If the length L of a rectangle is 3 meters more than twice its width and its perimeter is 300 meters, which of the following equations could be used to find L?

A) 3L + 3 = 300
B) 3L = 300
C) 3L - 3 = 300
D) 2L + 3 = 300
E) 4L = 300

Solution

Let L and W be the length and width of the rectangle. "length L of a rectangle is 3 meters more than twice its width " is translated mathematically as follows

L = 2 W + 3

Use perimeter to write

300 = 2 W + 2 L

We need to rewrite the above equation in terms of L only. Solve the equation L = 2 W + 3 for W to obtain

W = (1/2)(L - 3)

We now substitute W by (1/2)(L - 3) in the perimeter formula. Hence

300 = 2 ((1/2)(L - 3)) + 2 L = 3L - 3

Simplify

300 = 3L - 3

Which correspond to answer in C) above.

The average of two numbers is 50. Their difference is 40. Write an equation that may be used to find x the smallest of the two numbers.

A) x - 20 = 50
B) 2x + 20 = 50
C) x - 20 = 100
D) x + 20 = 40
E) x + 20 = 50

Solution

If x is the smallest number and the difference of the two numbers is 40, then the second number is 40 + x. The average of the two numbers is 50. Hence

(x + x + 40) / 2 = 50

Multiply both sides of the equation by 2 and group like terms

2x + 40 = 100

Divide all terms of the above equation by 2

x + 20 = 50

Which corresponds to the answer in E).

Pump A can fill a tank in 2 hours and pump B can fill the same tank in 3 hours. If t is the time, in hours, that both pump take to fill the tank, which of these equations could be used to find t?

A) 2t + 3t = 1
B) t/2 + t/3 = 1
C) t / (2 + 3) = 1
D) (2 + 3) / t = 1
E) t + 2 + 3 = 1

Solution

If pump A can fill the tank in 2 hours, its rate is 1/2 tank/hour and similarly the rate of pump B is 1/3 tank/hour. In t hours, pump A fills (1/2) t tank and pump B fills (1/3) t tank. If t is the time to fill the whole tank (1), then

(1/2)t + (1/3) t = 1

The above equation may also be written as

t/2 + t/3 = 1

and corresponds to the answer in B)

John drove for two hours at the speed of 50 miles per hour (mph) and another x hours at the speed of 55 mph. If the average speed of the entire journey is 53 mph, which of the following could be used to find x?

A) (55 + x) / 2 = 53
B) (50 + x) / 2 = 53
C) (55 + 53) / 2 = x
D) 100 + 55 x = 53 (2 + x)
E) 53 x = 2

Solution

The average of the entire journey is given by

total distance / total time

The total distance for this journey is

total distsnce = 50*2 + 55 x = 100 + 55x

The total time for this journey is

total time = 2 + x

The average speed is 53. hence

53 = (100 + 55x) / (2 + x)

The above equation may be written as

100 + 55x = 53(2 + x)

which corresponds to the answer in D)

The sum of 3 consecutive even numbers is 126. Which of these equations could be used to find x, the largest of these 3 numbers?

A) 3x - 6 = 126
B) 3x + 6 = 126
C) 3x + 3 = 126
D) 3x - 3 = 126
E) 3x - 9 = 126

Solution

The difference between any two consecutive even numbers is 2. Hence if x is the largest, then the other two numbers are

x - 2 and x - 4

The sum of these numbers is 126. Hence

x - 4 + x - 2 + x = 126

which may be written as

3x - 6 = 126

and corresponds to answer A) above.

The radius of a circle is 3 centimeters (cm) more than twice the side of a square. The circumference of the circle is 4 times the perimeter of the square. Write an equation that can be used to find the radius r of the circle.

A) 2 pi r = 16 (r - 3)
B) 2 pi r = 8 (r + 3)
C) 2 pi r = 16 (r + 3)
D) 2 pi r = 8 (r - 3)
E) 6 pi = 4r

Solution

If r is the radius of the circle and x is the side of the square, then

r = 2x + 3

If C is the circumference of the circle and P is the perimeter of the square, then

C = 4 P

Now using the formula for circumference (C = 2 pi r) of circle and perimeter (P = 4x) of square, we can write

2 pi r = 4 (4x)

We now solve the equation r = 2x + 3 for x

x = (1/2)(r - 3)

and substitute x by (1/2)(r - 3) in 2 pi r = 4 (4x), simplify to obtain

2 pi r = 4 (4((1/2)(r - 3)))

2 pi r = 8(r - 3)

which corresponds to D) above

The height of a trapezoid is h (mm). b is the length of one of the two bases of the trapezoid and is 2 mm longer than 3 times the length of the second base. The area of the trapezoid is 300 mm^{2}. Write an equation to find b.

A) (h / 2) (4b - 2) = 300
B) (h / 2) (4b + 2) = 300
C) h (4b - 2) = 300
D) h (4b - 2) = 600
E) (h / 3) (4b - 2) = 600

Solution

The area A of a trapezoid of height h and bases b and B is given by

A = (h/2)*(b + B)

"b is the length of one of the two bases of the trapezoid and is 2 mm longer than 3 times the length of the second base" is translated mathematically as

b = 3B + 2 which can also be written as B = (b - 2) / 3

Substitute in the formula of the area

300 = (h/2) [ b + (b-2)/3 ]

Which may rewritten as

600 = (h / 3)(4b - 2) , which corresponds to E) above

25% of one third of the sum of twice x and 3 is equal to half of the difference of x and its tenth. Write an equation to find x.

The price x of a car was first decreased by 15% and decreased a second time by 10%. Write an equation to find x if the car was bought at $12,000.

A) (x - 0.15 x) - 0.1 (x - 0.15 x) = 12,000
B) x - 0.15x - 0.1 x = 12,000
C) x - 0.15x - 0.1 x = 12,000
D) (x - 0.15 x) - 0.1 x = 12,000
E) x = 12,000 - 15% - 10%

A company produces a product for which the variable cost is $6.2 per unit and the fixed costs are $22,000. The company sells the product for $11.45 per unit. Write an equation to find x, the numbers of units sold if the total profit made by the company was $45,000.

A) 45,000 = 11.45 x + (6.2 x + 22,000)
B) 45,000 - 22,000 = 11.45 x - 6.2 x
C) 45,000 = 11.45 x - (6.2 x + 22,000)
D) 45,000 = 11.45 x - 6.2 x + 22,000
E) 45,000 + 11.45 x = 6.2 x + 22,000