PSAT Math - How to find the solution to an equation

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

$14$

$10$

$12$

$16$

$18$

Explanation

Let $x$ = 1st even number, $x+2$ = 2nd even number, and $x+4$ = 3rd even number.

Then the equation to solve becomes $x+(x+2)+(x+4)=42$ .

$3x+6=42$

Thus $x=12,x+2=14,\ and\ x+4=16$ , so the middle number is 14.

What property of arithmetic is demonstrated here?

Reflexive

Transitive

Commutative

Identity

Symmetric

Explanation

The statement expresses the idea that any number is equal to itself. This is the reflexive property of equality.

If 5 + x is 5 more than 5,what is the value of 2_x_?

Explanation

5 more than 5 = 10

5 + x = 10

Subtract 5 from each side of the equation: x = 5 → 2_x_ = 10

Give the lines y = 0.5x+3 and y=3x-2. What is the y value of the point of intersection?

Explanation

In order to solve for the x value you set both equations equal to each other (0.5x+3=3x-2). This gives you the x value for the point of intersection at x=2. Plugging x=2 into either equation gives you y=4.

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?

$$500$

$$250$

$$750$

$$300$

$$625$

Explanation

Let $x$ = the number of birdhouses sold each month.

$Revenue=40x$

$Costs=15x+750$

$Profit = Revenue-Costs$

$=40x-15x-750$

$=25x-750$

Substituting in 50 for $x$ gives an answer of 500, so the profit on 50 birdhouses is $500.

The Widget Company makes widgets. The monthly fixed costs are $750. It costs $45 to make each widget. The widgets sells for $75 a piece.

The Widget Company wants to make a profit of $3,000. How many widgets must be sold?

Explanation

Profits = Revenues - Costs

Revenue:

Costs:

Profit:

So the equation to solve becomes

So a $3,000 profit occurs when they sell 125 widgets

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?

$55$

$40$

$45$

$50$

$60$

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:

$\frac{d + 225}{4} = 70$

$d + 225 = 280$

$d = 55 lbs$

The sum of two consecutive positive even integers is two greater than one-tenth of the product of the two integers. What is the sum of the positive factors of the smaller integer?

Explanation

We need to translate the information given into an algebraic equation that will help us solve for the integers.

Let x represent the smaller of the two integers. Because they are consecutive even integers, the other integer is equal to x + 2.

We are told that the sum of the two integers is two greater than one-tenth of their product. We can represent the sum of the two integers as x + (x + 2).

To represent one-tenth of the product, we can write (1/10)(x)(x + 2). Two greater than this quantity would be equal to 2 + (1/10)(x)(x + 2). Remember that "two greater" requires us to add 2, while "two times greater" would have required us to multiply by 2.

Now, we can set the sum equal to two greater than one-tenth the product.

x + (x + 2) = 2 + (1/10)(x)(x + 2)

Multiply both sides by 10 to eliminate the fraction.

10(x + x + 2) = 10(2 + (1/10)x(x + 2))

Distribute the 10 on both sides.

10_x_ + 10_x_ + 20 = 20 + 10(1/10)x(x + 2)

Simplify both sides.

20_x_ + 20 = 20 + x(x + 2)

Distribute the x on the right side.

20_x_ + 20 = 20 + x_2 + 2_x

Subtract 20 from both sides.

20_x_ = x_2 + 2_x

Subtract 20_x_ from both sides.

x_2 – 18_x = 0

Factor out an x.

x(x – 18) = 0

Set each factor equal to zero.

x = 0, or x – 18 = 0, which means x = 0 or 18. However, because we are told that the integers are positive, the value of x must be 18.

The smaller integer is 18. We are asked to find the sum of its factors. To find the factors of 18, we must think of all the pairs of numbers that will multiply to give us 18. The pairs are as follows:

1 and 18, 2 and 9, 3 and 6

This means the factors of 18 are 1, 2, 3, 6, 9, and 18. The sum of these numbers is 1 + 2 + 3 + 6 + 9 + 18 = 39.

The answer is 39.

What number decreased by 7 is equal to 10 increased by 7?