SAT Math : How to find compound interest

Study concepts, example questions & explanations for SAT Math

varsity tutors app store varsity tutors android store varsity tutors ibooks store

Example Questions

Example Question #1 : How To Find Compound Interest

A five-year bond is opened with \(\displaystyle \$5000\) in it and an interest rate of \(\displaystyle 2.5\)%, compounded annually. This account is allowed to compound for five years. Which of the following most closely approximates the total amount in the account after that period of time?

Possible Answers:

\(\displaystyle \$5625\)

\(\displaystyle \$6143\)

\(\displaystyle \$5518\)

\(\displaystyle \$5657\)

\(\displaystyle \$5811\)

Correct answer:

\(\displaystyle \$5657\)

Explanation:

Each year, you can calculate your interest by multiplying the principle (\(\displaystyle \$5000\)) by \(\displaystyle 1.025\). For one year, this would be:

\(\displaystyle 1.025*5000=5125\)

For two years, it would be:

\(\displaystyle 5125*1.025\), which is the same as \(\displaystyle 1.025*1.025*5000\)

Therefore, you can solve for a five year period by doing:

\(\displaystyle 1.025^5*5000\)

Using your calculator, you can expand the \(\displaystyle 1.025^5\) into a series of multiplications. This gives you \(\displaystyle 5657.041064453125\), which is closest to \(\displaystyle \$5657\)

Example Question #1 : How To Find Compound Interest

If a cash deposit account is opened with \(\displaystyle \$7500\) for a three year period at \(\displaystyle 3.5\)% interest compounded once annually, which of the following is closest to the positive difference between the interest accrued in the third year and the interest accrued in the second year?

Possible Answers:

\(\displaystyle \$11.41\)

\(\displaystyle \$281.2\)

\(\displaystyle \$9.51\)

\(\displaystyle \$81.41\)

\(\displaystyle \$0\)

Correct answer:

\(\displaystyle \$9.51\)

Explanation:

It is easiest to break this down into steps. For each year, you will multiply by \(\displaystyle 1.035\) to calculate the new value. Therefore, let's make a chart:

After year 1: \(\displaystyle 7500*1.035=7762.5\); Total interest: \(\displaystyle 262.5\)

After year 2: \(\displaystyle 7762.5*1.035=8034.1875\); Let us round this to \(\displaystyle 8034.19\); Total interest: \(\displaystyle 271.69\)

After year 3: \(\displaystyle 8034.19 * 1.035 = 8315.38665\); Let us round this to \(\displaystyle 8315.39\); Total interest: \(\displaystyle 281.2\)

Thus, the positive difference of the interest from the last period and the interest from the first period is: \(\displaystyle 281.2-271.69=9.51\)

Example Question #2 : How To Find Compound Interest

Jack has \(\displaystyle \$15\),\(\displaystyle 000\) to invest. If he invests two-thirds of it into a high-yield savings account with an annual interest rate of \(\displaystyle 8\%\), compounded quarterly, and the other third in a regular savings account at \(\displaystyle 6\%\) simple interest, how much does Jack earn after one year?

Possible Answers:

\(\displaystyle \$1128.75\)

\(\displaystyle \$1051.32\)

\(\displaystyle \$1124.32\)

\(\displaystyle \$1081.98\)

\(\displaystyle \$901.43\)

Correct answer:

\(\displaystyle \$1124.32\)

Explanation:

First, break the problem into two segments: the amount Jack invests in the high-yield savings, and the amount Jack invests in the simple interest account (10,000 and 5,000 respectively).

Now let's work with the high-yield savings account. $10,000 is invested at an annual rate of 8%, compounded quarterly. We can use the compound interest formula to solve:

\(\displaystyle \text{Final Balance}=\text{ Principal} \cdot \bigg(1+\frac{\text{Interest Rate}}{c}\bigg)^{(\text{Time})(c)}\)

Plug in the values given:

\(\displaystyle =10,000\cdot \bigg(1+\frac{.08}{4}\bigg)^{(1)(4)}\)

\(\displaystyle =10,000 \cdot (1.02)^{4}\)

\(\displaystyle =10,824.32\)

Therefore, Jack makes $824.32 off his high-yield savings account. Now let's calculate the other interest:

\(\displaystyle \text{Interest}=\text{Principal} \cdot \text{ Interest Rate} \cdot \text{ Time}\)

\(\displaystyle =5,000 \cdot (0.06)\cdot (1)\)

\(\displaystyle =300\)

 

Add the two together, and we see that Jack makes a total of, \(\displaystyle \$1124.32\) off of his investments.

Example Question #2 : How To Find Compound Interest

A truck was bought for \(\displaystyle \$ 20,000\) in 2008, and it depreciates at a rate of \(\displaystyle 13\%\) per year. What is the value of the truck in 2016? Round to the nearest cent.

Possible Answers:

\(\displaystyle y= 6564.23\)

\(\displaystyle y= 16564\)

\(\displaystyle y= 5664.23\)

\(\displaystyle y= 6764.23\)

\(\displaystyle y= 6664.23\)

Correct answer:

\(\displaystyle y= 6564.23\)

Explanation:

The first step is to convert the depreciation rate into a decimal. \(\displaystyle 13\%=0.13\). Now lets recall the exponential decay model. \(\displaystyle y=C(1-r)^t\), where \(\displaystyle C\) is the starting amount of money, \(\displaystyle r\) is the annual rate of decay, and \(\displaystyle t\) is time (in years). After substituting, we get

 

\(\displaystyle y=20000(1-0.13)^8\)

\(\displaystyle y=20000(0.87)^8\)

\(\displaystyle y=20000\cdot 0.3282117\)

\(\displaystyle y= 6564.23\)

 

Learning Tools by Varsity Tutors