When a certain tree was first planted

[shohet.mark]

When a certain tree was first planted, 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. 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


Answer is D. I set it up as 4 + 6x = (6/5)(4+4x) and got x = 2/3. I don't have an issue with the problem, but what tripped me up initially was that I interpreted "constant amount each year" to mean that the height should be multiplied by x each year (but the correct way to solve is to add x each year).


So I just want to confirm:

1) If something is increasing by a CONSTANT RATIO, then we must MULTIPLY each year's result by X, but if something is increasing by a CONSTANT AMOUNT, then we must ADD X to each year. Is this accurate?
2) What if the language said something was increasing by a constant RATE--would we multiply each year by X or add X to each year?
3) Any other synonyms for these terms or things to look out for on these types of problems

Thank you.

[RonPurewal]

Your #1 is correct.
This is not some random nit-pick, by the way; it's the actual definition of "amount". Amounts are absolute, not relative.

I'm sure you know this already; you probably just aren't making connections.
E.g., if someone's weight goes from 200 pounds to 220 pounds, and I asked you "By what amount did his weight increase?", you'd say 20 pounds. You wouldn't say %10.

[RonPurewal]

2) What if the language said something was increasing by a constant RATE--would we multiply each year by X or add X to each year?

Look at the units of the rate.

Most rates are defined as (amount)/(time). E.g., miles per hour, dollars per week, calories per minute.
If one of these rates is constant, then the thing changes by a constant amount (in the same units as the numerator) per unit of time.

If a rate is explicitly defined as a relative rate, though, then it's multiplication.
Such rates will almost always be expressed as percent increases (not ratios or multipliers) per time unit.
E.g., if a bank account accrues interest at a rate of 5 percent/year, compounded annually, then multiply by 1.05 each year.

[RonPurewal]

3) Any other synonyms for these terms or things to look out for on these types of problems

Because you can use units as your guide (see #2 above), there's little reason to cobble together a vocabulary list.

[shohet.mark]

Thanks Ron. Very helpful.

[RonPurewal]

Thanks Ron. Very helpful.

No problem.