Question 1

Solve the problem and answer the question.
-A barge takes 6 hours to move (at a constant rate) downstream for 60 miles, helped by a current of 2 miles per hour. If the barge's engines are set at the same pace, find the time of its return trip against the current.

Accepted Answer

The barge's speed downstream is: 
speed = distance/time = 60/6 = 10 mph

The current's speed is 2 mph, so the barge's speed upstream is: 
speed = 10 - 2 = 8 mph

The distance to travel upstream is the same 60 miles, so the time it takes the barge to travel upstream is: 
time = distance/speed = 60/8 = 7.5

Since the answer choices are given in hours, the closest choice is 10 hours (B). Let the speed of the barge in still water be x miles per hour. Then, the speed downstream (with the current) is (x+2) miles per hour, and the speed upstream (against the current) is (x-2) miles per hour.
We can use the formula: Distance = Speed x Time
For the downstream trip: 60 = (x+2) x 6
Solving for x, we get x = 8
For the upstream trip: 60 = (x-2) x t (where t is the time taken)
Substituting x = 8, we get 60 = 6(t-2)
Solving for t, we get t = 10
Therefore, the time of the return trip against the current is 10 hours, which is given by choice B.

Question 2

Find the quantity indicated.
-s varies directly as r. Find k when r = 56 and s = 4.

Accepted Answer

The formula for direct variation is:

s = kr

To solve for k, we can plug in the given values:

4 = k(56)

Solving for k:

k = 4/56

k = 1/14

Therefore, the answer is:

Answer: C) 1/14

Question 3

Find the quantity indicated.
-g varies directly as f. Find f when g = 81 and k = 9.

Accepted Answer

The answer of Find the quantity indicated.
-g varies directly as...

Question 4

Solve the problem.
-The amount of water used to take a shower is directly proportional to the amount of time that the shower is in use. A shower lasting 17 minutes requires 13.6 gallons of water. Find the amount of water used in a shower
Lasting 5 minutes.

Accepted Answer

The amount of water used is directly proportional to the amount of time that the shower is in use. So, the ratio of water used to time is constant.Let's call this constant k. Then, we have:water used = k * timeWe can find the value of k by using the given information:13.6 gallons = k * 17 minutesk = 13.6 gallons / 17 minutesk = 0.8 gallons / minuteNow, we can use this value of k to find the amount of water used in a shower lasting 5 minutes:water used = k * timewater used = 0.8 gallons / minute * 5 minuteswater used = 4 gallonsTherefore, the amount of water used in a shower lasting 5 minutes is 4 gallons.

Question 5

Find the quantity indicated.
-y varies directly as  If y = 405 when z = 9, find y when z = 4.

Accepted Answer

The answer of Find the quantity indicated.
-y varies directly as...

Question 6

Write the variation.
-r varies directly as the square of t.

Accepted Answer

The variation r varies directly as the square of t means that r = kt^2, where k is a constant of proportionality. This is a direct variation because as t increases, r increases at the same rate. The correct variation is shown in choice B. The variation can be written as r = kt^2, where k is the constant of variation. As t increases, r will increase at a rate proportional to the square of t. Option B has a different value for the constant of variation k, which makes it a valid variation.

Question 7

Solve the problem and answer the question.
-A twin-engined aircraft can fly 1600 miles from city A to city B in 5 hours with the wind and make the return trip in 8 hours against the wind. What is the speed of the wind?

Accepted Answer

The speed of the plane in still air is the average of the speeds with and against the wind: (1600/5 + 1600/8)/2 = 200 mph. The speed with the wind is 320 mph (1600/5), and against the wind is 200 mph (1600/8). The wind speed is the difference between the plane's speed with the wind and in still air: 320 - 200 = 60 mph.

Question 8

Find the quantity indicated.
-y varies directly as the square of x. Find y when k = 5 and x = 3.

Accepted Answer

The answer of Find the quantity indicated.
-y varies directly as...

Question 9

Solve the problem and answer the question.
-Glenn and Paula want to dig an irrigation channel for their garden. Together it takes them 1.6 hours to dig an appropriately sized irrigation channel. If Paula can dig the channel by herself in 2.7 hours, how long would it
Take Glenn to dig the channel by himself? (Round your answer to the nearest tenth of an hour, if necessary.)

Accepted Answer

The answer of Solve the problem and answer the question.
-Glenn...

Question 10

Solve the problem.
-The distance that an object falls when it is dropped is directly proportional to the square of the amount of time since it was dropped. An object falls 39.2 meters in 2 seconds. Find the distance the object falls in 5 seconds.

Accepted Answer

Let D be the distance the object falls in 5 seconds. Then,

D/39.2 = (5^2)/(2^2)

D/39.2 = 25/4

D = (39.2*25)/4

D = 245 meters.

Therefore, the correct answer is (C). We can set up a proportion using the information given:

(distance)/(time^2) = k  (where k is the constant of proportionality)

Using the first data point, we can substitute in and solve for k:

39.2/2^2 = k
k = 9.8

Now we can use this value of k to find the distance for the second data point:

(distance)/(5^2) = 9.8

distance = 245 meters.

Therefore, the answer is choice C.

Question 11

Solve the problem and answer the question.
-One conveyor belt can move 1000 boxes in 11 minutes. Another can move 1000 boxes in 8 minutes. If another conveyor belt is added and all three are used, the boxes are moved in 3 minutes. How long would it take the
Third conveyor belt alone to do the same job?

