Video Explanation

Text Explanation

Notice that, since R is the annual percent as a decimal, we can form a multiplier simply by adding one: (1 + R). That's very handy!

Method One: Step-by-step

Starting amount = A

After one year, we multiply by the multiplier once

That's the total amount at the end of the first year. The amount A is the original principal, and AR is the interest earned.

At the end of the second year, that entire amount is multiplied by the multiplier. We need to FOIL.

That's the total amount at the end of the second year. The amount A is the original principal, and the rest is the interest earned.

At the end of the third year, this entire amount is again multiplied by the multiplier.

That's the total amount at the end of the third year. The amount A is the original principal, and the rest is the interest earned.

Answer = (C)

Method Two: some fancy algebra

Over the course of three years, the initial amount A is multiplied by the multiplier (1 + R) three times. Thus, after three years,

Now, if you happen to know it offhand, we can use the cube of a sum formula:

Thus,

and

Answer = (C)

FAQ: 

Wait, isn't the answer missing a 1? Shouldn't it be A(1 + 3R + 3R² + R³)? 

A: It would be A + 3AR + 3AR² + RA³ if we wanted the total amount of money in the account at the end of the year.  You could certainly factor out an A and end up with:

A(R³ + 3R² + R + 1)

However, that still represents the total amount of money in the account.  But the problem is asking us to find only the interest. Recall that A is the initial amount of money in the account. To find the interest in the account, we need to subtract the initial amount, A, from the total amount: 

[ A + 3AR + 3AR² + RA³ ]  – [ A ]  = 3AR + 3AR² + RA³

Thus, this is what we factor the A out of, and end up with our final answer for the interest:

A(3R + 3R² + R³)

Why do we use 1 + R? 

A: The 1+R portion of this question represents a percent multiplier. We add 1 to R in order to include the initial amount in our calculation. We need to include the initial amount because it affects how much interest is earned. Let's think about this in terms of numbers, rather than variables, to make this a bit more clear. 

Let's say we initially invest $100, at a .05 interest rate.

To find how much money we have in year 2, we have to add 1 to the interest rate for our multiplier:

$100(1+0.05) = $100(1.05) = $105

In the second year, we do this again:

$105(1.05) = $110.25

And so on.

So in summary, using a percent multiplier such as 1+R,  allows us to quickly calculate new amounts by simply multiplying it to our initial amount. 

Can you explain how we are 'treating R as a decimal'? 

A: Think of R as a variable or a place holder that can represent any number. In this question, the problem tells us that R hold a value of a decimal. So even though we have a whole R variable, the numerical value of R will be a decimal. For instance, we could have these relationships:

• R = 0.2
• R = 0.45
• R = 0.67

All of these values could be values of R because they are decimals. In the answer choices, the letter "R" is the place holder of the decimal value, so if we knew what decimal value R represented, we could plug that value in for R into the answer choice equations. However, the problem does not tell us what decimal value we are dealing with, so we place R in the equation as a place holder for the decimal value.

In other words, taking the whole value of R represents the percentage value as a decimal that the question describes without having to deal with the decimal itself.