When a certain tree was first planted, it was 4 feet tall

[condenach]

GMATprep. Exam 1. Question 31

When a certain tree was first planted, it was 4 feet tall, and the heigth 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 increased each year?

Answers: 3/10, 2/5, 1/2, 2/3, 6/5

The correct answer is 2/3 but I have no idea how to solve it. Any help?

thanks

[StaceyKoprince]

This is essentially a sequence problem in disguise. Let x = amount of yearly growth, in feet.

Yr0 = 4
Yr1 = 4+x
Yr2 = 4+x+x=4+2x
Yr3 = 4+x+x+x=4+3x
Yr4 = 4+x+x+x+x=4+4x
Yr5 = 4+x+x+x+x+x=4+5x
Yr6 = 4+x+x+x+x+x+x=4+6x

We are told the amount at the end of Year 6 is 6/5 of the amount at the end of year 4. Thus we can write:

4+6x = 6/5 (4+4x)
5(4+6x) = 6(4+4x)
20+30x = 24+24x
6x=4
x=2/3

[JadranLee]

Hi Condenach,

You're right, there was a typo in Stacey's original explanation. I edited her explanation, so I deleted your follow-up question.

Thanks.

-Jad

Thanks for the nice explanation Stacey. Although I´m getting lost when you say that at the end of the 6th year:

6th year: 4+x+x+x+x+x+x= (4+x) + (4+x)1/5

I dont get this second part of the ecuation... Why is it 4+x, shouldnt it be 4+x+x+x+x?

If we try with this ecuation:

6th year: 4+x+x+x+x+x+x= (4+x+x+x+x) + (4+x+x+x+x)1/5 and we solve we get the same result

4+6x= 4+4x + 4/5+ 4x/5

2x - 4x/5= 4/5 ----> 6x=4 ----> x=2/3

So I guess it must be the same.

Thanks a lot for such great help

[myt]

This is essentially a sequence problem in disguise. Let x = amount of yearly growth, in feet.

We are told the amount at the end of Year 6 is 6/5 of the amount at the end of year 4. Thus we can write:

How can we say 6/5 when the question mentions 1/5 ? Please explain :(

[dhoomketu]

This is essentially a sequence problem in disguise. Let x = amount of yearly growth, in feet.

We are told the amount at the end of Year 6 is 6/5 of the amount at the end of year 4. Thus we can write:

How can we say 6/5 when the question mentions 1/5 ? Please explain :(

I agree the statement 6th year is 1/5 tall 4th year is confusing.

For e.g. one could assume

6th year - 4th year = 1/5 ; which leads you to nowhere

actually the correct equation is similar to %change i.e. (6th year - 4th year)/4th year = 1/5 and this leads to 2/3

[RonPurewal]

For e.g. one could assume

6th year - 4th year = 1/5 ; which leads you to nowhere

wrong.
the gmat is extremely fastidious about words and details. if this were the intended meaning, then the problem would have to say "1/5 foot". it doesn't, so the 1/5 MUST refer to a fraction of the aforementioned original quantity.

think about other examples and you'll see that this is correct: you can't, for instance, say "tim is 4 older than joe" if you mean "tim is 4 years older than joe".

--

DIGRESSION - caveat lector: the rest of this post has nothing to do directly with the original problem

there are, however, a couple of instances of genuine ambiguity, in which foreign readers must simply learn the common interpretation of certain phrasings. for instance,
temperature X is more than 20 degrees below the melting point of substance Y is, strictly speaking, genuinely ambiguous.
it could be read as
(1) temperature X is more than 20 degrees below the melting point of substance Y
or as
(2) temperature X is more than 20 degrees below the melting point of substance Y

if the melting point of substance Y were 87 degrees, then (1) would mean X < 67, and (2) would mean X > 67.
frustratingly - and dangerously, if X is a dangerous chemical - you MUST know that the correct interpretation is #1. native english speakers, even if they aren't that smart, will understand this without even stopping to think about it, but second-language english learners will be understandably confused.

[gkhan]

This is essentially a sequence problem in disguise. Let x = amount of yearly growth, in feet.

Yr0 = 4
Yr1 = 4+x
Yr2 = 4+x+x=4+2x
Yr3 = 4+x+x+x=4+3x
Yr4 = 4+x+x+x+x=4+4x
Yr5 = 4+x+x+x+x+x=4+5x
Yr6 = 4+x+x+x+x+x+x=4+6x

We are told the amount at the end of Year 6 is 6/5 of the amount at the end of year 4. Thus we can write:

4+6x = 6/5 (4+4x)
5(4+6x) = 6(4+4x)
20+30x = 24+24x
6x=4
x=2/3

Thanks skoprince!

