Question

A plant nursery is growing a tree that is 3ft tall and grows at an average height of 1 ft per year. Another tree at the nursery is 4ft tall and grows at an average rate of 0.5 ft per year. After how many years will the trees be the same height?

Answer 1

See a solution process below:

Explanation:

Let's call the number of years we are looking for: `y`

We can write and expression for how the first tree grows as:

`3 + 1y`

We can write an expression for how the second trees grows as:

`4 + 0.5y`

Because we want to find when these two will be equal we can equate them and solve for `y`:

`3 + 1y = 4 + 0.5y`

First, subtract `color(red)(3)` and `color(blue)(0.5y)` from each side of the equation to isolate the `y` term while keeping the equation balanced:

`3 - color(red)(3) + 1y - color(blue)(0.5y) = 4 - color(red)(3) + 0.5y - color(blue)(0.5y)`

`0 + (1 - color(blue)(0.5))y = 1 + 0`

`0.5y = 1`

Now, multiply each side of the equation by `color(red)(2)` to solve for `y` while keeping the equation balanced:

`color(red)(2) xx 0.5y = color(red)(2) xx 1`

`1y = 2`

`y = 2`

After 2 years both trees will be the same height:

• Tree 1: `3 + 1y = 3 + (1 xx 2) = 3 + 2 = 5`

• Tree 2: `4 + 0.5y = 4 + (0.5 xx 2) = 4 + 1 = 5`

After two years both trees would be 5 feet tall.