On July 1, 2017, a certain tree was 128 centimeters tall.

Question

On July 1, 2017, a certain tree was 128 centimeters tall. Each year, the tree's height increases 50%. 
Given this growth rate, the tree's height on July 1, 2023 will be how many centimeters greater than the tree's height on July 1, 2022?

A) (2^2)(3^4)
B) (2)(3^4)
C) (2)(3^5)
D) (4)(3^5)
E) (2)(3^6)

*kudos for all correct solutions

EgmatQuantExpert · Accepted Answer

Solution

Given:  
 • On July 1, 2017, a certain tree was 128 cm tall
• Each year the tree’s height increases by 50%

To find:  
 • The tree’s height on July 1, 2023 will be how many cm more than the tree’s height on July 1, 2022
 
  Approach and Working:   
As the increase is 50% every year, we can write
 • Height on July 1, 2018 =  128 * 1.5 
• Height on July 1, 2019 =  128 * 1.5^2 
• Height on July 1, 2020 =  128 * 1.5^3 
• Height on July 1, 2021 =  128 * 1.5^4 
• Height on July 1, 2022 =  128 * 1.5^5 
• Height on July 1, 2023 =  128 * 1.5^6

Therefore, the difference in height on July 1, 2023 and July 1, 2022 =  128 * 1.5^6 – 128 * 1.5^5  =  128 * 1.5^5 * 0.5  =  2^7 * \frac{3}{2} * \frac{3}{2} * \frac{3}{2} * \frac{3}{2} * \frac{3}{2} * \frac{1}{2}  =  2 * 3^5

Hence, the correct answer is option C.

Answer: C

SSM700plus · Answer

Every year tree's height increases by 50%.
Height on July 1, 2018 =  128 * 1.5 
Height on July 1, 2019 =  128 * 1.5^2 
Similarly 
Height on July 1, 2022 =  128 * 1.5^5 
Height on July 1, 2023 =  128 * 1.5^6

Therefore, the difference in height of tree from July 1, 2022 to July 1, 2023 cab be calculated as below
=  128 * 1.5^6 – 128 * 1.5^5  
=  128 * 1.5^5 * 0.5  =  2^7 * \frac{3}{2} * \frac{3}{2} * \frac{3}{2} * \frac{3}{2} * \frac{3}{2} * \frac{1}{2}  =  2 * 3^5

Hence C is Ans.

pushpitkc · Answer

On July 1st, 2017 the tree was 128 or  2^7  centimeters tall. Every year, the tree's height increase by 50%.

When there is a quantity increase of 50%, it becomes  \frac{3}{2} th of the original quantity.

On July 1st, 2018 the tree is  2^7 * \frac{3}{2}  or  2^6 * 3  centimeter tall.
On July 1st, 2019 the tree is  2^6 * 3 * \frac{3}{2}  or  2^5 * 3^2  centimeter tall.

3 years later, On July 1st, 2022 the tree is  2^3 * 3^4 * \frac{3}{2}  or  2^2 * 3^5  centimeter tall.
4 years later, On July 1st, 2023 the tree is  2^2 * 3^5 * \frac{3}{2}  or  2 * 3^6  centimeter tall.

The tree's height difference is  2 * 3^6 - 2^2 * 3^5 = 2 * 3^5(3 - 2) = 2 * 3^5

Therefore, the tree will be  2 * 3^5  centimeters (Option C)  greater on July 1, 2023 than July 1, 2022.

BrentGMATPrepNow · Answer

Let's create a  growth table  and look for a  pattern

year | height in cm 
2017: 128
2018: 128(1.5)
2019: 128(1.5)^2
2020: 128(1.5)^3
2021: 128(1.5)^4
2022:  128(1.5)^5 
2023:  128(1.5)^6

The tree's height on  July 1, 2023  will be how many centimeters greater than the tree's height on  July 1, 2022 ?   
Difference =  128(1.5)^6  -  128(1.5)^5 
Factor out 128(1.5^5) to get: difference = 128(1.5^5) 
Simplify: difference = 128(1.5^5) 
Rewrite with fractions: difference = (2^7)(3/2)^5)(1/2)
Expand: difference = (2^7)(3^5)/(2^6)
Simplify: difference = (2)(3^5)

Answer: C

Cheers,
Brent

LevanKhukhunashvili · Answer

I wonder how much time does it take to find the correct approach. it appeared too hard for me

GMATGuruNY · Answer

Formula for EXPONENTIAL GROWTH:
final amount = original * (multiplier)^(number of changes)

Here:
original =  128 = 2^7 
Since the height increases each year by 50%, the multiplier  = \frac{3}{2}

From 2017 to 2022 -- a period of 5 years -- there are total of 5 changes.
Thus:
final amount  = 2^7 * (\frac{3}{2})^5 = 2^23^5

The tree's height on July 1, 2023 will be how many centimeters greater than the tree's height on July 1, 2022 ?
Since the height increases by 50% each year, the increase from 2022 to 2023 is equal to half the height in 2022:
 \frac{1}{2} * 2^23^5 = 2*3^5

C

dine5207 · Answer

it depends on the approach you take. I solved using Compound interest formulae it hardy take 1 minute to solve
Compound Interest=principal∗(1+interestC)time∗C

If C=1, meaning that interest is compounded once a year

MathRevolution · Answer

July 1, 2017, a certain tree was 128 centimeters tall: 128 =  \frac{2^7

50% increase:   150 / 100}  =  \frac{3}{2}

The common ratio is 3/2. And this forms a G.P with the first term as a = a *  2^7  and r =  \frac{3}{2}

July 1, 2022 (5 years): a *  r^5

July 1, 2023 (6 years): a *  r^6

We have to find a difference between both: a *  r^6  - a *  r^5

=> a *  r^6  - a *  r^5  = a *  r^5  ( r-1)

=> a *  r^6  - a *  r^5  = a *  r^5  (  \frac{3}{2}  - 1)

=> a *  r^6  - a *  r^5  = a *  r^5  (  \frac{1}{2} )

=> a *  r^5  =  2^7  *  (\frac{3}{2})^5  *  \frac{1 }{ 2}

=> a *  r^5  = 2 *  (3)^5

Answer C

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On July 1, 2017, a certain tree was 128 centimeters tall.

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