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Question with Manhattan Functions Strategy section

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Question with Manhattan Functions Strategy section [#permalink] New post 16 Feb 2013, 20:13
I am having some difficulty understanding one of the prep questions in the Functions Strategy section of the Manhattan GMAT Prep book, 4th Edition. The problem, on page 77, is as follows:

"Jake was 4 1/2 feet tall on his 12th birthday, when he began to have a growth spurt. Between his 12th and 15th birthdays, he grew at a constant rate. If Jake was 20% taller on his 15th birthday than on his 13th birthday, how many inches per year did Jake grow during his growth spurt? (12 inches = 1 foot)"

The explanation of the answer is as follows:

In this problem, the constant growth does not begin until Jake has reached his twelfth birthday, so in order to use the constant growth function y = mx + b, let time x = 0 (the initial state) stand for Jake's twelfth birthday. Therefore, x = 1 stands for his 13th birthday, x = 2 stands for his 14th birthday, and x = 3 stands for his 15th birthday.

The problem asks for an answer in inches but gives you information in feet. Therefore, it is convenient to convert to inches at the beginning of the problem: 4 1/2 feet = 54 inches = b. Since the growth rate m is unknown, the growth function can be written as y = mx + 54. Jake's height on his 13th birthday, when x = 1, was 54 + m, and his height on his 15th birthday, when x = 3, was 54 + 3m, which is 20% more than 54 + m. Thus we have:

54 + 3m = (54 + m) + 0.20(54 + m)
54 + 3m = 1.2(54 + m)
54 + 3m = 64.8 + 1.2 m
1.8 m = 10.8
m = 6
Therefore Jake grew at a rate of 6 inches each year.

My question is in the red above, I don't understand how they worked out that 3m is 20% more than 3, I would never think of answering the problem that way. I would really appreciate it if any of you math gurus could give me an explanation for why this is the case. Thanks a ton!

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Re: Question with Manhattan Functions Strategy section [#permalink] New post 17 Feb 2013, 09:16
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That explanation seems a bit complicated. If he's growing at a constant rate, then we can assume he grows g inches per year. Then in inches, his heights on each birthday are:

12th bday: 54
13th bday: 54 + g
14th bday: 54 + 2g
15th bday: 54 + 3g

We know his height on his 15th birthday is 20% greater than on his 13th birthday. If one thing is 20% greater than another, it is 1.2 times the other. So we get an equation:

54 + 3g = 1.2(54 + g)
270 + 15g = 6(54 + g)
270 + 15g = 324 + 6g
9g = 54
g = 6

I multiplied both sides by 5 to get rid of decimals in the second step, but that's not necessary if you're happy working with decimals.

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Re: Question with Manhattan Functions Strategy section [#permalink] New post 20 Jun 2014, 22:37
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Re: Question with Manhattan Functions Strategy section   [#permalink] 20 Jun 2014, 22:37
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