Direct Proportion problem

If y = kx, where k is a constant such that x = 5 when y = 1/3, what is the value of x when y = 12?

A. 36/5
B. 15/12
C. 180
D. 12/15


First find the value of k: Using the known y and x-values, we have 1/3 = 5k, making k = 1/15. When y = 12, we have 12 = (1/15)x, making x = 180. The correct choice is (C).

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Linear Inequality problem

A motel charges a $100 flat fee per one night of stay plus 8.5% tax. Before tax, senior citizens receive a 20% discount, and children under 10 receive a 15% discount. If the motel wishes to collect at least $1,000 per day while having an average of x children under 10, y adults, and z senior citizens staying each night, which inequality shows the average daily total collected by the motel?

A. 100(0.085)[1.085x + y +  1.080z] ≥ 1000

B. 100(1.085)[0.85x + y +  0.80z] ≤ 1000

C. 100(1.085)[0.85x + y +  0.80z] ≥ 1000

D. 100(0.085)[0.85x + y +  0.80z] ≥ 1000

A senior citizen gets a 20% discount, so his rate is 100(1 – 0.20) = 100(0.80) per night of stay. After tax, a senior citizen will pay 100(0.80)(1 + 0.085) = 100(0.80)(1.085) dollars per night. Thus the motel will receive 100(0.80)(1.085)z from senior citizens if z senior citizens stay per night. An adult gets no discount, so they will pay 100(1 + 0.085) = 100(1.085) dollars per night. A total of 100(1.085)y is collected per night by the motel if y adults stay per night. A child receives a 15% discount, so his rate including tax is 100(1 – 0.15)(1 + 0.085) = 100(0.85)(1.085) dollars per night. The motel will collect 100(0.85)(1.085)x if x children stay per night. Thus the total collected by the motel for x children, y adults, and z senior citizens staying for the night is 100(0.85)(1.085)x + 100(1.085)y + 100(0.80)(1.085)z = 100(1.085)[0.85x + y +  0.80z] dollars. Since this sum must be at least 1000 dollars, it must be greater than or equal to 1000. Thus the correct answer is (C).

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Systems of linear equations

For the following system of equations, find the value of A which produces no solution for the system.

-4w/3 + 2z = -3

5Aw + 5z/8 = -3

A. 6

B. -12

C. 1/3

D. -1/12

First try to think graphically. Each of the equations above is linear. A solution to a system of two linear equations means that there is a point of intersection between the two lines corresponding to the two equations (there can only be one such point). Having no such point means that the lines are parallel (they never meet at any point). If the lines are parallel, they must have the same slope. To have the same slope, coefficients in front of z in each equation are such that 2 = 5k/8 for some constant k. In other words, the constant k transforms a variable coefficient in one equation into the same exact variable coefficient in the second equation. This means that 2 is a scalar multiple of 5/8 for some k. Solving for k we get k = 16/5. Similarly, coefficients in front of w are such that -4/3 = 5A • k, and using the known k we have -4/3 = 5A • 16/5, which gives A = -1/12. Thus the correct answer is (D).

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Complex number problem

If i = √-1, which of the following expressions shows the square of the sum (2i + 3) + (2 – 6i)?

A. 41 + 40i

B. 9 + 40i

C. 41 – 40i

D. 9 – 40i

You may be tempted to square the sum right away, but it is simpler to add the numbers before squaring them. The sum is 2 + 3 – 6i + 2i = 5 – 4i. Squaring this result produces 5 • 5 + 2 • 5 • (-4i) + (-4i) • (-4i) = 25 – 40i + 16i² = 25 – 40i + 16(-1) = 25 – 16 – 40i = 9 – 40i. Thus the correct choice is (D).

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Geometry problem

In the following figure, an isosceles triangle ABC is perfectly enclosed in a semi-circle of radius r. If AB is a diameter of the semi-circle, which of the following expressions shows the combined area of the shaded regions in terms of r?

A. r²(π – 1) / 2

B. r²(π – 2)

C. r²(π – 2) / 2

D. r²(π – 1) 

The combined area of the shaded regions is determined from subtracting the area of the triangle from the area of the semi-circle. AB = 2r, and this segment must pass through the center of the circle. This means that the height of the triangle (to the vertex C) must be equal to the radius r. Thus the area of the triangle is 2r(r)/2 = r². Area of the semi-circle is πr²/2. Thus the combined area of the shaded regions is πr²/2 – r² = r²(π – 2) / 2. The correct answer choice is (C).

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Nonlinear function word problem #2

Perimeter of a rectangular playing field measures 1,500 feet. Find the maximum area of the field.

Let the length of the field be l and width w. Then, 1,500 = 2(l + w), meaning that 750 = l + w. This means that w = 750 - l, and since the area is given by A = lw, we have
A = l(750 - l)
= -l² + 750l
= -(l² – 750l)
= -[l² – 750l + (-750/2)² – (-750/2)²]
= -(l² – 750l + 140,625)  + 140,625
= -(l – 375)² + 140,625. 

This is a vertex form of a parabola. Since there is a negative coefficient in front of l, the parabola expands downward, and the vertex represents its maximum value. The vertex is given by (375, 140625), thus for the length 375 feet, the maximum area of the playing field is 140,625 square feet with the given perimeter of the field. Note that you could have expressed the area in terms of the width w, and it would not have changed the result.
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Graphical Analysis problem

If the curve shown on the graph is shifted 2 units to the left and then 3 units down, what will be the coordinates of its new x-intercept?

A. (0, 0)
B. (1, 0)
C. (-1, 0)
D. (2, 0)


The easiest method to solve this problem is to find a point on the graph that will land on the x-axis (where y = 0) when performing the horizontal and vertical shifts. The point (2, 3) on the curve is a good choice. The shifted curve will intercept the origin. Thus the correct choice is (A).

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