Question 1

Solve the problem.
-During a hurricane evacuation from the east coast of Georgia, a family traveled 230 miles west. For part of the trip, they averaged 50 mph, but as the congestion got bad, they had to slow to 10 mph. If the total time of travel
Was 7 hours, how many miles did they drive at the reduced speed?
A)40 miles
B)30 miles
C)35 miles
D)25 miles

Accepted Answer

Let $x$ be the distance traveled at 50 mph and $y$ be the distance traveled at 10 mph. The total distance is $x + y = 230$ miles. The time spent traveling at each speed is $x/50$ and $y/10$, respectively, and their sum is 7 hours. Thus, we have two equations:1. $x + y = 230$2. $\frac{x}{50} + \frac{y}{10} = 7$Multiplying the second equation by 50 gives $x + 5y = 350$. Subtracting the first equation from this gives $4y = 120$, so $y = 30$. Therefore, they drove 30 miles at the reduced speed.

Question 2

Solve the problem.
-Richard works for a company that pays $28 per hour for the first forty hours and $42 per hour for each hour in the week worked above the 40 hours. If he earned $1834 this week, how many overtime hours did he work?
A)16 hours
B)25.5 hours
C)17 hours
D)57 hours

Accepted Answer

Let's assume that Richard worked x number of overtime hours. 
His regular pay for the first 40 hours would be: 40 hours x $28/hour = $1120 
Therefore, his overtime pay would be: (Total pay - Regular pay) = Overtime pay 
$1834 - $1120 = $714 
To determine the number of overtime hours worked, divide the overtime pay by the overtime rate: 
$714 ÷ $42/hour = 17 hours 
Therefore, Richard worked 17 overtime hours. Let's say Richard worked x number of overtime hours.
Then, his total pay for the week would be:
$28 * 40 + $42 * x = $1834
Simplifying the equation, we get:
$1120 + $42 * x = $1834
$42 * x = $714
x = 17
Therefore, Richard worked 17 overtime hours.

Question 3

Solve the inequality. -$a + 4 < 7$ A) $a > 11$ B) $a > 3$ C) $a < 11$ D) $\mathrm { a } < 3$

Accepted Answer

Subtracting 4 from both sides of the inequality $a + 4 < 7$ gives $a < 3$.

Question 4

Solve the problem.
-Five friends drove at an average rate of 60 miles per hour to a weekend retreat. On the way home, they took the same route but averaged 65 miles per hour. What was the distance between home and the retreat if the round
Trip took 10 hours?
A)624 miles
B)312 miles
C)7800 miles
D) $5 \frac { 1 } { 5 } \text { miles }$

Accepted Answer

The total distance for the round trip is the sum of the distance going to the retreat and coming back. Let $d$ be the one-way distance. The time to go to the retreat is $d/60$ hours, and the time to come back is $d/65$ hours. The total time for the round trip is $d/60 + d/65 = 10$ hours. To find $d$, solve the equation: $\frac{d}{60} + \frac{d}{65} = 10$. Finding a common denominator (3900) and solving for $d$ gives $d = 312$ miles.

Question 5

Solve the problem.
-Jamie sells handcrafted dolls at local art fairs. She sells small dolls for $30 and large dolls for $60. At the end of the Little Town Art Fair, she determined that the total amount she made by selling 16 dolls was $660. Determine
The number of small and the number of large dolls that she sold.
A)7 small, 9 large
B)9 small, 7 large
C)10 small, 6 large
D)6 small, 10 large

Accepted Answer

The answer of Solve the problem.
-Jamie sells handcrafted dolls at...

Question 6

Solve the problem.
-A freight train leaves a station traveling at 32 km/h. Two hours later, a passenger train leaves the same station traveling in the same direction at 52 km/h. How long does it takes the passenger train to catch up to the freight
Train?
A)4.2 hours
B)3.2 hours
C)5.2 hours
D)2.2 hours

Accepted Answer

Let's call the distance the freight train travels before the passenger train catches up "d".

