# ACT Math : Equations / Inequalities

## Example Questions

### Example Question #141 : Algebra

How much pure water must be added to 2 gallons of 90% pure cleaning solution to yield a 30% pure cleaning solution?

Explanation:

Let pure water be 0% and pure solution be 100%.

So the general equation to solve is:

where  is the volume and the  is percent solution.

So the equation to solve becomes

Solving shows that we need to add 4 gallons of pure water to 2 gallons of 90% pure cleaning solution to get a 30% pure solution.

Susan got a new piggy bank and counted the change she put into it.  She had one more nickel than dimes and two fewer quarters than nickles.  The value of her change was $1.40. How many total coins did she have? Possible Answers: Correct answer: Explanation: Let = number of dimes, = number of nickels, and = number of quarters. The general equation to use is: where is the money value and is the number of coins So the equation to solve becomes Thus, solving the equation shows that she had five nickels, four dimes, and three quarters giving a total of 12 coins. ### Example Question #42 : Equations / Inequalities How much pure water should be added to of 80% cleaning solution to dilute it to 25% cleaning solution. Possible Answers: Correct answer: Explanation: Pure water is 0% and pure solution is 100% where is the volume and is the percent. So the equation to solve becomes So we need to add pure water to of 80% cleaning solution to yield 25% cleaning solution. ### Example Question #41 : Linear / Rational / Variable Equations Luke purchased a tractor for$1200. The value of the tractor decreases by 25 percent each year. The value, , in dollars, of the tractor at  years from the date of purchase is given by the function .

In how many years from the date of purchase will the value of the tractor be $675? Possible Answers: 1 3 5 2 4 Correct answer: 2 Explanation: We are looking for the value of t that gives$675 as the result when plugged in V (t ). While there are many ways to do this, one of the fastest is to plug in the answer choices as values of t .

When we plug = 1 into V (t ), we get V (1) = 1200(0.75)1 = 1000(0.75) = $900, which is incorrect. When we plug = 2 into V (t ), we get V (2) = 1200(0.75)2 =$675, so this is our solution.

The value of the tractor will be $675 after 2 years. Finally, we can see that if = 3, 4, or 5, the resulting values of the V (t ) are all incorrect. ### Example Question #43 : Equations / Inequalities Solve for : Possible Answers: Correct answer: Explanation: First combine like terms. In this case, 4x and 9x can be added together: 13x + 13 = 0 Subtract 13 from both sides: 13x = -13 Divide both sides by 13 to isolate x: x = -13/13 x = -1 ### Example Question #56 : How To Find The Solution To An Equation Ben is walking three dogs that weigh an average of 75 pounds each. Ben begins to walk a fourth dog, and the average weight of the dogs decreases to 70 pounds. What is the weight in pounds of the fourth dog? Possible Answers: Correct answer: Explanation: The total weight of the first three dogs is 225 pounds. This amount, plus the weight of the fourth dog, divided by total number of dogs, is the new average weight: ### Example Question #152 : Algebra Pets Plus makes bird houses. Their monthly fixed expenses are$750.  The cost for each bird house is $15. The bird houses sell for$40.

What is the monthly break-even point at Pets Plus?

Explanation:

Let  the number of bird houses sold each month.

The break-even point is where the revenue is the same as the costs:

Solve for :

Therefore, Pets Plus must sell 30 bird houses to break-even.

### Example Question #58 : How To Find The Solution To An Equation

Pets Plus makes bird houses.  Their monthly fixed expenses are $750. The cost for each bird house is$15.  The bird houses sell for $40. If Pets Plus sells 50 bird houses, what is the profit? Possible Answers: Correct answer: Explanation: Let = the number of birdhouses sold each month. Substituting in 50 for gives an answer of 500, so the profit on 50 birdhouses is$500.

### Example Question #91 : How To Find The Solution To An Equation

George is three times older than Joey.  The sum of their ages is 16.  What is the product of their ages?

Explanation:

Let  = Joey's age and  = George's age.

Then the equation to solve becomes .

Therefore, Joey is 4 years old and George is 12 years old, so the product of their ages is 48.

### Example Question #91 : How To Find The Solution To An Equation

Three consecutive even numbers add to 42.  What is the middle number?