[RonPurewal]

Thanks skoprince!

glad it helped

[vijaykumar.kondepudi]

I also made the same mistake on the exam..

Height of tree in 6th yr - Height of tree in 4th yr = 1/5

And since the resultant answer(1/10) dind't match any of the given answers, I resorted to a random guess :)

But, regarding the example Ron gave:

the gmat is extremely fastidious about words and details. if this were the intended meaning, then the problem would have to say "1/5 foot". it doesn't, so the 1/5 MUST refer to a fraction of the aforementioned original quantity.

think about other examples and you'll see that this is correct: you can't, for instance, say "tim is 4 older than joe" if you mean "tim is 4 years older than joe".

"tim is 4 older than joe" doesn't mean that Tim is 4 times as old as Joe..Right?

In the original question is the word "times" implied?
How do non-native speakers recognize such statements?

Thanks

[mschwrtz]

Interesting question vijaykumar.kondepudi, but no. Integers work differently than either fractions or percents. Note that the particuar values used below are arbitrary.

FRACTIONS:
1/5 greater than x = x + (1/5)x=(6/5)x

PERCENTS
15% less than y=y-%15y=85%y

INTEGERS
7 more than z=z+7

And your example, "tim is 4 older than joe" doesn't mean anything, alas. If you put in a unit, though, it'll signal addition.

[agha79]

This is essentially a sequence problem in disguise. Let x = amount of yearly growth, in feet.

Yr0 = 4
Yr1 = 4+x
Yr2 = 4+x+x=4+2x
Yr3 = 4+x+x+x=4+3x
Yr4 = 4+x+x+x+x=4+4x
Yr5 = 4+x+x+x+x+x=4+5x
Yr6 = 4+x+x+x+x+x+x=4+6x

We are told the amount at the end of Year 6 is 6/5 of the amount at the end of year 4. Thus we can write:

4+6x = 6/5 (4+4x)
5(4+6x) = 6(4+4x)
20+30x = 24+24x
6x=4
x=2/3

i am still confused why the amount at the end of year 6 is 6/5 of the amount at the end of 4th year?
can some one please help on this?

[jnelson0612]

This is essentially a sequence problem in disguise. Let x = amount of yearly growth, in feet.

Yr0 = 4
Yr1 = 4+x
Yr2 = 4+x+x=4+2x
Yr3 = 4+x+x+x=4+3x
Yr4 = 4+x+x+x+x=4+4x
Yr5 = 4+x+x+x+x+x=4+5x
Yr6 = 4+x+x+x+x+x+x=4+6x

We are told the amount at the end of Year 6 is 6/5 of the amount at the end of year 4. Thus we can write:

4+6x = 6/5 (4+4x)
5(4+6x) = 6(4+4x)
20+30x = 24+24x
6x=4
x=2/3

i am still confused why the amount at the end of year 6 is 6/5 of the amount at the end of 4th year?
can some one please help on this?

I am quoting from the problem: "At the end of the 6th year, the tree was 1/5 taller than it was at the end of the 4th year."

Let's make up an example to illustrate this. Let's say that you are 5 feet tall when you are twelve years old. I then tell you that when you are sixteen years old you will be 1/5th taller.

How do we calculate your new height? We take 5 feet * (1 + 1/5). We take 5 feet * 1 to illustrate your prior height PLUS 5 feet * 1/5 to illustrate the amount you have grown. Thus, we take 5 feet * 6/5. You would then be 6 feet tall.

[agha79]

Thanks Nelson! Great illustration. got the point!

[RonPurewal]

Thanks Nelson! Great illustration. got the point!

glad it helped.

[rachelhong2012]

I like how in the strategy book the incremental increase is being described as "jump".

So from the difference between each number is a "jump" from the previous number to the following number. Or the gap between two numbers

Here,

we are given two numbers:

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.

the number for at the end of 6 "jumps" from the starting point is:

4 + 6 jumps

the number for at the end of 4 "jumps" from teh starting point is:

4 + 4 jumps

4 + 6 jumps = (1 + .2) (4 + 4 jumps)
4 + 6 jumps = 1.2 (4 + 4 jumps"
distribute and you'll get

jump = .8/1.2 = 2/3