In the figure below, what is the measure of ∠α?

Solution
The triangle pictured is isosceles, meaning that the two angles opposite the sides that have equal length have equal measure. Since the sum of the interior angles of a triangle is 180◦, the measure of ∠α is 180◦ − 2(55◦), or 180◦ − 110◦, which is 70◦.
If x = 3yz^{2}, what is y in terms of x and z?

Solution
To get y in terms of x and z, solve x = 3yz^{2} for y, as shown below:
x = 3yz^{2}
\(\frac{x}{3z^{2}}=y,\: or\: y=\frac{x}{3z^{2}}\)
A rectangular garden has a length of x and a width of y. The garden has its length reduced by 3 feet and its width extended by 2 feet. What is the area of the new garden?

Solution
You are given that a rectangular garden has a length of x and a width of y, and has its length reduced by 3 feet and its width extended by 2 feet. Therefore, the length becomes x − 3 and its width becomes y + 2. The area of the new garden is then (x − 3)(y + 2).
In the standard (x, y) coordinate plane, if the xcoordinate of each point on a line is 5 more than half the ycoordinate, what is the slope of the line?

Solution
If the xcoordinate of each point on a line is 5 more than half the ycoordinate, then x = y 2 + 5. To find the slope of the line, solve for y and put the equation in slopeintercept form (y = mx + b, where m is the slope).
To do so, first subtract 5 from both sides, then multiply the entire equation by 2 to get y = 2x−10. The slope is 2.
In the parallelogram PQRS shown below, PS is 7 centimeters long. If the parallelogram’s perimeter is 40 centimeters, how many centimeters long is PQ?

Solution
The perimeter is the distance around the parallelogram. In parallelograms, opposing sides have equal lengths. If PS is 7 cm long, so is QR. Also, PQ and SR will have the same length. Set the length of PQ and SR to l, and solve.
7 + 7 + l + l = 40
14 + 2l = 40
14 + 2l = 40
l = \(\frac{26}{2}\) , or 13.
Which of the following is a value of r for which (r + 2)(r − 3) = 0?

Solution
The expression (r + 2)(r − 3) will equal zero when either r + 2 or r − 3 equals zero. Thus r = −2 or r = 3.
Bus X travels 40 miles per hour for 2 hours; Bus Y travels 60 miles per hour for 1^{1}⁄_{2}hours. What is the difference, in miles, between the number of miles traveled by Bus X and the number of miles traveled by Bus Y?

Solution
To find the difference in the distance traveled, first find the distance traveled by each bus. Distance equals rate multiplied by time. Since Bus X travels 40 miles per hour for 2 hours, it traveled 40(2) = 80 miles. Likewise, since Bus Y travels 60 miles per hour for 1^{1}⁄_{2} hours, it traveled 60(1.5) = 90 miles. Therefore, the difference is 90 − 80, or 10 miles.
Contributions to the school dance fund are made by each of 4 student groups according to the table below.
What is the average dollar amount of the contributions made by the 4 student groups?

Solution
To find the average, divide the sum of the contributions by the number of contributions, as follows:
\(\frac{(25 + 40 + 30 + 15)}{4}=\frac{110}{4}= 27.50\)
To keep up with rising costs, a carpenter needs to increase his $30.00 per hour rate by 18%. What will be his new hourly rate?

Solution
To calculate the new hourly rate, multiply the current rate by 18%, or 0.18, its decimal equivalent, and add the result to the current rate:
30(0.18) + 30 = 30 + 5.40, or $35.40
One foot is equivalent to approximately 0.3048 meters.If a building is 65feet long, what is the length of the building in meters, to the nearest tenth?

Solution
You are given that one foot is approximately 0.3048 meters. Therefore, a building that is 65 feet long will be 65(0.3048), or 19.8 meters long.