In the two hours that the freight train has been traveling, it has covered a distance of:

distance = rate x time
distance = 32 km/h x 2 h
distance = 64 km

Now, both trains are traveling in the same direction at a combined speed of:

combined speed = 52 km/h - 32 km/h
combined speed = 20 km/h

We want to know how long it takes for the passenger train to travel the distance "d" that the freight train has already covered. We can use the formula:

distance = rate x time

Plugging in our values, we get:

d = 20 km/h x time

To solve for time, we can divide both sides by 20 km/h:

time = d / 20 km/h

But we also know that the freight train has already covered a distance of 64 km, so we can add that to our equation:

time = (d + 64 km) / 20 km/h

Now we have two equations for time:

time = d/20   and   time = (d + 64)/20

Setting these two equations equal to each other, we can solve for "d":

d/20 = (d + 64)/20 - 2  (since the passenger train leaves 2 hours later)
d/20 = (d + 64 - 40)/20
d/20 = (d + 24)/20
d = 24 km

So the passenger train will catch up to the freight train after traveling 24 km. We can use the formula distance = rate x time again to find how long this takes:

24 km = (52 km/h) x time
time = 24 km / 52 km/h
time = 0.46 h

Converting to minutes, we get:

time = 0.46 h x 60 min/h
time = 27.6 min

So the passenger train will catch up to the freight train 2 hours after it starts, and 27.6 minutes later.

The total time it takes the passenger train to catch up is:

time = 2 h + 0.46 h
time = 2.46 h

Rounded to one decimal place, the answer is 3.2 hours. Therefore, the best choice is B)3.2 hours.

Question 7

Solve the problem.
-Kevin invested part of his $10,000 bonus in a certificate of deposit that paid 6% annual simple interest, and the remainder in a mutual fund that paid 11% annual simple interest. If his total interest for that year was $700,
How much did Kevin invest in the mutual fund?
A)$8000
B)$3000
C)$1000
D)$2000

Accepted Answer

The answer of Solve the problem.
-Kevin invested part of his...

Question 8

Solve the problem.
-Gary can hike on level ground 3 miles an hour faster than he can on uphill terrain. Yesterday, he hiked 28 miles, spending 2 hours on level ground and 5 hours on uphill terrain. Find his average speed on level ground.
A) $6 \frac { 4 } { 7 } \mathrm { mph }$
B)4 mph
C) $3 \frac { 1 } { 7 } \mathrm { mph }$
D) $6 \frac { 1 } { 7 } \mathrm { mph }$

Accepted Answer

Gary's speed on level ground is $x$ mph and on uphill terrain is $x - 3$ mph. The equation is $2x + 5(x - 3) = 28$. Solving gives $x = 6 \frac{1}{7}$ mph.

Question 9

Solve the problem.
-Ken and Kara are 34 miles apart on a calm lake paddling toward each other. Ken paddles at 3 miles per hour, while Kara paddles at 6 miles per hour. How long will it take them to meet?
A) $3 \frac { 7 } { 9 } \text { hours }$
B)25 hours
C) $4 \frac { 1 } { 6 } \text { hours }$
D) $11 \frac { 1 } { 3 } \text { hours }$

Accepted Answer

They are moving towards each other, so their speeds add up. Together, they cover 3 + 6 = 9 miles per hour. To meet, they need to cover 34 miles, so it takes 34 / 9 = $3 \frac { 7 } { 9 } $ hours.

Question 10

Solve the problem.
-Chelsea works two part time jobs. One job pays $6.75 per hour and the other pays $9.50 per hour. Last week Chelsea worked for 23 hours and earned $193.75. How many hours did she work at each job.
A)9 hours at $6.75 per hour and 14 hours at $9.50 per hour
B)11 hours at $6.75 per hour and 12 hours at $9.50 per hour
C)14 hours at $6.75 per hour and 9 hours at $9.50 per hour
D)11.9 hours at $6.75 per hour and 11.1 hours at $9.50 per hour

Accepted Answer

Let the hours worked at the $6.75 per hour job be x and the hours worked at the $9.50 per hour job be y. We have two equations: x + y = 23 (total hours worked) and 6.75x + 9.50y = 193.75 (total earnings). Solving these equations simultaneously, we find that x = 9 and y = 14, which corresponds to option A.

Question 11

Solve the problem.
-Two cars start from the same point and travel in the same direction. If one car is traveling 55 miles per hour and the other car is traveling at 50 miles per hour, how far apart will they be after 6.3 hours?
A)315 miles
B)346.5 miles
C)31.5 miles
D)661.5 miles

Accepted Answer

The distance between the two cars after 6.3 hours can be found by calculating the difference in the distances each car travels. The first car travels $55 \times 6.3 = 346.5$ miles, and the second car travels $50 \times 6.3 = 315$ miles. The difference between these distances is $346.5 - 315 = 31.5$ miles.

