# ACT Math : Equations / Inequalities

## Example Questions

### Example Question #81 : Algebra

Jen and Karen are travelling for the weekend. They both leave from Jen's house and meet at their destination. Jen drives 45mph the whole way. Karen drives 60mph but leaves a half hour after Jen. How long after Jen leaves does Karen catch up with her?

She can't catch up.

Explanation:

For this type of problem, we use the formula:

When Karen catches up with Jen, their distances are equivalent. Thus,

We then make a variable for Jen's time, . Thus we know that Karen's time is  (since we are working in hours).

Thus,

### Example Question #81 : Algebra

Bill and Bob are working to build toys. Bill can build  toys in 6 hours. Bob can build  toys in 3 hours. How long would it take Bob and Bill to build  toys working together?

Explanation:

Bill builds  toys an hour. Bob builds  toys an hour. Together, their rate of building is . Together they can build  toys in 2 hours. They would be able to build  toys in 8 hours.

### Example Question #1856 : Sat Mathematics

Audrey, Penelope and Clementine are all sisters. Penelope is 8 years older than Clementine and 2 years younger than Audrey. If the sum of Penelope and Clementine's age is Audrey's age, how old is Clementine's age?

Explanation:

Let  = Audrey's age,  = Penelope's age, and  = Clementine's age.

Since , then .

Furthermore, , and .

Through substitution, .

### Example Question #1857 : Sat Mathematics

If   and , what is the value of ?

Explanation:

We could use the substitution or elimination method to solve the system of equations. Here we will use the elimination method.

To solve for , combine the equations in a way that makes the  terms drop out. The first equation has  and the second , so multiplying the first equation times 2 then adding the equations will eliminate the  terms.

Multiplying the first equation times 2:

Adding this result to the second equation:

Isolate  by dividing both sides by 7:

### Example Question #51 : Linear / Rational / Variable Equations

If   and , then what is the value of  ?

Explanation:

Since the expression we want just involves z and x, but not y, we start by solving  for y .

Then we can plug that expression in for y in the first equation.

Multiply everything by 12 to get rid of fractions.

### Example Question #1852 : Sat Mathematics

If , what is  in terms of ?

Explanation:

Use inverse operations to isolate x. Working from the outermost part on the left side, we first divide both sides by 5.

To isolate the x term, subtract y from both sides.

Finally, isolating just x, divide both sides by 3.

### Example Question #81 : Linear / Rational / Variable Equations

If , then, in terms of ,

Cannot be determined

Explanation:

You can solve this problem by plugging in random values or by simply solving for k. To solve for k, put the s values on one side and the k values on the other side of the equation. First, subtract 4s from both sides. This gives 4s – 6k = –2k. Next, add 6k to both sides. This leaves you with 4s = 4k, which simplifies to s=k. The answer is therefore s.

### Example Question #1851 : Sat Mathematics

The sum of two consecutive odd integers is 32. What is the value of the next consecutive odd integer?

Cannot be determined