GMATPrep: Find the constant increase per year

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GMATPrep: Find the constant increase per year

by nandinitaneja » Sun Aug 24, 2014 8:57 am
Hello,

I am approaching the Q below in a very math way. Manually calculating the height at each year by plugging in the answer choices as the constant rate of increase, finding 1/5th the height in Year 4, adding to year 4 height and then checking to see if that matches the value I got for Year 6. I'm certain that I am missing the logic/fastest way to approach the question, appreciate the help!

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by abhasjha » Sun Aug 24, 2014 9:02 am
We have to read the question carefully to make sure we see that the end of the 6th year is being compared to the end of the 4th year. Also note that the tree grows by a constant amount each year. This is different than the same % each year.

Let x = the constant annual growth in feet.

4ft is the original height of the tree, so after the 6th year, we have

4 + x + x + x + x + x + x....aslo 4 + 6x

After the 4th year we have 4 + x + x + x+ x or 4 + 4x

Lets compare the two values algebraically as the problem does:

6th year is 1/5 taller than the 4th year.

The difference in growth from the 4th to the 6th year will be 1/5 (or 0.2) of the height after the 4th year.

So...(4+6x)-(4+4x) = 1/5(4+4x)...now solve for x

4 + 6x -4 - 4x = 4/5 + 4/5x
2x = 4/5 + 4/5x
Subtract 4/5x from both sides. I've changed 2x into 10/5x to easily do the subtraction of fractions.

10/5x - 4/5x = 4/5

6/5x = 4/5

Divide both sides by 6/5, which is the same as multiplying by 5/6

X = 4/5 * 5/6. (Reduce the 5's or multiply out and) you have 20 / 30 or x = 2/3.

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by [email protected] » Sun Aug 24, 2014 10:29 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 6th year, the tree was 1/5 taller than it was at the end of 4th year. By how many feet the height of the tree increase each year?

1) 3/10
2) 2/5
3) 1/2
4) 2/3
5) 6/5
Height of tree on day 0 = 4
Let d = the height increase each year
Height of tree at the end of the 1st year = 4+d
Height of tree at the end of the 2nd year = 4+d+d = 4 + 2d
Height of tree at the end of the 3rd year = 4+d+d+d = 4 + 3d
Height of tree at the end of the 4th year = 4+d+d+d+d = 4 + 4d
Height of tree at the end of the 5th year = 4+d+d+d+d+d = 4 + 5d
Height of tree at the end of the 6th year = 4+d+d+d+d+d+d = 4 + 6d

At the end of the 6th year, the tree was 1/5 taller than it was at the end of the 4th year
In other words, 6th year height = 4th year height + 1/5(4th year height)
Or we can write 4 + 6d = (4 + 4d) + 1/5(4 + 4d)
Simplify: 4 + 6d = 6/5(4 + 4d)
Multiply both sides by 5 to get: 5(4 + 6d) = 6(4 + 4d)
Expand: 20 + 30d = 24 + 24d
Simplify: 6d = 4
d = 4/6 = [spoiler]2/3[/spoiler] = D

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by [email protected] » Sun Aug 24, 2014 12:05 pm
Hi nandinitaneja,

TESTing the ANSWERS is a great way to tackle this question. The "fast" way to solve a problem can still sometimes take time, but regardless of how you approach a prompt, you still need to take notes and stay organized.

From the screen capture, you chose answer C (1/2). If you jot down some quick notes, here's what you'd have:

Start = 4 ft
Yr. 1 = 4 1/2
Yr. 2 = 5
Yr. 3 = 5 1/2
Yr. 4 = 6
Yr. 5 = 6 1/2
Yr. 6 = 7

It doesn't make much time/effort to take these notes. Now, compare Year 6 to Year 4....Is it 1/5 greater? 7 to 6 is 1/6 greater, so answer C is not what we're looking for. It also gives us a "nudge" in the right direction. We need a 1/5 increase, but we only have a 1/6 increase right now....so we need a bigger increase.....so we need a bigger absolute increase each year. The correct answer has to be D or E.

Looking at all 5 choices as a group, I'm pretty sure the answer is D, but we can certainly prove it...