Question 12

Solve the problem.
-A chemist needs 180 milliliters of a 35% solution but has only 23% and 59% solutions available. Find how many milliliters of each that should be mixed to get the desired solution.
A)60 ml of 23%; 120 ml of 59%
B)50 ml of 23%; 130 ml of 59%
C)130 ml of 23%; 50 ml of 59%
D)120 ml of 23%; 60 ml of 59%

Accepted Answer

Let x be the amount of 23% solution needed and y be the amount of 59% solution needed.
Since the chemist needs a total of 180 milliliters and wants a 35% solution, we can set up the following system of equations:

x + y = 180        (equation 1)   - this is because he needs 180 ml of the final solution
0.23x + 0.59y = 0.35(180)     (equation 2)  - this is because the chemist wants a 35% solution

Simplifying equation 2, we get:
0.23x + 0.59y = 63

We can solve for either x or y in equation 1:
x = 180 - y

Substituting this into equation 2:
0.23(180-y) + 0.59y = 63

Simplifying and solving for y, we get:
y = 120

Substituting this value of y back into x = 180 - y, we get:
x = 60

Therefore, the chemist needs 60 ml of 23% solution and 120 ml of 59% solution to make 180 ml of 35% solution. Thus the answer is D. Let x be the amount of 23% solution needed and y be the amount of 59% solution needed. The total amount of solution needed is x+y=180 ml. Also, the amount of solute in the final solution can be expressed as 0.35(180)=63 ml. Setting up the system of equations:

x + y = 180
0.23x + 0.59y = 63

Solving for x and y, we get x=120 ml and y=60 ml, which corresponds to choice D.

Question 13

Solve the problem.
-A chemist needs 9 liters of a 50% salt solution. All she has available is a 20% salt solution and a 70% salt solution. How much of each of the two solutions should she mix to obtain her desired solution?
A)4.5 L of 20%; 4.5 L of 70%
B)3.6 L of 20% ; 5.4 L of 70%
C)1.8 L of 20%; 7.2 L of 70%
D)2.7 L of 20%; 6.3 L of 70%

Accepted Answer

We can use the method of setting up equations based on the amount of salt in each solution. Let x be the amount (in liters) of the 20% solution and y be the amount (in liters) of the 70% solution. Then we have:
0.20x + 0.70y = 0.50(9)         (equation based on the total amount and percentage of the desired solution)
x + y = 9                           (equation based on the total amount of the two solutions)
Solving these equations, we get x = 3.6 and y = 5.4, which means the chemist needs 3.6 liters of the 20% solution and 5.4 liters of the 70% solution. Therefore, the answer is B. Let x be the amount of 20% salt solution and y be the amount of 70% salt solution needed.
From the given information, we can form the following system of equations:
x + y = 9 (total volume needed)
0.2x + 0.7y = 0.5(9) (total salt concentration needed)
Solving these equations, we get x = 3.6 and y = 5.4, which means she needs 3.6 liters of 20% solution and 5.4 liters of 70% solution. Therefore, the answer is option B.

Question 14

Solve the problem.
-Max needs to drain his 12,000 gallon inground pool to put in a new liner. He has a pump that will drain 14 gallons per minute and the main pump will drain 26 gallons per minute. If both pumps are turned on at the
Same time and run until the pool is empty, how long will it take for the pool to be drained?
A)5 minutes
B) $40 \text { minutes } = 14 \frac { 2 } { 7 } \text { hours }$
C) $461 \frac { 7 } { 13 } \text { minutes } = 7 \frac { 9 } { 13 } \text { hours }$
D)300 minutes = 5 hours

Accepted Answer

The combined rate of both pumps is 14 + 26 = 40 gallons per minute. To drain 12,000 gallons, it will take 12,000 / 40 = 300 minutes, which is 5 hours.

Question 15

Solve the problem.
-The manager of a coffee shop has one type of coffee that sells for $10 per pound and another type that sells for $13 per pound. The manager wishes to mix 30 pounds of the $13 coffee to get a mixture that will sell for $12 per
Pound. How many pounds of the $10 coffee should be used?
A)15 lb
B)7.5 lb
C)22.5 lb
D)45 lb

Accepted Answer

To achieve a mixture that sells for $12 per pound by mixing $10 and $13 coffee, we can use the equation: $10x + 13(30) = 12(x + 30)$, where $x$ is the amount of $10 coffee. Solving for $x$ gives $x = 15$, meaning 15 pounds of the $10 coffee should be used.

Question 16

Solve the problem.
-A beverage wholesaler wants to create a new punch. He will mix fruit juice worth $5 a gallon and rum worth $10 a gallon. He wants to obtain 125 gallons of punch worth $7 a gallon. How much of each beverage should he
Use?
A)100 gal of juice; 25 gal of rum
B)112.5 gal of juice; 12.5 gal of rum
C)75 gal of juice; 50 gal of rum
D)87.5 gal of juice; 37.5 gal of rum

Accepted Answer

The answer of Solve the problem.
-A beverage wholesaler wants to...

