### All SAT Math Resources

## Example Questions

### Example Question #61 : How To Find Order Of Operations

Consider the expression

If the values of all of the variables are known which operation is carried out *third*?

**Possible Answers:**

Addition

Subtraction

Multiplication

Division

Squaring

**Correct answer:**

Division

The order of operations dictates that any operations within grouping symbols is carried out first, going from the innermost symbols to the outermost symbols. The innermost symbols - the parentheses - flank an addition, so that operation is carried out first.

What remains within the outermost symbols - the brackets - are a multiplication and a division. Between the two, the one that is at left is carried out first; this means the multiplication is performed second, then the division is worked out third.

Division is the correct answer.

### Example Question #61 : Order Of Operations

Define an operation on the set of real numbers as follows:

For all real ,

Evaluate .

**Possible Answers:**

**Correct answer:**

Set in the expression in the definition, then evaluate according to the order of operations:

Square first, remembering that the square of any number is nonnegative:

Add going left to right:

### Example Question #62 : Order Of Operations

Define an operation on the set of real numbers as follows:

For all real ,

Evaluate the expression .

**Possible Answers:**

None of the other responses gives a correct answer.

**Correct answer:**

Set and in the expression in the definition, then evaluate according to the order of operations:

Multiplication and division take precedence over subraction; carry out the multiplication first, since it is the operation at left:

Divide next:

Now subtract:

### Example Question #63 : Order Of Operations

Define an operation on the set of real numbers as follows:

For all real ,

Evaluate: .

**Possible Answers:**

None of the other responses gives a correct answer.

**Correct answer:**

Set and in the expression in the definition, then evaluate according to the order of operations:

By order of operations, carry out the exponentiation first:

The division is carried out next:

Now add:

### Example Question #61 : How To Find Order Of Operations

Consider the expression

If the values of all of the variables are known, which operation is carried out *third*?

**Possible Answers:**

Addition

Division

Squaring

Multiplication

Subtraction

**Correct answer:**

Multiplication

The order of operations dictates that any operations within grouping symbols, such as parentheses, be carried out first; in this expression. There are two expressions - an exponentiation (squaring) and an addition - that fall between the parentheses; by the order of operations, the squaring is carried out, then the addition.

This leaves a multiplication, a division, and a subtraction. Multiplication and division have precedence over subtraction; since the multiplication appears first in left-to-right order, the multiplication is carried out next. This makes multiplication the correct response.

### Example Question #61 : How To Find Order Of Operations

Evaluate the following expression:

**Possible Answers:**

**Correct answer:**

We refer to the order of operations to determine how to properly interpret a mathematical statement involving more than one operation. In this case, the operations involved are multiplication, exponentiation, addition, subtraction, and division, as can be seen from the reproduction of the expression below:

According to the order of operations, we focus our attention on the term with exponents first:

Next, we deal with the multiplication and division operations from left to right:

Finally, we deal with the addition and subraction operations from left to right:

Hence, the correct answer to the evaluation of the given expression is .

### Example Question #61 : Arithmetic

Which expression is equivalent to the following quotient? For .

**Possible Answers:**

**Correct answer:**

All we need to do is remember the quotient rule for exponents.

We apply this to each term and we get the following.

### Example Question #62 : Arithmetic

25 *( 47 + 72 + 19^{2})/12 + 65 = ?

**Possible Answers:**

1065

156

172

2273

87

**Correct answer:**

1065

Order of operations: "PEMDAS” or "Please Excuse My Dear Aunt Sally"

"Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction".

25 *( 47 + 72 + 19^{2})/12 + 65 =

25 * (47 + 72 + 361) / 12 + 65 =

25 * ( 480) / 12 + 65 =

12000 / 12 + 65 =

1000 + 65 = 1065

### Example Question #72 : New Sat Math Calculator

Find the Ratio of Solution 1 to Solution 2 in the following quadratic equation, where Solution 1 is and is a constant.

**Possible Answers:**

**Correct answer:**

The first step is to solve for , so we will plug in the solution.

So our quadratic equation becomes

Since we know one solution, we can do synthetic division to figure out the other solution.

We can guess and check for the other solution which is , so in equation form we have

The ratio is then .

### Example Question #63 : Arithmetic

(28 / 14) ^{3 }+ ( 144 – 12 )*17 + 36^{2 }= ?

**Possible Answers:**

3548

1652

2288

22032

2452

**Correct answer:**

3548

Order of operations: "PEMDAS” or "Please Excuse My Dear Aunt Sally"

"Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction".

(28 / 14) ^{3 }+ ( 144 – 12 )*17 + 36^{2 }=

(2) ^{3 }+ ( 132 )*17 + 36^{2 }=

8 + ( 132 )*17 + 1296 =

8 + 2244 + 1296 =

3548