Start = 4 ft
Yr. 1 = 4 2/3
Yr. 2 = 5 1/3
Yr. 3 = 6
Yr. 4 = 6 2/3
Yr. 5 = 7 1/3
Yr. 6 = 8

This comparison requires a bit more math, but isn't "crazy" by any definition.

6 2/3 = 20/3
8 = 24/3

Ignore the denominators....24 to 20..... 1/5 of 20 = 4.....24 IS 1/5 greater than 20.

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by j_shreyans » Sat Feb 07, 2015 9:37 pm
Hi ,

Ii am bit confused, shouldn't be like below equation?

4+6x=1/5 + (4+4x)

Thanks

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by GMATGuruNY » Sun Feb 08, 2015 4:18 am
When a certain tree was first planted, it was 4 feet tall, and the height of the tree increased by a certain 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
Let x = the yearly increase.
Height after 4 years = 4 + 4x.
Height after 6 years = 4 + 6x.

At the end of the 6th year, the tree was 1/5 taller than it was at the end of the 4th year.
The phrase in red implies the following:
The height in the 6th year was 6/5 of the height in the 4th year.
An equivalent statement:
At the end of the 6th year, the tree was 20% taller than it was at the end of the 4th year.
Here, the phrase in red implies the following:
The height in the 6th year was 120% of the height in the 4th year.

Since the height in the 6th year was 6/5 of the height in the 4th year, we get:
(4 + 6x) = (6/5)(4 + 4x)
20 + 30x = 24 + 24x
6x = 4
x = 4/6 = 2/3.

An alternate approach is to plug in the answers, which represent the increase each year.
When the correct answer choice is plugged in, (6th-year height)/(4th-year height) = 6/5.

Height after 4 years = 4 + 4(2/3) = 20/3.
Height after 2 more years = 20/3 + 2(2/3) = 24/3.
(6th-year height)/(4th-year height) = (24/3) / (20/3) = 24/20 = 6/5.
Success!

j_shreyans wrote:Hi ,

Ii am bit confused, shouldn't be like below equation?

4+6x=1/5 + (4+4x)
Here, the value in red implies that the height increased by 1/5 FOOT from the 4th year to the 6th year.
But the prompt does not state that the tree 1/5 foot taller.
Rather, it states that the tree was 1/5 TALLER.
1/5 taller means 20% taller.
In other words, the 6th-year height was 6/5 -- or 120% -- of the 4th-year height.
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by [email protected] » Sun Feb 08, 2015 10:02 pm
j_shreyans wrote:Hi ,

Ii am bit confused, shouldn't be like below equation?

4+6x=1/5 + (4+4x)
That's close, but it's 1/5 greater, which means we need to take something and increase it by 1/5 of itself. So our equation should be 4 + 6x = (1 + 1/5) * (4 + 4x), or (4 + 6x) = (6/5)(4 + 4x).

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by Poisson » Thu Sep 01, 2016 4:43 am
j_shreyans wrote:Hi ,

Ii am bit confused, shouldn't be like below equation?

4+6x=1/5 + (4+4x)

Thanks

Shreyans
Here's my translation of the sentence:

At the end of the 6th year, the tree was 1/5 taller than it was at the end of the 4th year

4+ 6H = (4 + 4H) + 0.2(4 + 4H)

Distribute the 0.2 to both values in the parentheses:

4 + 6H = 4 + 4H + 0.8 + 0.8H

Combine like terms:

4 + 6H = 4.8 + 4.8H

4 + 1.2H = 4.8

1.2H = 0.8

Finally divide by 1.2 to get[spoiler] H = 2/3[/spoiler]

Translating the above sentence correctly was the hardest part for me

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by [email protected] » Thu Sep 01, 2016 4:12 pm
Poisson wrote: At the end of the 6th year, the tree was 1/5 taller than it was at the end of the 4th year

4+ 6H = (4 + 4H) + 0.2(4 + 4H)
This is a nice way of doing it, I think: some students are confused about how adding 1/5 can be the same as multiplying by 6/5, but this shows why in a friendly way. Some color coding could help too:

x + (1/5)x

is

1*x + (1/5)*x

is

x * (1 + (1/5))

is

x * (6/5)

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