Let x = the yearly increase.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
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.
The correct answer is D.
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.
Answer choice D: 2/3
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!
The correct answer is D.
