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height of a tree

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height of a tree

by LevelOne » Sat Jun 20, 2009 1:15 am
Hi guys,

What's the quickest way to solve these 2 problems? I might be missing some of the logic here...

1) When a certain tree was first planted, it was 4 feet tall, and the height of the tree increased by a constant amount each year for the next 6 years. At the end of the 6th year, the tree was 1/5 taller than it was at the end of the 4th year. By how many feet did the height of the tree increase each year?

A. 3/10
B. 2/5
C. 1/2
D. 2/3
E. 6/5

2) If 2^x -2^x-2 = 3(2^13), what is the value of x?

A. 9
B. 11
C. 13
D. 15
E. 17

Thanks and the OAs will follow.
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by nitya34 » Sat Jun 20, 2009 1:37 am
1) When a certain tree was first planted, it was 4 feet tall, and the height of the tree increased by a constant amount each year for the next 6 years. At the end of the 6th year, the tree was 1/5 taller than it was at the end of the 4th year. By how many feet did the height of the tree increase each year?

A. 3/10
B. 2/5
C. 1/2
D. 2/3
E. 6/5
===============
draw a diagram

tree height was 4-------->becomes 4+4x---------->then 4+6x


now 4+6x=(4+4x)(1+ 1/5)
=>x=2/3
IMO D
===============
2) If 2^x -2^(x-2) = 3(2^13), what is the value of x?

A. 9
B. 11
C. 13
D. 15
E. 17
========
2^x(1 - 1/4) = 3 (2^13)

=>2^x=2^15

IMO D
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by LevelOne » Sat Jun 20, 2009 1:44 am
Thanks, D is correct. However, I don't understand this bit: (1 - 1/4).
can you explain, please?
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by nitya34 » Sat Jun 20, 2009 1:49 am
If 2^x -2^(x-2) = 3(2^13)
=>2^x - (2^x/ 2^2) = 3 (2^13)

=>2^x(1 - 1/4) = 3 (2^13)

=>2^x=2^15

hope it clears doubt :)
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by shanmugam.d » Sat Jun 20, 2009 1:52 am
Hi the solutions are as follows:

1)
1st year the tree growth = 4 + r (r is the growth)
2nd year = 4+2r
3rd year = 4+3r
-
-
-6th year = 4 + 6r
the problem says: (4+6r) = (4+4r) (1+1/5)
-> 4+6r = 24/5+24r/5
->6r=4
->r=2/3

2) 2^x -2^x-2 = 3(2^13)
-> 2^x -2^x/2^2 = 3(2^13)
->(4*2^x -2^x)/2^2 = 3(2^13)
->(4*2^x -2^x) = 3(2^13)* 2^2
->3*2^x = 3(2^15)
->x =15
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by rahulg83 » Sat Jun 20, 2009 2:27 am
When a certain tree was first planted, it was 4 feet tall, and the height of the tree increased by a constant amount each year for the next 6 years. At the end of the 6th year, the tree was 1/5 taller than it was at the end of the 4th year. By how many feet did the height of the tree increase each year?

A. 3/10
B. 2/5
C. 1/2
D. 2/3
E. 6/5

Can somebody explain how this statement leads to equation

(4+6x) = (4+4x)(1+1/5) ??
what does 1/5 taller mean? 1/5 times or what? :?
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by abcdefg » Sun Jul 19, 2009 7:57 am
Damn this tree problem had me spinning like crazy because of the wording. I understand how to do it but can someone clarify for me how to recognize what the heck gmat is asking.

1. "grew by a constant amount each year" - so is this saying that they grew by 3 inches every yr or saying grew 3 times larger than did they the previous year?
2. "it was 1/5 taller"- okay I know where this is going but it got me confused again because I wasn't exactly sure whether they were asking for +1/5 inches or (1/5) times!

Can someone clarify for me how to recognize this? thanks.
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by ghacker » Sun Jul 19, 2009 9:11 am
Lets assume that the tree grow by a constant "a" feet per year

Then after 6 years the height of the tree is 1/5 more than the height of the tree at 4 years

so 2a = (4+4a)/5 -----> 10a= 4+4a -------> 6a=4

a = 2/3

ans = 2/3
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by Stuart@KaplanGMAT » Sun Jul 19, 2009 9:21 am
If 2^x -2^(x-2) = 3(2^13)
If we're multiplying or dividing powers, the rules are straightforward. However, there is no quick way to add or subtract exponents. For example:

3(2^9) + 7(2^8) = ?

