12 Days of Christmas GMAT Competition - Day 9: If a certain tree reach

If a certain tree reached a height of 78 inches by the end of 2023, did it grow by less than 15 inches that year?

(1) By the end of 2023, the tree was less than 30 percent higher than by the end of 2022.
(2) In 2023, the tree grew by more than 20 percent of its height by the end of 2022.

Solution: Let height of tree before 2023 was 'x' inches
We need to find if x > 63

Statement 1: By the end of 2023, the tree was less than 30 percent higher than by the end of 2022.
or 1.3x > 78
x > 60
x can be 61, 62, 63 etc
we cannot conclusively say that x > 63
INSUFFICIENT

Statement 2: In 2023, the tree grew by more than 20 percent of its height by the end of 2022.
or 1.2x < 78
x < 65
x can 63, 64 etc
we cannot conclusively say that x > 63
INSUFFICIENT

Combining Statement 1 and 2
60 < x < 65
x can be 61, 62, 63, 64 or even non integer values as well.
INSUFFICIENT

Option E is the right answer

If a certain tree reached a height of 78 inches by the end of 2023, did it grow by less than 15 inches that year?


Let the growth of the tree by the end of 2022 = x.
For the tree to grow by less than 15 inches in 2023, x>63

(1) By the end of 2023, the tree was less than 30 percent higher than by the end of 2022.
Using this statement we get,
-> 78 < x + x*30%
-> 78 < 13x/10
-> x>60
x can be 61,62,63,64,65, etc. giving us both yes and no as answers.
Therefore, insufficient.


(2) In 2023, the tree grew by more than 20 percent of its height by the end of 2022.
Using this statement we get,
-> x + x*20/100 < 78
-> 6x/5 < 78
-> x<65, Not sufficient since we can get both yes and no as answers.
Therefore, insufficient.

(1)+(2) -> 60<x<65, x can still be 62, giving the answer as no and can be 64 giving the answer as yes.
Insufficient.

Ans=E.

(1) By the end of 2023, the tree was less than 30 percent higher than by the end of 2022. 78 / 130% = 60; 78 - 60 = 18. growth < 18 Insufficent alone
(2) In 2023, the tree grew by more than 20 percent of its height by the end of 2022. 78 / 120% = 65; 78 - 65 = 13; growth > 13
Combined, 13 < growth < 18. Insufficient together, need more information.

A) the length at E' of 22 is > 60, so can't be concluded.
[insufficient]
B) the length at E' of 22 is < 65, so can't be concluded.
Hence, E

E.

1. if the tree grew by 30% in 2023, height at end of 2022=60
growth in 2023 <30%=<18 NOT SUFFICIENT

2. if it grew by 20%, original height=65
growth in 2023>20%=>13 NOT SUFFICIENT

1+2 ALSO NOT SUFFICIENT

Statement 1:
Lets calculate the height in 2022 from given information
1.3 * H = 78
H = 60
Height in 2022 was more than 60 inches

78-60 = 18
It grew less then 18 inches
We cannot say if it grew less than 15 inches

Insufficient

Statement 2:
1.2 *H = 78
H = 65
Height in 2022 was less than 65 inches

78 - 65 = 13
It grew more then 13 inches

Insufficient.

Both together,
it grew between 13 and 18 inches
Still insufficient.

E

OA is E

Posted from my mobile device

The height of tree grew in 2022 = a
The height of tree grew in 2023 = b
A certain tree reached a height of 78 inches by the end of 2023 -> a + b = 78

(1) By the end of 2023, the tree was less than 30 percent higher than by the end of 2022.
-> (78 - a)/a <30% -> 78/a - 1 < 0.3 -> 78/a <1.3 -> a> 60
-> b = 78 - a -> b < 18 -> insufficient

(2) In 2023, the tree grew by more than 20 percent of its height by the end of 2022.
-> (78 - a)/a >20% -> 78/a - 1 > 0.2 -> 78/a > 1.2-> a < 65
-> b = 78 - a -> b> 13 -> insufficient

(1) and (2) -> 13<b<18 -> insufficient

Answer E

The question asks whether height at the end of 2022 (let x) was > 63 inches

1. 1.3 x < 78
x< 60 .. No height not more than 63 .. Sufficient
2. 78 > 1.2 x
x< 65 ... if x = 64, Yes, if x = 62 .. No... InSufficient

Ans A

1 tree grew between 100% to 130% which is means initial height could be between 56'' IN to nearly 73''. not sufficient height.
2 tree grew at least to 120% which means initial height could be 51'' or less. 73''-51'' = 22''>15''

answer B

If a certain tree reached a height of 78 inches by the end of 2023, did it grow by less than 15 inches that year?

Let the height of tree by the end of 2022 = h
Height of tree by the end of 2023 = 78

(1) By the end of 2023, the tree was less than 30 percent higher than by the end of 2022.

