High School Math : Equations

Example Questions

← Previous 1 3 4

Example Question #1 : Equations

Tom is painting a fence  feet long. He starts at the West end of the fence and paints at a rate of  feet per hour. After  hours, Huck joins Tom and begins painting from the East end of the fence at a rate of  feet per hour. After  hours of the two boys painting at the same time, Tom leaves Huck to finish the job by himself.

If Huck completes painting the entire fence after Tom leaves, how many more hours will Huck work than Tom?

Explanation:

Tom paints for a total of  hours (2 on his own, 2 with Huck's help). Since he paints at a rate of  feet per hour, use the formula

(or )

to determine the total length of the fence Tom paints.

feet

Subtracting this from the total length of the fence  feet gives the length of the fence Tom will NOT paint:  feet. If Huck finishes the job, he will paint that  feet of the fence. Using , we can determine how long this will take Huck to do:

hours.

If Huck works  hours and Tom works  hours, he works  more hours than Tom.

Example Question #2 : Equations

Simplify the fraction to the lowest terms:

Cannot be simplified

Explanation:

Find the common multiple between the numerator and denominator.

divide numerator and denominator by 3:

divide numerator and denominator by 7:

divide numerator and denominator by 4:

Cannot be divided any more- lowest terms.

Example Question #1 : Equations

Solve the following equation for x in terms of the other variables:

Explanation:

Multiply both sides by to get:

Distribute the :

Combine like terms:

Divide both sides by :

Example Question #4 : Equations

Solve the following equation for x in terms of the other variables:

Explanation:

Divide both sides by :

Example Question #1 : Equations

If given the equation , with a positive integer, the result must be an integer multiple of:

2

8

10

12

5

5

Explanation:

The mathematical expression given in the question is . Adding together like terms, , this can be simplified to . The expression  can be factored as . For every positive integer ,  must be a multiple of 5. If , then , which is not an integer multiple of 2, 8, 10, or 15. Therefore, the correct answer is 5.

Example Question #1 : Equations

Cindy's Cotton Candy sells cotton candy by the bag.  Her monthly fixed costs are . It costs to make each bag and she sells them for .

What is the monthly break-even point?

Explanation:

The break-even point occurs when the .

The equation to solve becomes

so the break-even point is .

Example Question #7 : Equations

Cindy's Cotton Candy sells cotton candy by the bag.  Her monthly fixed costs are . It costs to make each bag and she sells them for .

To make a profit of , how many bags of cotton candy must be sold?

Explanation:

So the equation to solve becomes , or must be sold to make a profit of .

Example Question #8 : Equations

Solve for  and  to satisfy both equations in the system:

Explanation:

The two equations in this system can be combined by addition or subtraction to solve for  and . Isolate the  variable to solve for it by multiplying the top equation by  so that when the equations are combined the  term disappears.

Divide both sides by  to find  as the value for .

Substituting  for  in both of the two equations in the system and solving for  gives a value of  for

Example Question #9 : Equations

Solve for :

Explanation:

Rewrite  as a compound statement and solve each part separately:

The solution set is

Example Question #10 : Equations

and . What is the value of ?

130

25

33

112

65

65

Explanation:

First, notice that we can factor  into the form (a-b)(a+b). We are told that a-b=3, so we can substitute that into the first equation.

If we divide both sides by 3, we can obtain the value of a+b.

We now have a system of equations: a-b = 3, a+b = 11. We will solve this system by elimination. If we add the two equations together, we obtain the following:

.

Divide both sides by 2.

Going back to the equation a-b = 7, we can solve for b.

Subtract 3 from both sides.

Ultimately, the question asks us to determine the value of .

= .