### All SAT Math Resources

## Example Questions

### Example Question #1 : Pattern Behaviors In Exponents

A five-year bond is opened with in it and an interest rate of %, 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:**

**Correct answer:**

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

For two years, it would be:

, which is the same as

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

Using your calculator, you can expand the into a series of multiplications. This gives you , which is closest to .

### Example Question #1 : How To Find Compound Interest

If a cash deposit account is opened with for a three year period at % interest compounded once annually, which of the following is closest to the positive difference of the interest accrued between the last two years and the first two years?

**Possible Answers:**

**Correct answer:**

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

After year 1: ; Total interest:

After year 2: ; Let us round this to ; Total interest:

After year 3: ; Let us round this to ; Total interest:

Thus, the positive difference of the interest from the last period and the interest from the first period is:

### Example Question #2 : How To Find Compound Interest

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

**Possible Answers:**

**Correct answer:**

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:

Plug in the values given:

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

Add the two together, and we see that Jack makes a total of, off of his investments.

### Example Question #11 : New Sat Math No Calculator

A truck was bought for in 2008, and it depreciates at a rate of per year. What is the value of the truck in 2016? Round to the nearest cent.

**Possible Answers:**

**Correct answer:**

The first step is to convert the depreciation rate into a decimal. . Now lets recall the exponential decay model. , where is the starting amount of money, is the annual rate of decay, and is time (in years). After substituting, we get

### Example Question #1 : Pattern Behaviors In Exponents

If a^{x}·a^{4} = a^{12} and (b^{y})^{3} = b^{15}, what is the value of x - y?

**Possible Answers:**

6

-9

3

-2

-4

**Correct answer:**

3

Multiplying like bases means add the exponents, so x+4 = 12, or x = 8.

Raising a power to a power means multiply the exponents, so 3y = 15, or y = 5.

x - y = 8 - 5 = 3.

### Example Question #2 : Pattern Behaviors In Exponents

If p and q are positive integrers and 27^{p }= 9^{q}, then what is the value of q in terms of p?

**Possible Answers:**

p

2p

(2/3)p

3p

(3/2)p

**Correct answer:**

(3/2)p

The first step is to express both sides of the equation with equal bases, in this case 3. The equation becomes 3^{3p }= 3^{2q}. So then 3p = 2q, and q = (3/2)p is our answer.

### Example Question #1 : How To Find Patterns In Exponents

Simplify 27^{2/3}.

**Possible Answers:**

9

729

27

125

3

**Correct answer:**

9

27^{2/3} is 27 squared and cube-rooted. We want to pick the easier operation first. Here that is the cube root. To see that, try both operations.

27^{2/3} = (27^{2})^{1/3} = 729^{1/3} OR

27^{2/3} = (27^{1/3})^{2} = 3^{2}

Obviously 3^{2} is much easier. Either 3^{2} or 729^{1/3} will give us the correct answer of 9, but with 3^{2} it is readily apparent.

### Example Question #661 : Algebra

If and are integers and

what is the value of ?^{ }

**Possible Answers:**

**Correct answer:**

To solve this problem, we will have to take the log of both sides to bring down our exponents. By doing this, we will get .

To solve for we will have to divide both sides of our equation by to get .

will give you the answer of –3.

### Example Question #1 : How To Find Patterns In Exponents

If and , then what is ?

**Possible Answers:**

**Correct answer:**

We use two properties of logarithms:

So

### Example Question #571 : Algebra

Evaluate:

**Possible Answers:**

**Correct answer:**

, here and , hence .