78-h<0.3h
h>60

if h=61 then the tree grew by more than 15 inches
if h=70 then tree grew by less than 15 inches

Multiple answers. Insufficient.

(2) In 2023, the tree grew by more than 20 percent of its height by the end of 2022.

Let's say the tree grew by 20 percent of its height by the end of 2022

h+0.2h = 78
h = 65

Since the tree grew by more than 20 percent of its height by the end of 2022, its height by the end of 2022 must be less than 65 inches.

If its height was 64 inches by the end of 2022, then it grew by less than 15 inches.
If its height was 60 inches by the end of 2022, then it grew by more than 15 inches.

Insufficient

(3) Combining both statements

By the end of 2023 its height grew by more than 20% and less than 30%. We still don't get a definite answer. Hence, E IMO.

Height of tree at the end of 2022 = h

(1) By the end of 2023, the tree was less than 30 percent higher than by the end of 2022.

h*1.3 = 78

h = 78/1.3 = 60

h > 60

Not sufficient.

If h = 77 => the difference was not atleast 15
If h = 61 => the difference is more than 15

Not sufficient.

(2) In 2023, the tree grew by more than 20 percent of its height by the end of 2022.

1.2h = 78

h > 65

Not sufficient.

If h = 65.000001 => the difference was atleast 15
If h = 77 => the difference is not more than 15

(1) + (2)

60 < h < 65

adding 15 on both sides

75 < h < 80

As 78 is this range, we cannot if the tree grew by less than 15 inches that year.

Statements combined are not sufficient.

IMO E

Height by end of year 2023 = 78 inches
From starting of 2023 to end of 2023 did it grow by 15 inches
Let height by the end of 2022 is 'a'

(1) By the end of 2023, the tree was less than 30 percent higher than by the end of 2022.

=> a * 130/100 > 78
=> a > 78 * 10/13
=> a > 60 inches

Growth in year 2023 is less than = 78 - 60 = 18 inches. Hence it may or may not have been less than 15

Insufficient

(2) In 2023, the tree grew by more than 20 percent of its height by the end of 2022.

=> 78 > a(120/100)
=> a < 780/12
=> a < 65
=> difference in growth can be greater than 78 - 65
=> Growth > 13

Insufficient as it may or may not be less than 15

Even with Statement (1) and (2) together we get

13 > Growth > 18, hence Insufficient

Option E

If option 1 is considered, then if no limit is given, it might grown 10% also, thus it may be less than 15 and more than 20%, more than 15 inches. The lower limit of grow is given in option 2.

The correct option is C

Height at the end of 2023 = 78 inches

(1) By the end of 2023, the tree was less than 30 percent higher than by the end of 2022.
Less than 30% higher can range from 0% to to 29%.Both more than and less than 15 inches increase value will satisfy this. Insufficient.

(2) In 2023, the tree grew by more than 20 percent of its height by the end of 2022.
Let tree height in 2022 is x
If it grows by 20% in 2023 than the height in 2023 than 1.2x = 78cm
x= 65cm
Height is grown by more than 13cm.
For 30% growth, 1.3x=78cm
x=60
Height is grown by more than 18cm.
Insufficient.

Combining 1 and 2 , we get the range from (20-30)% in which both more than and less than 15 inches increase value will exist.

IMO E.

If a certain tree reached a height of 78 inches by the end of 2023, did it grow by less than 15 inches that year?

(1) By the end of 2023, the tree was less than 30 percent higher than by the end of 2022.
Let's Hx=Height at the end of X

So it means that 78-H2022<0,3*H2022 => H2022>78/1,3=60. So the tree grew by 18 inches at most but we don't know if it grew more of less than 15 inch between end of 2022 and end of 2023 (H2022 could be 70 or could be 62)=> so (1) is not sufficient to answer the question.

(2) In 2023, the tree grew by more than 20 percent of its height by the end of 2022.
so it means that 78-H2022>0,2*H2022 => H2022<78/1,2=65 so the tree grew by at least 13 inches but we don't know if it grew more of less than 15 inches. (H2022 could be 64 therefore the tree would have grown by 14 inches, or H2022 could be 60 therefore the tree would have grown by 18 inches) => so (2) is not sufficient to answer the question.



(1) and (2) together. 60<H2022<65 => 18>78-H2022>13 => we still can't say if the tree grew by more or less 15 inches (it could be 14, it could be 16...)

Therefore Answer E both statement aren't sufficient.

According to statement 1,
Tree's height at end of 2022=x, 13x/10>78 (as per the statement)
x>60, therefore max difference = 18 cm. cannot conclude anything, insufficient.

According to statement 2,
6x/5<78, x<65. Minuimum difference can be 13 cm, cannot conclude anything, insufficient.

Taking both together 13<difference<18. Not sufficient.

IMO, answer is E