Jake was 4 1/2 feet tall on his 12th birthday, when he began to have

Question

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)

A. 4
B. 5
C. 6
D. 7
E. 8

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 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!

IanStewart · Accepted Answer

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.

Green2k1 · Answer

Hi  Bunuel

in this question - it is mentioned "he grew at a constant rate" - I interpreted it as constant percentage. Could you please help me out in understanding when to consider % and when to consider a fix number as mentioned above in one of the solution.

rocky620 · Answer

Green2k1

I also had the same dilemma but the question says    how many inches per year  did Jake grow during his growth spurt? (12 inches = 1 foot)

So, it translates to a constant rate in terms of  number of inches increase per year  and  not to a constant rate in terms of percentages .

I hope I am correct.

Green2k1 · Answer

rocky620 
i am not much concern about this particular question but mainly i have asked from concept clarification view point. And my question is related to bold face section.

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)

Although @buniel has not posted this question, hope he will resolve this doubt..

sahibr · Answer

I used a different approach from the one that Ian used.

The word   constant   in the question jumps out at me.
If the growth is constant, it must be a straight line and we can form a straight line equation.

Let there be 3 points, where x coordinate represents the age and y coordinate represents the height:
Point A : (12, 54) ............ {As 4 1/2 feet is 54 inches}
Point B : (13, a) 
Point C : (15, 1.2a) ............. Given in the question that height at age 15 is 20% higher than age 13, hence 1.2 times.

Now using point B and C to form a straight line equation {y-y1= \frac{y2-y1}{x2-x1}  * (x-x1) }

So y-a= \frac{1.2a-a}{15-13} *(x-13)

Solving the above we get,  10y+3a=ax

Substituting values of point A(12,54) for x,y in the above we get,

a=60.

At age 15, height is 1.2a=1.2*60=72

Growth per year= \frac{Height at 15-Height at 12}{3} 
= \frac{72-60}{3} =6 inches

Hence Ans is C

Riri15 · Answer

bb   Bunuel  How do we know the constant rate has to be taken as a number and not a percentage? wouldn't constant rate be a percentage?

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Jake was 4 1/2 feet tall on his 12th birthday, when he began to have

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