### All SAT Math Resources

## Example Questions

### Example Question #1 : Variables

Multiply:

**Possible Answers:**

**Correct answer:**

Distribute the monomial through the polynomial.

The answer is:

### Example Question #2 : Variables

Evaluate:

**Possible Answers:**

**Correct answer:**

Distribute the monomial through each term inside the parentheses.

### Example Question #1 : Monomials

Multiply the monomials:

**Possible Answers:**

**Correct answer:**

In order to multiply this, we can simply multiply the coefficients together and the variables together. Anytime we multiply a variable of the same base, we can add the exponents.

Simplify the right side.

The answer is:

### Example Question #3 : Variables

Distribute:

**Possible Answers:**

**Correct answer:**

Distribute the monomial to each part of the polynomial, paying careful attention to signs:

### Example Question #4 : Variables

Find the product:

**Possible Answers:**

**Correct answer:**

Find the product:

Step 1: Multiply the numerators and denominators using the properties of exponents. (When multiplying exponents, add them.)

Step 2: Simplify the expression.

### Example Question #1 : How To Use The Direct Variation Formula

Phillip can paint square feet of wall per minute. What area of the wall can he paint in 2.5 hours?

**Possible Answers:**

**Correct answer:**

Every minute Phillip completes another * square feet of painting. To solve for the total area that he completes, we need to find the number of minutes that he works.*

There are 60 minutes in an hour, and he paints for 2.5 hours. Multiply to find the total number of minutes.

If he completes * square feet per minute, then we can multiply ** by the total minutes to find the final answer.*

### Example Question #1 : Variables

The value of varies directly with the square of and the cube of . If when and , then what is the value of * when and ?*

**Possible Answers:**

**Correct answer:**

Let's consider the general case when *y* varies directly with *x*. If *y* varies directly with *x*, then we can express their relationship to one another using the following formula:

*y* = *kx*, where *k* is a constant.

Therefore, if *y* varies directly as the square of *x* and the cube of *z*, we can write the following analagous equation:

*y* = *kx*^{2}*z*^{3}, where *k* is a constant.

The problem states that *y* = 24 when *x* = 1 and *z* = 2. We can use this information to solve for *k* by substituting the known values for *y*, *x*, and *z*.

24 = *k*(1)^{2}(2)^{3} = *k*(1)(8) = 8*k*

24 = 8*k*

Divide both sides by 8.

3 = *k*

*k* = 3

Now that we have *k*, we can find *y* if we know *x* and *z*. The problem asks us to find *y* when *x* = 3 and *z* = 1. We will use our formula for direct variation again, this time substitute values for *k*, *x*, and *z*.

*y* = *kx*^{2}*z*^{3}

*y* = 3(3)^{2}(1)^{3} = 3(9)(1) = 27

*y* = 27

The answer is 27.

### Example Question #1 : How To Use The Direct Variation Formula

In a growth period, a population of flies triples every week. If the original population had 3 flies, how big is the population after 4 weeks?

**Possible Answers:**

**Correct answer:**

We know that the initial population is 3, and that every week the population will triple.

The equation to model this growth will be , where is the initial size, is the rate of growth, and is the time.

In this case, the equation will be .

Alternatively, you can evaluate for each consecutive week.

Week 1:

Week 2:

Week 3:

Week 4:

### Example Question #2 : How To Use The Direct Variation Formula

and are the diameter and circumference, respectively, of the same circle.

Which of the following is a true statement? (Assume all quantities are positive)

**Possible Answers:**

varies directly as the fourth root of .

varies directly as the fourth power of .

varies inversely as the fourth power of .

varies directly as .

varies inversely as the fourth root of .

**Correct answer:**

varies directly as .

If and are the diameter and circumference, respectively, of the same circle, then

.

By substitution,

Taking the square root of both sides:

Taking as the constant of variation, we get

,

meaning that varies directly as .

### Example Question #1 : How To Use The Direct Variation Formula

is the radius of the base of a cone; is its height; is its volume.

; .

Which of the following is a true statement?

**Possible Answers:**

varies directly as the third power of .

varies directly as the fifth power of .

varies directly as the fifth root of .

varies directly as the cube root of .

varies directly as .

**Correct answer:**

varies directly as the fifth power of .

The volume of a cone can be calculated from the radius of its base , and the height , using the formula

, so .

, so .

, so by substitution,

Square both sides:

If we take as the constant of variation, then

,

and varies directly as the fifth power of .

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