

MANHATTAN GMAT OFFICIAL SOLUTION:

The form of the function we are comparing involves only multiplication, division, and exponents—there is no addition or subtraction. This is why a particular answer will be generated from percent changes in the variables (otherwise, you’d need to know more than percent changes).

All the comparisons are to Councilor Plant’s numbers, so we can pick values for his or her numbers: say, 1 throughout for Q, a, b, and c. Thus the numbers for Mayor Flower will be as follows, converting to fractions on the fly (we’ll fully reduce later):
Q = 25/10
a = 16/10
b = 14/10
c = 8/10

Since the problem asks for d, rearrange the formula to isolate d, or d^2 (we can take the square root at the end):
d^2 = (41ab^2c^3) / Q

We can ignore the 41, so that the “starting” value (Plant’s value) of d would be 1 (all of Plant’s other values are 1). Then Flower’s value of d^2 would be this:
d^2 = (ab^2c^3) / Q = (16/10)*(14/10)^2*(8/10)^3/ (25/10)

Turn 16/10 into 8/5. 14/10 squared becomes 196/100, or with a little rounding, 2. Meanwhile, 8/10 cubed becomes 512/1,000, or with a little more rounding, 0.5. This rounded 0.5 and the rounded 2 multiply to 1, leaving this:
d^2 = ab^2c^3/ Q = (8/5)/(25/10)= 80/125 .

Multiply top and bottom by 8 to get 640/1000, or 64/100. Finally, take the square root to get d = 8/10, or 0.8. Since d started off at an assumed 1 (for Councilor Plant), we can see that Mayor Flower’s 0.8 is 20% less.

The correct answer is D.

I understand, the solution but isnt it just too much calculation for a PS quant question?

You can also look at the proportions to solve the problem. Mayor F's score is 2.5 Council P's score. And the follow are the proportions of a, b, c and d to the score Q:

a is directly proportional (10% increase in a = 10% increase in Q)
b is directly proportional squared (10% increase in b = 10%^2 increase in Q)
c is directly proportional cubed (10% increase in c = 10%^3 increase in Q)
d is inversely proportional squared (10% increase in d = 1/10%^2 decrease in Q)

So... 2.5 = (1.6 * 1.4^2 * 0.8^3) / d^2
= 1.6 * 1.96 * 0.51) / d^2
or approx = (1.6 * 2 * .5) / d^2
= (3.2 * .5 ) / d^2

So... 2.5 = 1.6 / d^2

Cross multiply... d^2 = 1.6/2.5
Multiply by 10/10... d^2 = 16/25
Take the square root of each side... d = 4/5

So, d = 8/10 or 80%, which is a decrease of 20%

Answer D

Given: In the country of Celebria, the Q-score of a politician is computed from the following formula:

\(Q = \frac{(41ab^2c^3)}{d^2}\), in which the variables a, b, c, and d represent various perceived attributes of the politician, all of which are measured with positive numbers. Mayor Flower’s Q-score is 150% higher than that of Councilor Plant; moreover, the values of a, b, and c are 60% higher, 40% higher, and 20% lower, respectively, for Mayor Flower than for Councilor Plant.

Asked: By approximately what percent higher or lower than the value of d for Councilor Plant is the corresponding value for Mayor Flower?

\(Q = \frac{(41ab^2c^3)}{d^2}\)
\(Q_1/Q_2 = a_1b_1^2c_1^3d_2^2/a_2b_2^2c_2^3d_1^2\)
\(2.5 = 1.6(1.4)^2(.8)^3 (d_2/d_1)^2\)
\((\frac{d_2}{d_1})^2 = \frac{2.5}{1.6(1.4)^2(.8)^3}\)
\(\frac{d_2}{d_1}^2 = \frac{5}{2} * \frac{5}{8} * \frac{5^2}{7^2} * \frac{5^3}{4^3} = \frac{5^7}{2^{10}7^2}\)
\(\frac{d_2}{d_1} = \frac{5^3*\sqrt{5}}{2^5*7} = \frac{125*2.23}{32*7} = 275/224 = 55/45 = 11/9\)
\(d_1/d_2 = 9/11 = 1 - 2/11\)

IMO D

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