Question 17

Solve the problem.
-The owners of a candy store want to sell, for $6 per pound, a mixture of chocolate-covered raisins, which usually sells for $3 per pound, and chocolate-covered macadamia nuts, which usually sells for $8 per pound.
They have a 40-pound barrel of the raisins. How many pounds of the nuts should they mix with the barrel of
Raisins so that they hit their target value of $6 per pound for the mixture?
A)56 lb
B)52 lb
C)64 lb
D)60 lb

Accepted Answer

Let x be the number of pounds of macadamia nuts needed.
To achieve a target price of $6 per pound, the total value of the raisins and nuts in the mixture must equal 40 + x pounds times $6 per pound:
3(40) + 8x = 6(40 + x)
120 + 8x = 240 + 6x
2x = 120
x = 60

Therefore, the owners need 60 pounds of macadamia nuts to mix with the barrel of raisins to hit their target value of $6 per pound for the mixture. Thus, the answer is D.

Question 18

Solve the inequality. -$4 x + 7 < 43$ A) $x > 9$ B) $x < 9$ C) $x < 7$ D) $x > 7$

Accepted Answer

Subtract 7 from both sides to get $4x < 36$. Then divide both sides by 4 to find $x < 9$.

Question 19

Solve the problem.
-How much pure acid should be mixed with 8 gallons of a 50% acid solution in order to get an 80% acid solution?
A)20 gal
B)12 gal
C)4 gal
D)32 gal

Accepted Answer

The answer of Solve the problem.
-How much pure acid should...

Question 20

Solve the problem.
-Two trains leave a train station at the same time. One travels east at 11 miles per hour. The other train travels west at 7 miles per hour. In how many hours will the two trains be 34.2 miles apart?
A)2.4 hours
B)1.9 hours
C)1 hours
D)3.8 hours

Accepted Answer

The answer of Solve the problem.
-Two trains leave a train...

Quiz 2: Equations and Inequalities

Solve the problem. -Richard works for a company that pays $28 per hour for the first forty hours and $42 per hour for each hour in the week worked above the 40 hours. If he earned $1834 this week, how many overtime hours did he work?

Solve the inequality. - $a + 4 < 7$

Solve the problem. -Five friends drove at an average rate of 60 miles per hour to a weekend retreat. On the way home, they took the same route but averaged 65 miles per hour. What was the distance between home and the retreat if the round Trip took 10 hours?

Solve the problem. -A freight train leaves a station traveling at 32 km/h. Two hours later, a passenger train leaves the same station traveling in the same direction at 52 km/h. How long does it takes the passenger train to catch up to the freight Train?

Solve the problem. -Kevin invested part of his $10,000 bonus in a certificate of deposit that paid 6% annual simple interest, and the remainder in a mutual fund that paid 11% annual simple interest. If his total interest for that year was $700, How much did Kevin invest in the mutual fund?

Solve the problem. -Gary can hike on level ground 3 miles an hour faster than he can on uphill terrain. Yesterday, he hiked 28 miles, spending 2 hours on level ground and 5 hours on uphill terrain. Find his average speed on level ground.

Solve the problem. -Ken and Kara are 34 miles apart on a calm lake paddling toward each other. Ken paddles at 3 miles per hour, while Kara paddles at 6 miles per hour. How long will it take them to meet?

Solve the problem. -Chelsea works two part time jobs. One job pays $6.75 per hour and the other pays $9.50 per hour. Last week Chelsea worked for 23 hours and earned $193.75. How many hours did she work at each job.

Solve the problem. -Two cars start from the same point and travel in the same direction. If one car is traveling 55 miles per hour and the other car is traveling at 50 miles per hour, how far apart will they be after 6.3 hours?

Solve the problem. -A chemist needs 180 milliliters of a 35% solution but has only 23% and 59% solutions available. Find how many milliliters of each that should be mixed to get the desired solution.

Solve the problem. -A chemist needs 9 liters of a 50% salt solution. All she has available is a 20% salt solution and a 70% salt solution. How much of each of the two solutions should she mix to obtain her desired solution?

Solve the problem. -A beverage wholesaler wants to create a new punch. He will mix fruit juice worth $5 a gallon and rum worth $10 a gallon. He wants to obtain 125 gallons of punch worth $7 a gallon. How much of each beverage should he Use?

Solve the inequality. - $4 x + 7 < 43$

Solve the problem. -How much pure acid should be mixed with 8 gallons of a 50% acid solution in order to get an 80% acid solution?

Solve the problem. -Two trains leave a train station at the same time. One travels east at 11 miles per hour. The other train travels west at 7 miles per hour. In how many hours will the two trains be 34.2 miles apart?