### All ACT Math Resources

## Example Questions

### Example Question #1 : Rational Expressions

Simplify the following:

**Possible Answers:**

**Correct answer:**

To simplify the following, a common denominator must be achieved. In this case, the first term must be multiplied by (x+2) in both the numerator and denominator and likewise with the second term with (x-3).

### Example Question #2 : Rational Expressions

Simplify the following

**Possible Answers:**

**Correct answer:**

Find the least common denominator between x-3 and x-4, which is (x-3)(x-4). Therefore, you have . Multiplying the terms out equals . Combining like terms results in .

### Example Question #3 : Rational Expressions

Simplify the following expression:

**Possible Answers:**

**Correct answer:**

In order to add fractions, we must first make sure they have the same denominator.

So, we multiply by and get the following:

Then, we add across the numerators and simplify:

### Example Question #4 : Rational Expressions

Combine the following two expressions if possible.

**Possible Answers:**

**Correct answer:**

For binomial expressions, it is often faster to simply FOIL them together to find a common trinomial than it is to look for individual least common denominators. Let's do that here:

FOIL and simplify.

Combine numerators.

** **

Thus, our answer is

### Example Question #5 : Rational Expressions

Select the expression that is equivalent to

**Possible Answers:**

**Correct answer:**

To add the two fractions, a common denominator must be found. With one-term denominators, it is easier to simply find the least common denominator between them and multiply each side to obtain it.

In this case, the least common denominator between and is . So the first fraction needs to be multiplied by and the second by :

Now, we can add straight across, remembering to combine terms where we can.

So, our simplified answer is

### Example Question #6 : Rational Expressions

Find the product of and .

**Possible Answers:**

**Correct answer:**

Solve the first equation for .

Solve the second equation for .

The final step is to multiply and .

### Example Question #7 : Rational Expressions

The following table shows the temperature of a cup of coffee at different times

Time 1:09 1:11 1:13 1:15 1:17

Temperature (ºF) 187.1 184.4 181.7 179.0 176.3

If this trend continues, what will the temperature of the coffee at minute 1:25?

**Possible Answers:**

162.9°F

171.0°F

160.2°F

165.5°F

168.3°F

**Correct answer:**

165.5°F

The table shows that for every two minutes, the temperature of the coffee lowers 2.7ºF. At 1:25, 16 minutes, or eight 2-minute intervals have passed, and the temperature of the coffee has lowered by 8*2.7ºF, reaching a temperature of 165.5ºF.

### Example Question #8 : Rational Expressions

Amy buys concert tickets for herself and her friends. She initially buys them at $40/ticket. Weeks later, her other friends ask her to buy them tickets, but the prices have increased to $54. Amy buys 7 tickets total and spends $350. How many tickets has she paid $40 on?

**Possible Answers:**

**Correct answer:**

Amy has bought 7 tickets, x of them at $40/ticket, and the remaining 7-x at $54/ticket. She spends at total of

### Example Question #9 : Rational Expressions

What is

?

**Possible Answers:**

**Correct answer:**

To find an equivalency we must rationalize the denominator.

To rationalize the denominator multiply the numerator and denominator by the denominator.

To simplify completely, factor out a three from the numerator and denominator resulting in the final solution.

### Example Question #10 : Rational Expressions

Simplify:

**Possible Answers:**

**Correct answer:**

The common denominator of these two fractions simply is the product of the two denominators, namely:

Thus, you will need to multiply each fraction's numerator and denominator by the opposite fraction's denominator:

Let's first simplify the numerator:

, which is the simplest form you will need for this question.

However, the correct answer has the denominator multiplied out. Merely FOIL :

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