Question 1

Solve the problem.
-Megan is having her yard landscaped. She obtained an estimate from two landscaping companies. Company A gave an estimate of $240 for materials and equipment rental plus $55 per hour for labor.
Company B gave an estimate of $320 for materials and equipment rental plus $45 per hour for labor.
Determine how many hours of labor will be required for the two companies to cost the same.

Accepted Answer

To find the number of hours where the cost is the same for both companies, set the two cost equations equal to each other and solve for the number of hours (h):Company A: $240 + $55hCompany B: $320 + $45hEquation: $240 + $55h = $320 + $45hSolving for h: $55h - $45h = $320 - $240$10h = $80h = 8 hours

Question 2

Solve the problem.
-During a road trip, Tony drove one-third the distance that Lana drove. Mark drives 12 more miles than Lana drove. The total distance they drove on the trip was 558 miles. How many miles did each person
Drive?

Accepted Answer

Let the distance Lana drove be $L$, Tony drove $\frac{1}{3}L$, and Mark drove $L + 12$. The equation is $\frac{1}{3}L + L + (L + 12) = 558$. Solving gives $L = 234$, so Tony drove $78$ miles, and Mark drove $246$ miles.

Question 3

Solve the problem.
-You inherit $70,000 from a very wealthy grandparent, with the stipulation that for the first year, the money must be invested in two stocks paying 4% and 10% annual interest, respectively. How much should be
Invested at each rate if the total interest earned for the year is to be $4000?

Accepted Answer

To earn $4000 in interest, let x be the amount invested at 4% and y be the amount invested at 10%. We have the equations: x + y = 70,000 and 0.04x + 0.10y = 4000. Solving these gives x = 50,000 and y = 20,000.

Question 4

Solve the problem.
-A train ticket in a certain city is $1.50. People who use the train also have the option of purchasing a frequent rider pass for $16.50 each month. With the pass, each ticket costs only $0.75. Determine the
Number of times in a month the train must be used so that the total monthly cost without the pass is the
Same as the total monthly cost with the pass.

Accepted Answer

To find the break-even point, set the total cost without the pass equal to the total cost with the pass: 1.50x = 16.50 + 0.75x. Solving for x gives x = 22.

Question 5

Solve the problem.
-There are 22 more sophomores than juniors in an 8 AM algebra class. If there are 46 students in this class, find the number of sophomores and the number of juniors in the class.

Accepted Answer

Let the number of juniors be $j$. Then, the number of sophomores is $j + 22$. The total number of students is $j + (j + 22) = 46$. Solving this equation, $2j + 22 = 46$, we get $2j = 24$, so $j = 12$. Therefore, there are 12 juniors and $12 + 22 = 34$ sophomores.

Question 6

Solve the problem.
-The president of a certain university makes three times as much money as one of the department heads. If the total of their salaries is $200,000, find each worker's salary.

Accepted Answer

Let x be the department head's salary. Then the president's salary is 3x. The total of their salaries is $200,000, so we can write an equation:
x + 3x = $200,000
Simplifying this, we get 4x = $200,000, or x = $50,000. Therefore, the department head's salary is $50,000 and the president's salary is 3x = $150,000. Choice A gives those exact salaries.

Question 7

Solve the formula for the specified variable.
-$\mathrm { A } = \frac { 1 } { 2 } \mathrm { bh } \quad$ for $h$

Accepted Answer

To solve for $h$, multiply both sides by 2 and divide by $b$, resulting in $h = \frac { 2 A } { b }$.

Question 8

Solve the problem.
-A car rental agency charges $175 per week plus $0.20 per mile to rent a car. How many miles can you travel in one week for $295?

Accepted Answer

The total cost is $295, and the base charge is $175. Subtracting the base charge from the total gives $295 - $175 = $120 for miles. At $0.20 per mile, divide $120 by $0.20 to get 600 miles.

Question 9

Solve the problem.
-You inherit $10,000 with the stipulation that for the first year the money must be invested in two stocks paying 6% and 11% annual interest, respectively. How much should be invested at each rate if the total
Interest earned for the year is to be $700?

Accepted Answer

To earn $700 in interest, $8000 should be invested at 6% ($480 interest) and $2000 at 11% ($220 interest), totaling $700.

Question 10

Solve the problem.
-A bookcase is to be constructed as shown in the figure below. The height of the bookcase is 3 feet longer than the length of a shelf. If 24 feet of lumber is available for the entire unit (including the shelves, but
NOT the back of the bookcase), find the length and height of the unit.

Accepted Answer

The answer of Solve the problem.
-A bookcase is to be...

Question 11

Solve the problem.
-An auto repair shop charged a customer $280 to repair a car. The bill listed $80 for parts and the remainder for labor. If the cost of labor is $50 per hour, how many hours of labor did it take to repair the
Car?

Accepted Answer

The cost of labor is $50 per hour, and the total cost of labor is $280 - $80 = $200. So, the number of hours of labor is $200 / $50 = 4 hours.

Question 12

Solve the problem.
-The sum of the angles of a triangle is 180°. Find the three angles of the triangle if one angle is three times the smallest angle and the third angle is 30° greater than the smallest angle.

Accepted Answer

The answer of Solve the problem.
-The sum of the angles...

Question 13

Solve the problem.
-After a 9% price reduction, a boat sold for $30,940. What was the boat's price before the reduction? (Round to the nearest cent, if necessary.)

Accepted Answer

The boat's price after a 9% reduction is 91% of its original price. To find the original price, divide the sale price by 0.91: $30,940 / 0.91 = $34,000.