So, we need to be sneaky. We recognize that when the base and the power are the same, we simply add or subtract the coefficients. For example:

3(2^8) + 7(2^8) = 10(2^8)

Therefore, in order to add or subtract, we need to equalize the exponents. Before going to the actual question, let's look at the simpler example I posted above:

3(2^9) + 7(2^8)

Most people find it easiest to reduce the bigger exponent to match the smaller one. So, in this case, let's reduce 2^9 to 2^8.

To do so, we use our multiplication rule:

x^a * x^b = x^(a+b)

in reverse. We can expand 2^9 to:

2^8 * 2^1

and rewrite the entire expression as:

3(2^1)(2^8) + 7(2^8)

or

6(2^8) + 7(2^8)

Finally, we add the coefficients to get:

13(2^8)

In essence, we've factored out the common term so we can add the powers together.

Let's focus on the left side of the equation posted:

2^x - 2^(x-2)

This question is a bit trickier because of the x-2 in the exponent, but here's a fundamental rule of math: no matter how strange things look, the same basic rules apply. So, let's equalize the exponents to (x-2), the smaller power.

We can rewrite 2^x as (2^(x-2)) * 2^2. Now we have:

2^2 * 2^(x-2) - 2^(x-2)

4(2^(x-2)) - 1(2^(x-2))

3(2^(x-2))

going back to the full equation:

3(2^(x-2)) = 3(2^13)

2^(x-2) = 2^13

x-2 = 13

x=15

Once you understand the theory and approach, you can answer these questions very quickly.
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by shibal » Sun Jul 19, 2009 10:34 am
why do I have to add 1 to 1/5?
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by Stuart@KaplanGMAT » Sun Jul 19, 2009 11:54 am
shibal wrote:why do I have to add 1 to 1/5?
Because 1/5 taller means the same height plus 1/5 more.

So, if we want to find the height that's 1/5 taller than x:

x + (1/5)x = (1 + 1/5)x

Similarly, 20% taller would mean your current height plus 20%. So, if we wanted to know what height is 20% taller than 5 feet, the answer would be:

100%(5) + 20%(5) = 120%(5)
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Re: height of a tree

by doclkk » Fri Jul 31, 2009 1:25 pm
LevelOne wrote:Hi guys,

What's the quickest way to solve these 2 problems? I might be missing some of the logic here...

1) When a certain tree was first planted, it was 4 feet tall, and the height of the tree increased by a constant amount each year for the next 6 years. At the end of the 6th year, the tree was 1/5 taller than it was at the end of the 4th year. By how many feet did the height of the tree increase each year?

A. 3/10
B. 2/5
C. 1/2
D. 2/3
E. 6/5

2) If 2^x -2^x-2 = 3(2^13), what is the value of x?

A. 9
B. 11
C. 13
D. 15
E. 17

Thanks and the OAs will follow.
I don't know about quickest - but I know when I was reading the explanations and people had 4 + 6x = (4+4x)(1+1/5) - had me confused.

So for the people that didn't understand that part. It means this:

4 + 6x = (4+4x) + [(4+4x)/5]

The equation is really simple to do then. It's the same thing but this way is clearer.
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by shahdevine » Fri Jul 31, 2009 2:58 pm
First step: Break down years. always put down year 0 because that's how they trip you up in problems via overcounting or undercounting.

year 0 = 4
year 1 = 4+x
year 2 = 4+2x
year 3= 4+3x
year 4 = 4+4x
year 5 = 4+5x
year 6 = 4+6x

2nd step: Translate

year 6 = 1/5 more than year 4 --> 1/5 means 1+1/5 or 6/5

so 4+6x=6/5(4+4x)

x=2/3


--------------------------------------------------------------------

2^x - 2^(x-2) = 3(2^13)

key here is to understand properties.

2^(x-2) --> 2^x*2^-2

then distribute out 2^x --> 2^x(1-2^-2)

rest is cake

cheers,

sd
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