### All ACT Math Resources

## Example Questions

### Example Question #21 : Algebraic Fractions

Evaluate the following equation when and round your answer to the nearest hundredth.

**Possible Answers:**

**Correct answer:**

1. Plug in wherever there is an in the above equation.

2. Perform the above operations.

### Example Question #1 : How To Evaluate A Fraction

Mary walked to school at an average speed of 2 miles per hour and jogged back along the same route at an average speed of 6 miles per hour. If her total traveling time was 1 hour, what was the total number of miles in the round trip?

**Possible Answers:**

**Correct answer:**

Since Mary traveled 3 times as quickly coming from school as she did going to school (6 miles per hour compared to 2 miles per hour), we know that Mary spent only a third of the time coming from school as she did going. If *x* represents the number of hours it took to get to school, then *x*/3 represents the number of hours it took her to return.

Knowing that the total trip took 1 hour, we have:

*x *+ *x*/3 = 1

3x/3 + 1x/3 = 1

4*x*/3 = 1

*x * = 3/4

So we know it took Mary 3/4 of an hour to travel to school (and the remaining 1/4 of an hour to get back).

Remembering that distance = rate * time, the distance Mary traveled on her way to school was (2 miles per hour) * (3/4 of an hour) = 3/2 miles. Furthermore, since she took the same route coming back, she must have traveled 3/2 of a mile to return as well.

Therefore, the the total number of miles in Mary's round trip is 3/2 miles + 3/2 miles = 6/2 miles = 3 miles.

### Example Question #1 : How To Evaluate A Fraction

If then which of the following is equal to ?

**Possible Answers:**

**Correct answer:**

To raise to the exponent , square and then take the cube root.

### Example Question #2 : How To Evaluate A Fraction

Solve

**Possible Answers:**

infinitely many solutions

–1

0

no solution

**Correct answer:**

infinitely many solutions

The common denominator of the left side is x(x–1). Multiplying the top and bottom of 1/x by (x–1) yields

Since this statement is true, there are infinitely many solutions.

### Example Question #1 : How To Evaluate A Fraction

For this question, the following trigonometric identities apply:

,

Simplify:

**Possible Answers:**

**Correct answer:**

To begin a problem like this, you must first convert everything to and alone. This way, you can begin to cancel and combine to its most simplified form.

Since and , we insert those identities into the equation as follows.

From here we combine the numerator and denominators of each fraction together to easily see what we can combine and cancel.

Since there is a in the numerator and the denominator, we can cancel them as they divide to equal 1. All we have left is , the answer.

### Example Question #1 : How To Evaluate A Fraction

If 3x = 12, y/4 = 10, and 4z = 9, what is the value of (10xyz)/xy?

**Possible Answers:**

360

160

10

22 ^{1}/2

4 ^{1}/_{2}

**Correct answer:**

22 ^{1}/2

Solve for the variables, the plug into formula.

x = 12/3 = 4

y = 10 * 4 = 40

z= 9/4 = 2 ^{1}/_{4}

10xyz = 3600

Xy = 160

3600/160 = 22 ^{1}/2

### Example Question #8 : How To Evaluate A Fraction

If , , and , find the value of .

**Possible Answers:**

**Correct answer:**

In order to solve , we must first find the values of , , and using the initial equations provided. Starting with :

Then:

Finally:

With the values of , , and in hand, we can solve the final equation:

### Example Question #9 : How To Evaluate A Fraction

If and , then which of the following is equal to ?

**Possible Answers:**

**Correct answer:**

In order to solve , first substitute the values of and provided in the problem:

Find the Least Common Multiple (LCM) of the fractional terms in the denominator and find the equivalent fractions with the same common denominator:

Finally, in order to divide by a fraction, we must multiply by the reciprocal of the fraction:

### Example Question #2 : How To Evaluate A Fraction

Find the value of if and .

**Possible Answers:**

**Correct answer:**

In order to solve for , first substitute into the equation for :

Then, find the Least Common Multiple (LCM) of the two fractions and generate equivalent fractions with the same denominator:

Finally, simplify the equation:

### Example Question #11 : How To Evaluate A Fraction

**Possible Answers:**

**Correct answer:**

Factor out 7 from the numerator:

This simplifies to 7.

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