1. Write the expression for three times the sum of twice a number and one minus 6.
- Answer: 3(2x+1) – 6
- Explanation: The expression is three times the sum of twice a number and 1, which is 3(2x+1). Then, 6 is subtracted from this expression.
2. Which of the following inequalities is equivalent to 3 – (1/2)x ≥ 2?
- Answer: x ≤ 2
- Explanation: To simplify this inequality, subtract 3 from both sides to get:
Then, multiply both sides by -2 (remembering this flips the direction of the inequality) to get x ≤ 2.
3. 6 is 30% of what number?
- Answer: 20
- Explanation: 30% is 3/10. The number itself must be 10/3 of 6, or (10/3) × 6 = 20.
4. Solve the following equation:
- Answer: 0.0523
- Explanation: Multiplying by 10^(-3) means moving the decimal point three places to the left, putting in zeroes as necessary.
5. A closet is filled with red, blue, and green shirts. If 1/3 of the shirts are green and 2/5 are red, what fraction of the shirts are blue?
- Answer: 4/15
- Explanation: The total fraction taken up by green and red shirts will be:
The remaining fraction is:
6. On Monday, Robert mopped the floor in 4 hours. On Tuesday, he did it in 3 hours. If on Monday, his average rate of mopping was p sq. ft. per hour, what was his average rate on Tuesday?
- Answer: (4/3)p sq. ft. per hour
- Explanation: Robert accomplished his task on Tuesday in 3/4 the time compared to Monday. He must have worked 4/3 as fast.
7. A rectangle has a length that is 5 feet longer than three times its width. If the perimeter is 90 feet, what is the length in feet?
- Answer: 35
- Explanation: Denote the width as w and the length as l. Then, l=3w+5. The perimeter is 2w+2l = 90. Substituting the first expression for l into the second equation yields 2(3w+5)+2w = 90, or 8w = 80, so w = 10. Putting this into the first equation, it yields:
8. A company invests $50,000 in a building where they can produce saws. If the cost of producing one saw is $40, then which function expresses the amount of money the company pays? The variable y is the money paid and x is the number of saws produced.
- Answer: y = 40x+50,000
- Explanation: For manufacturing costs, there is a linear relationship between the cost to the company and the number produced, with a y-intercept given by the base cost of acquiring the means of production, and a slope given by the cost to produce one unit. In this case, that base cost is $50,000, while the cost per unit is $40. So, y = 40x+50,000.
9. How do you solve V = lwh for h?
- Answer: h = V/lw
- Explanation: The formula can be manipulated by dividing both sides by the length, l, and the width, w. The length and width will cancel on the right, leaving height by itself.
10. The square and circle have the same center. The circle has a radius of r. What is the area of the shaded region?
- Answer: (4 – π)r^2
- Explanation: The area of the shaded region is the area of the square, minus the area of the circle. The area of the circle will be πr^2. The side of the square will be 2r, so the area of the square will be 4r^2. Therefore, the difference is: