Question 1

To reach a sales meeting, Carrie traveled by plane for 1 hour and by car for an additional 2 hours. The plane travels 82 mph faster than the car. If the total distance traveled is 217 miles, what are the Speeds of the plane and the car? (Assume that the plane and the car travel at constant rates.)
A) plane: 132 mph; car: 40 mph
B) plane: 122 mph; car: 50 mph
C) plane: 117 mph; car: 35 mph
D) plane: 127 mph; car: 45 mph

Accepted Answer

Let the speed of the car be x mph. Then, the speed of the plane is x + 82 mph. The total distance is given by the sum of the distances traveled by plane and car: (x + 82) * 1 + x * 2 = 217. Solving this equation gives x = 45 mph for the car and x + 82 = 127 mph for the plane.

Question 2

Lorenzo invested $22,000 into two accounts; one pays 4% simple interest and the other pays 7% simple interest. At the end of the first year, Lorenzo's total return was $1150. How much did he Invest in each account?
A) $13,000 at 4%; $9000 at 7%
B) $9000 at 4%; $13,000 at 7%
C) $10,000 at 4%; $12,000 at 7%
D) $12,000 at 4%; $10,000 at 7%

Accepted Answer

Let x be the amount invested at 4% and y be the amount invested at 7%.We have the following system of equations:x + y = 220000.04x + 0.07y = 1150Solving this system of equations, we get:x = 13000y = 9000Therefore, Lorenzo invested $13,000 at 4% and $9000 at 7%.

Question 3

Paula invested into two accounts; one pays 7% simple interest and the other pays 8% simple interest. She invested $2000 more in the account paying 8% than in the account paying 7%. At the End of the first year, Paula's total return was $1810. How much did she invest in each account?
A) $14,000 at 7%; $10,000 at 8%
B) $11,000 at 7%; $13,000 at 8%
C) $13,000 at 7%; $11,000 at 8%
D) $10,000 at 7%; $14,000 at 8%

Accepted Answer

Let x be the amount invested at 7%. Then, x + 2000 is the amount invested at 8%. The equation for total interest is 0.07x + 0.08(x + 2000) = 1810. Solving this gives x = 11000, so she invested $11,000 at 7% and $13,000 at 8%.

Question 4

A chemist wants to mix a 23% acid solution with a 43% acid solution to get 15 L of a 31% acid solution. How many liters of the 23% solution and how many liters of the 43% solution should be Mixed?
A) 5 L of the 23% solution; 10 L of the 43% solution
B) 9 L of the 23% solution; 6 L of the 43% solution
C) 10 L of the 23% solution; 5 L of the 43% solution
D) 6 L of the 23% solution; 9 L of the 43% solution

Accepted Answer

To solve this, we use the equation for mixture problems: (amount of first solution * its percentage + amount of second solution * its percentage) / total amount = desired percentage. Let x be the amount of the 23% solution, so 15 - x is the amount of the 43% solution. The equation is: (0.23x + 0.43(15 - x)) / 15 = 0.31. Solving for x gives 9 L of the 23% solution and 6 L of the 43% solution.

Question 5

A plane flew 450 miles in 3 hours with the wind. It took 5 hours to travel the same distance against the wind. What is the rate of the plane in still air and the rate of the wind?
A) plane: 150 mph; wind: 90 mph
B) plane: 120 mph; wind: 30 mph
C) plane: 130 mph; wind: 40 mph
D) plane: 130 mph; wind: 30 mph

Accepted Answer

The plane's speed in still air is the average of its speed with the wind and its speed against the wind. With the wind, its speed is 450 miles / 3 hours = 150 mph. Against the wind, its speed is 450 miles / 5 hours = 90 mph. The plane's speed in still air is (150 mph + 90 mph) / 2 = 120 mph. The wind's speed is the difference between the plane's speed with the wind and its speed in still air, which is 150 mph - 120 mph = 30 mph.

Question 6

Which is not a solution of $10 > x - 7 y$ ?
A) $( 0,7 )$
B) $( 2,10 )$
C) $( 12,0 )$
D) $( - 3,10 )$

Accepted Answer

Plugging the values from each option into the inequality $10 > x - 7y$, option C $(12, 0)$ results in $10 > 12 - 7(0)$, which simplifies to $10 > 12$, a false statement. Hence, $(12, 0)$ is not a solution.

Question 7

Decide which inequality symbol (<, >, $\leq$ , $\geq$) should be used by looking at the graph. x - y ____2 A) $\leq$ B) $<$ C) $>$ D) $\geq$

Accepted Answer

The answer of Decide which inequality symbol (<, >, $\leq$...

Question 8

The Walnut Cove Ruritan Club sells hot dogs and drinks from a concession stand at the annual 4th of July parade. Carl bought 6 hot dogs and 4 drinks and paid $17.00. Susan purchased 3 hot dogs And 4 drinks and was charged $12.50. What is the cost of 1 hot dog and what is the cost of 1 drink?
A) $1.50 for a hot dog; $2.00 for a drink
B) $2.00 for a hot dog; $1.50 for a drink
C) $1.60 for a hot dog; $1.85 for a drink
D) $1.40 for a hot dog; $2.15 for a drink

Accepted Answer

Let x be the cost of a hot dog and y be the cost of a drink.From the given information, we have:6x + 4y = 173x + 4y = 12.5Solving these equations, we get:x = 1.50y = 2.00Therefore, the cost of a hot dog is $1.50 and the cost of a drink is $2.00.

Question 9

Two trains leave the depot at the same time, one traveling east and the other traveling west. The speed of one train is 14 mph slower than the other. If after 3 hours the distance between the trains is 150 miles, find the speed of each train.
A) 18 mph; 32 mph
B) 17 mph; 33 mph
C) 20 mph; 30 mph
D) 19 mph; 31 mph

Accepted Answer

The speed of the slower train is x mph, so the speed of the faster train is x + 14 mph. After 3 hours, the slower train has traveled 3x miles and the faster train has traveled 3(x + 14) miles. The distance between the trains is 3(x + 14) - 3x = 150 miles. Solving for x gives x = 18 mph, so the speed of the slower train is 18 mph and the speed of the faster train is 18 + 14 = 32 mph.

Question 10

Identify the inequalities that match the given graph.  
A) $x \geq 0$ and $y \geq 0$
B) $x \leq 0$ and $y \leq 0$
C) $x \leq 0$ and $y \geq 0$
D) $x \geq 0$ and $y \leq 0$

Accepted Answer

The answer of Identify the inequalities that match the given...

Question 11

It takes a barge 1 hour to travel 8 miles downstream with the current and 4 hours to return against the current. Find the speed of the barge in still water and the speed of the current.
A) barge: 8 mph; current: 2 mph
B) barge: 5 mph; current: 3 mph
C) barge: 6 mph; current: 4 mph
D) barge: 6 mph; current: 3 mph

Accepted Answer

The speed of the barge in still water is the average of its speed downstream and upstream. Downstream speed is 8 miles/hour, and upstream speed is 8 miles / 4 hours = 2 miles/hour. The average speed in still water is (8 + 2) / 2 = 5 miles/hour. The speed of the current is the difference between the barge's speed in still water and its speed upstream, or the difference between its speed in still water and its speed downstream, divided by 2. So, the current's speed is (8 - 2) / 2 = 3 miles/hour.

Question 12

A rectangle has the perimeter of 24 inches. The length is 4 in longer than the width. Find the dimensions of the rectangle.
A) 2 inches by 6 inches
B) 7 inches by 11 inches
C) 5 inches by 9 inches
D) 4 inches by 8 inches

Accepted Answer

The perimeter of a rectangle is given by 2(length + width). If the length is 4 inches longer than the width, let the width be x. Then the length is x + 4. The equation becomes 2(x + x + 4) = 24. Solving for x gives x = 4, so the width is 4 inches and the length is 8 inches.

Question 13

An isosceles triangle has two angles of the same measure (see figure). If the angle represented by y measures 60° more than the angle x, find the measures of all angles of the triangle. (Recall that the sum of the measures of the angles of a triangle is 180°.)  [Figure is not necessarily drawn to scale.]
A) 37°; 37°; 97°
B) 35°; 35°; 110°
C) 45°; 45°; 90°
D) 40°; 40°; 100°

Accepted Answer

The answer of An isosceles triangle has two angles of...

Question 14

In a right triangle, one acute angle measures 30° more than 4 times the other. If the sum of the measures of the two acute angles must equal 90°, find the measures of the acute angles.
A) 12°, 78°
B) 14°, 76°
C) 11°, 79°
D) 13°, 82°

Accepted Answer

Let the measure of the smaller angle be x degrees. Then, the measure of the larger angle is 4x + 30 degrees. Since the sum of the angles in a right triangle is 90 degrees (excluding the right angle), we have x + 4x + 30 = 90. Solving this equation gives x = 12 degrees for the smaller angle and 4(12) + 30 = 78 degrees for the larger angle.

Question 15

A total of $4.80 consisting of dimes and quarters was collected from a soft drink vending machine. If there are 1 more quarters than dimes, find the number of dimes and the number of quarters.
A) 12 dimes; 13 quarters
B) 14 dimes; 15 quarters
C) 18 dimes; 12 quarters
D) 13 dimes; 14 quarters

Accepted Answer

Let the number of dimes be x. Then the number of quarters is x + 1. The total value is 10x + 25(x + 1) = 480 cents. Solving 10x + 25x + 25 = 480 gives x = 13, so there are 13 dimes and 14 quarters.

Question 16

The Hillsville Little Theater is currently presenting a new stage production of the animated feature Hoodwinked. At the opening, 70 tickets were sold and ticket sales totaled $672. If adult tickets cost $12 and children's tickets cost $8, how many of each type of ticket were sold?
A) 28 adult tickets; 42 child tickets
B) 26 adult tickets; 44 child tickets
C) 27 adult tickets; 43 child tickets
D) 25 adult tickets; 45 child tickets

Accepted Answer

Let the number of adult tickets be $x$ and the number of child tickets be $y$. We have two equations based on the given information: $x + y = 70$ (total tickets sold) and $12x + 8y = 672$ (total sales). Solving these equations simultaneously, we find that $x = 28$ and $y = 42$, which corresponds to option A: 28 adult tickets and 42 child tickets.

Question 17

Solve the system.
$$\begin{array} { l } 
9 x + 9 y = 37 \
- 7 x + 11 y = 37
\end{array}$$
A) $\left( \frac { 37 } { 81 } , \frac { 296 } { 81 } \right)$
B) $\left( \frac { 37 } { 18 } , - \frac { 148 } { 9 } \right)$
C) $\left( \frac { 37 } { 18 } , - \frac { 2 } { 9 } \right)$
D) $\left( \frac { 37 } { 81 } , - \frac { 2 } { 9 } \right)$

Accepted Answer

To solve the system, we can use the method of elimination or substitution. Here, we'll use elimination for simplicity. First, we can multiply the first equation by 7 and the second equation by 9 to make the coefficients of $x$ in both equations equal but opposite in sign:$$63x + 63y = 259$$$$-63x + 99y = 333$$Adding these two equations eliminates $x$, giving us:$$162y = 592$$Solving for $y$, we get:$$y = \frac{592}{162} = \frac{296}{81}$$Substituting $y$ back into one of the original equations to solve for $x$, we use the first equation:$$9x + 9(\frac{296}{81}) = 37$$$$9x = 37 - 9(\frac{296}{81})$$$$9x = \frac{3333 - 2664}{81}$$$$9x = \frac{669}{81}$$$$x = \frac{669}{729} = \frac{37}{81}$$Therefore, the solution is $\left( \frac { 37 } { 81 } , \frac { 296 } { 81 } \right)$, which corresponds to choice A.

Question 18

Which is a solution of $4 x + 5 y \leq 20$ ?
A) $( 0,5 )$
B) $( 1,1 )$
C) $( 5,7 )$
D) $( 8,0 )$

Accepted Answer

Substituting $(1,1)$ into the inequality: $4(1) + 5(1) = 9 \leq 20$, which is true.

Question 19

One angle measures 66° more than 2 times another. If the two angles are complementary, find the measures of the angles.
A) 10°; 80°
B) 18°; 72°
C) 13°; 77°
D) 8°; 82°

Accepted Answer

The answer of One angle measures 66° more than 2...

Question 20

Nail polish remover is essentially a mixture of water and a chemical called acetone. How much pure acetone must be combined with a solution that is 10% acetone to make 30 oz of a 37% solution?
A) 15 oz
B) 9 oz
C) 21 oz
D) 14 oz

Accepted Answer

To create a 37% solution from a 10% solution by adding pure acetone, we can use the equation: (amount of pure acetone) + 0.10(amount of 10% solution) = 0.37(total amount of solution). Let x be the amount of pure acetone added. Since the total solution is 30 oz, the amount of 10% solution is (30 - x) oz. The equation becomes: x + 0.10(30 - x) = 0.37(30). Solving for x gives x = 9 oz.

Quiz 3: Systems of Linear Equations and Inequalities

To reach a sales meeting, Carrie traveled by plane for 1 hour and by car for an additional 2 hours. The plane travels 82 mph faster than the car. If the total distance traveled is 217 miles, what are the Speeds of the plane and the car? (Assume that the plane and the car travel at constant rates.)

Lorenzo invested $22,000 into two accounts; one pays 4% simple interest and the other pays 7% simple interest. At the end of the first year, Lorenzo's total return was $1150. How much did he Invest in each account?

Paula invested into two accounts; one pays 7% simple interest and the other pays 8% simple interest. She invested $2000 more in the account paying 8% than in the account paying 7%. At the End of the first year, Paula's total return was $1810. How much did she invest in each account?

A chemist wants to mix a 23% acid solution with a 43% acid solution to get 15 L of a 31% acid solution. How many liters of the 23% solution and how many liters of the 43% solution should be Mixed?

A plane flew 450 miles in 3 hours with the wind. It took 5 hours to travel the same distance against the wind. What is the rate of the plane in still air and the rate of the wind?

Which is not a solution of $10 > x - 7 y$ ?

Decide which inequality symbol (<, >, $\leq$ , $\geq$ ) should be used by looking at the graph. x - y 2 $Decide which inequality symbol (<, >, \leq , \geq ) should be used by looking at the graph. x - y 2 A) \leq B) < C) > D) \geq$

The Walnut Cove Ruritan Club sells hot dogs and drinks from a concession stand at the annual 4th of July parade. Carl bought 6 hot dogs and 4 drinks and paid $17.00. Susan purchased 3 hot dogs And 4 drinks and was charged $12.50. What is the cost of 1 hot dog and what is the cost of 1 drink?

Two trains leave the depot at the same time, one traveling east and the other traveling west. The speed of one train is 14 mph slower than the other. If after 3 hours the distance between the trains is 150 miles, find the speed of each train.

Identify the inequalities that match the given graph. $Identify the inequalities that match the given graph. A) x \geq 0 and y \geq 0 B) x \leq 0 and y \leq 0 C) x \leq 0 and y \geq 0 D) x \geq 0 and y \leq 0$

It takes a barge 1 hour to travel 8 miles downstream with the current and 4 hours to return against the current. Find the speed of the barge in still water and the speed of the current.

A rectangle has the perimeter of 24 inches. The length is 4 in longer than the width. Find the dimensions of the rectangle.

In a right triangle, one acute angle measures 30° more than 4 times the other. If the sum of the measures of the two acute angles must equal 90°, find the measures of the acute angles.

A total of $4.80 consisting of dimes and quarters was collected from a soft drink vending machine. If there are 1 more quarters than dimes, find the number of dimes and the number of quarters.

The Hillsville Little Theater is currently presenting a new stage production of the animated feature Hoodwinked. At the opening, 70 tickets were sold and ticket sales totaled $672. If adult tickets cost $12 and children's tickets cost $8, how many of each type of ticket were sold?

Solve the system. $\begin{array} { l } 9 x + 9 y = 37 \\- 7 x + 11 y = 37\end{array}$

Which is a solution of $4 x + 5 y \leq 20$ ?

One angle measures 66° more than 2 times another. If the two angles are complementary, find the measures of the angles.

Nail polish remover is essentially a mixture of water and a chemical called acetone. How much pure acetone must be combined with a solution that is 10% acetone to make 30 oz of a 37% solution?