Question 14

Solve the problem.
-Sergio's internet provider charges its customers $15 per month plus 4¢ per minute of on-line usage. Sergio received a bill from the provider covering a 5-month period and was charged a total of $101.00. How
Many minutes did he spend on-line during that period? (Round to the nearest whole minute, if necessary.)

Accepted Answer

Sergio's fixed charges for 5 months are 5 * $15 = $75. Subtracting this from the total bill, $101 - $75 = $26, is for online usage. At 4¢ per minute, the number of minutes is $26 / $0.04 = 650 minutes.

Question 15

Solve the problem.
-In a recent International Gymnastics competition, the U.S., China, and Romania were the big winners. If the total number of medals won by each team are three consecutive integers whose sum is 51 and the U.S.
Won more than China who won more than Romania, how many medals did each team win?

Accepted Answer

Let the number of medals won by Romania be x. Then the number of medals won by China is x + 1, and the number of medals won by the U.S. is x + 2. Since the sum of the number of medals won by each team is 51, we have x + (x + 1) + (x + 2) = 51. Solving for x, we get x = 16. Therefore, Romania won 16 medals, China won 17 medals, and the U.S. won 18 medals.

Question 16

Solve the formula for the specified variable.
-$\mathrm { S } = 2 \pi \mathrm { rh } + 2 \pi \mathrm { r } ^ { 2 } \quad$ for $h$

Accepted Answer

To solve for $h$, we need to isolate $h$ on one side of the equation. First, we can factor out $2\pi r$ to get:
$$\mathrm { S } = 2 \pi r (h + r) $$
Next, we can divide both sides by $2 \pi r$ to get:
$$\frac{S}{2 \pi r} = h + r$$
Finally, we can subtract $r$ from both sides to get:
$$h = \frac{S}{2 \pi r} - r = \frac{S-2\pi r^2}{2\pi r}$$
Therefore, the answer is choice letter A.

Question 17

Solve the formula for the specified variable.
-$$F = \frac { 9 } { 5 } C + 32 \quad \text { for } C$$

Accepted Answer

To solve for $C$, subtract 32 from both sides to get $F - 32 = \frac{9}{5}C$, then multiply both sides by $\frac{5}{9}$ to isolate $C$, resulting in $C = \frac{5}{9}(F - 32)$.

Question 18

Solve the problem.
-Inclusive of a 6.8% sales tax, a diamond ring sold for $2349.60. Find the price of the ring before the tax was added. (Round to the nearest cent, if necessary.)

Accepted Answer

The total cost of the ring including sales tax is $2349.60. To find the original price before tax, divide the total cost by 1 plus the tax rate (expressed as a decimal). The tax rate is 6.8%, or 0.068 as a decimal. So, the original price is $2349.60 / (1 + 0.068) = $2200.

Question 19

Solve the formula for the specified variable.
-$$V = \frac { 1 } { 3 } B h \quad \text { for } B$$

Accepted Answer

To solve for $B$, multiply both sides by 3 and divide by $h$, resulting in $B = \frac{3V}{h}$.

Question 20

Solve the problem.
-The length of a rectangular room is 6 feet longer than twice the width. If the room's perimeter is 156 feet, what are the room's dimensions?

Accepted Answer

The problem can be solved by setting up equations based on the given information. Let the width be $w$ feet. Then, the length is $2w + 6$ feet. The perimeter of a rectangle is given by $2(length + width)$, so we have $2(w + 2w + 6) = 156$. Solving this equation gives $w = 24$ feet and the length $2(24) + 6 = 54$ feet.

Quiz 1: Equations and Inequalities

Solve the problem. -During a road trip, Tony drove one-third the distance that Lana drove. Mark drives 12 more miles than Lana drove. The total distance they drove on the trip was 558 miles. How many miles did each person Drive?

Solve the problem. -There are 22 more sophomores than juniors in an 8 AM algebra class. If there are 46 students in this class, find the number of sophomores and the number of juniors in the class.

Solve the problem. -The president of a certain university makes three times as much money as one of the department heads. If the total of their salaries is $200,000, find each worker's salary.

Solve the formula for the specified variable. - $\mathrm { A } = \frac { 1 } { 2 } \mathrm { bh } \quad$ for $h$

Solve the problem. -A car rental agency charges $175 per week plus $0.20 per mile to rent a car. How many miles can you travel in one week for $295?

Solve the problem. -You inherit $10,000 with the stipulation that for the first year the money must be invested in two stocks paying 6% and 11% annual interest, respectively. How much should be invested at each rate if the total Interest earned for the year is to be $700?

Solve the problem. -An auto repair shop charged a customer $280 to repair a car. The bill listed $80 for parts and the remainder for labor. If the cost of labor is $50 per hour, how many hours of labor did it take to repair the Car?

Solve the problem. -The sum of the angles of a triangle is 180°. Find the three angles of the triangle if one angle is three times the smallest angle and the third angle is 30° greater than the smallest angle.

Solve the problem. -After a 9% price reduction, a boat sold for $30,940. What was the boat's price before the reduction? (Round to the nearest cent, if necessary.)

Solve the formula for the specified variable. - $\mathrm { S } = 2 \pi \mathrm { rh } + 2 \pi \mathrm { r } ^ { 2 } \quad$ for $h$

Solve the formula for the specified variable. - $F = \frac { 9 } { 5 } C + 32 \quad \text { for } C$

Solve the problem. -Inclusive of a 6.8% sales tax, a diamond ring sold for $2349.60. Find the price of the ring before the tax was added. (Round to the nearest cent, if necessary.)

Solve the formula for the specified variable. - $V = \frac { 1 } { 3 } B h \quad \text { for } B$

Solve the problem. -The length of a rectangular room is 6 feet longer than twice the width. If the room's perimeter is 156 feet, what are the room's dimensions?