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The graph models the hypothetical mass, in kilograms, of a Tyrannosaur

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Re: The graph models the hypothetical mass, in kilograms, of a Tyrannosaur [#permalink]
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Official Explanation

RO1: Infer

The most straightforward way to calculate the arithmetic mean of the masses of T. rex for the 19 integer values of its age from 12–30 is to read the values of the mass corresponding to each of these ages from the graph, add them together, and divide the result by 19.

But since each option in the menu is a 1,000-year range, a rough approximation should be sufficient for this problem. Since the graph is approximately linear from age 12 to age 20, the value of the mass corresponding to the midpoint of that interval (i.e., 3,000, the vertical coordinate of point B) serves as a good approximation of the arithmetic mean for the 9 corresponding integer values of the age. From age 21 to age 30, the range of the mass is quite narrow (approximately 5,400–5,600). Using 5,500 to approximate the mass from age 21 to age 30 leads to the following estimate for the arithmetic mean:

$$\small \frac{1}{19}[9(3,000) + 10(5,500)] = \small \frac{1}{19}(82,000) = 4,315\small \frac{15}{19}$$

This value falls between 4,000 and 5,000.

The correct answer is 4,000 and 5,000.

RO2: Infer

The masses at ages 12, 16, and 20 are equal to the vertical coordinates of points A, B, and C—approximately 900, 3,000, and 5,100—respectively. Since 3,000 – 900 and 5,100 – 3,000 are both equal to 2,100, the numerical change in the mass over each 4-year period is roughly the same (as can also be seen from the symmetry of the graph).

But the percent change in mass from age 12 to age 16 must be larger than that from age 16 to age 20, because the mass at age 12 is less than the mass at age 16. Carrying out the calculation, the percent change in mass from age 12 to age 16 is equal to

$$\small \frac{Mass\: at\: age\: 16-Mass\: at \: age \: 12}{Mass\: at\: age\: 12}(100)$$

which is approximately $$\small \frac{2,100}{900}(100)$$, or $$\small 233\tfrac{1}{3}.$$ Likewise, the percent change in mass from age 16 to age 20 is equal to

$$\small \frac{Mass\: at\: age\: 20-Mass\: at \: age \: 16}{Mass\: at\: age\: 16}(100)$$

which is approximately $$\small \frac{2,100}{3,000}(100)$$, or 70. Since $$\small \frac{233\tfrac{1}{3}}{70} = \small \small 3\tfrac{1}{3}$$, the percent change in mass from age 12 to age 16 is roughly 3 times the corresponding percent change from age 16 to age 20.

The correct answer is 3 times.
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Re: The graph models the hypothetical mass, in kilograms, of a Tyrannosaur [#permalink]
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a: 4000-4500 KG
b: % change from 12 to 16 is approximately three times of % change from 16 to 20..

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Re: The graph models the hypothetical mass, in kilograms, of a Tyrannosaur [#permalink]
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Video solution from Quant Reasoning starts at 18:20
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Re: The graph models the hypothetical mass, in kilograms, of a Tyrannosaur [#permalink]
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Kudos
Quote:
Here is the most optimal method to solve this question.

­Now, let us solve the first question that is asking us for integer values of the age from 12 to 30, the average (arithmetic mean) mass falls approximately between ___________ and _____________ kilograms.

First of all, we have been given a graph with a curve. So, to determine the exact value is almost impossible. However, we can divide the graph into two straight lines: one from A to C and then from C to 30.

Now, we have to remember a very simple thing: The average of a straight line is equal to its median. So, the average of the fist line is 3000 and the average of the second line is approximately 5500.

Now, we will find the weighted mean of both of them, which will be (3000*8 + 5500*10)/18 = (24000 + 55000)/18 = (79000)/18 = (39500)/9 = 4800 approximately. Hence, the answer must be between 4000 to 5000.

Okay, this question is easier. All you have to find is percentage change between 12 to age 16 and 16 to age 20.

%age change between 12 to 16 = (3000-1000)/1000 = 2
%age change between 16 to 20 = (5000-3000)/3000 = 2/3

Clearly, 2 is 3 times of 2/3. Hence, three times is our answer.
Re: The graph models the hypothetical mass, in kilograms, of a Tyrannosaur [#permalink]
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