# ACT Math : Algebraic Fractions

## Example Questions

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

Solve

0

–1

infinitely many solutions

no solution

infinitely many solutions

Explanation:

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 #5 : How To Evaluate A Fraction

For this question, the following trigonometric identities apply:

,

Simplify:

Explanation:

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 #2 : How To Evaluate A Fraction

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

22 1/2

1/2

10

160

360

22 1/2

Explanation:

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 #1 : How To Evaluate A Fraction

If  , , and , find the value of .

Explanation:

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 #8 : How To Evaluate A Fraction

If    and , then which of the following is equal to

Explanation:

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 .

Explanation:

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 #10 : How To Evaluate A Fraction

Explanation:

Factor out 7 from the numerator:

This simplifies to 7.

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

If  pizzas cost  dollars and  sodas cost  dollars, what is the cost of  pizzas and  sodas in terms of  and ?

Explanation:

If 10 pizzas cost x dollars, then each pizza costs x/10. Similarly, each soda costs y/6. We can add the pizzas and sodas together by finding a common denominator:

### Example Question #31 : Algebraic Fractions

According the pie chart, the degree measure of the sector representing the number of workers spending 5 to 9 years in the same role is how much greater in the construction industry chart than in the financial industry chart?

Explanation:

Since the values in the pie charts are currently in terms of percentages (/100), we must convert them to degrees (/360, since within a circle) to solve the question. The "5 to 9 years" portion for the financial and construction industries are 18 and 25 percent, respectively. As such, we can cross-multiply both:

18/100 = x/360

x = 65 degrees

25/100 = y/360

y = 90 degrees

Subtract: 90 – 65 = 25 degrees

Alternatively, we could first subtract the percentages (25 – 18 = 7), then convert the 7% to degree form via the same method of cross-multiplication.

### Example Question #32 : Algebraic Fractions

6 contestants have an equal chance of winning a game.  One contestant is disqualified, so now the 5 remaining contestants again have an equal chance of winning.  How much more likely is a contestant to win after the disqualification?