Accepted Answer

Let x be the time taken by the third conveyor belt alone to move 1000 boxes.
Then, the rate of the first conveyor belt is 1000/11 boxes per minute.
The rate of the second conveyor belt is 1000/8 boxes per minute.
The rate of the three conveyor belts combined is 1000/3 boxes per minute.
Using the formula rate x time = work, the total work done by the three conveyor belts is 3000 boxes.
Thus, we can write the equations:
(1000/11) * 3 + (1000/8) * 3 + (1000/x) * 3 = 3000
Simplifying this equation, we get:
x = 22/3 minutes or 7.33 minutes
Therefore, the third conveyor belt alone would take 7.33 minutes to move 1000 boxes. 
Choice B is correct because 7.33 is closest to answer choice 7.3. Let x be the rate of the third conveyor belt (in boxes/minute).
Working together, the three conveyor belts can move 1000 boxes in 3 minutes:

3(1000/11 + 1000/8 + 1000/x) = 1000

Multiplying both sides by (11)(8)(x) gives:

24x(240 + 330x + 88x) = 11(8)x(1000)

Simplifying, we get:

792x^2 + 825x - 2750 = 0

Factoring, we get:

(22x - 25)(36x + 110) = 0

Since x represents a rate, it must be positive. Therefore:

x = 25/22

So the third conveyor belt can move 1000 boxes in:

1000/(25/22) = 880 seconds = 14 minutes and 40 seconds.

Therefore, the correct answer is B.

Question 12

Solve the problem.
-If the resistance in an electrical circuit is held constant, the amount of current flowing through the circuit is directly proportional to the amount of voltage applied to the circuit. When 11 volts are applied to a circuit,
220 milliamperes (mA) of current flow through the circuit. Find the new current if the voltage is increased to
12 volts.

Accepted Answer

The answer of Solve the problem.
-If the resistance in an...

Question 13

Solve the problem and answer the question.
-Julie and Eric row their boat (at a constant speed) 40 miles downstream for 4 hours, helped by the current. Rowing at the same rate, the trip back against the current takes 10 hours. Find the rate of the current.

Accepted Answer

Let the rate at which they row in still water be x and the rate of the current be y.
When rowing downstream, their effective speed is x+y, so distance traveled is (x+y) * 4 = 40 miles. This gives us the equation 4x + 4y = 40.
When rowing upstream, their effective speed is x-y, so distance traveled is (x-y) * 10 = 40 miles. This gives us the equation 10x - 10y = 40.
Solving these two equations simultaneously gives us x=7 mph (rate at which they row in still water) and y=3 mph (rate of the current). Therefore, the answer is choice C, 3 mph. Let the speed of the boat be B and the speed of the current be C. Then, the speed downstream is B + C and the speed upstream is B - C.

Using the formula Distance = Rate x Time, we get:

Distance downstream = (B + C) x 4 = 40
Distance upstream = (B - C) x 10 = 40

Simplifying these equations, we get:

B + C = 10
B - C = 4

Adding these equations gives us:

2B = 14
B = 7

Substituting B back into one of the equations gives us:

7 + C = 10
C = 3

Therefore, the rate of the current is 3 mph.

Question 14

Solve the problem.
-The amount of gas that a helicopter uses is directly proportional to the number of hours spent flying. The helicopter flies for 2 hours and uses 22 gallons of fuel. Find the number of gallons of fuel that the helicopter uses
To fly for 6 hours.

Accepted Answer

Let x be the number of gallons of fuel the helicopter uses to fly for 6 hours.
From the problem, we know that the amount of gas used is directly proportional to the number of hours spent flying. So we can set up the proportion:
2/22 = 6/x
Solving for x gives:
x = 66
Therefore, the helicopter would use 66 gallons of fuel to fly for 6 hours, which corresponds to choice B. We can set up a proportion to solve for the number of gallons of fuel used for 6 hours of flying time. Let x be the number of gallons of fuel used for 6 hours of flying time. Then we can write:

2/22 = 6/x

Cross-multiplying gives:

2x = 6*22 = 132

Dividing both sides by 2 gives:

x = 66

Therefore, the helicopter uses 66 gallons of fuel to fly for 6 hours. The answer is choice B).

Question 15

Solve the problem and answer the question.
-BJ can overhaul a boat's diesel inboard engine in 15 hours. His apprentice takes 30 hours to do the same job. How long would it take them working together assuming no gain or loss in efficiency?

Accepted Answer

Let's assume that the job is to overhaul one engine completely. 
BJ's rate of work = 1/15 of the job per hour
Apprentice's rate of work = 1/30 of the job per hour

When they work together, their rates of work are additive:
Rate of work when working together = BJ's rate of work + Apprentice's rate of work
= 1/15 + 1/30 (LCM of 15 and 30 is 30)
= 2/30 + 1/30
= 3/30
= 1/10 of the job per hour

Therefore, they can overhaul the engine working together in 10 hours:
time = job/rate of work = 1 / (1/10) = 10 hours.

So, the answer is C) 10 hr. BJ can complete 1/15 of the job in 1 hour.
His apprentice can complete 1/30 of the job in 1 hour.
Together they can complete (1/15 + 1/30) of the job in 1 hour.
Simplifying, (1/15 + 1/30) = 1/10.
Therefore, they can complete the job in 10 hours working together.

Question 16

Write the variation.
-c varies directly as b.

Accepted Answer

This is the standard variation formula for direct variation: c = kb, where k is the constant of variation. Since c varies directly as b, we can use this formula, and k will depend on the nature of the problem at hand. The statement "c varies directly as b" means that as b increases, c also increases in a constant proportion. This relationship can be expressed as c = kb, where k is the constant of proportionality. Therefore, choice B is the correct variation equation as it represents this relationship between c and b. Choices A, C, and D are incorrect as they do not accurately represent the direct variation relationship.

Question 17

Solve the problem and answer the question.
-A painter can finish painting a house in 7 hours. Her assistant takes 9 hours to finish the same job. How long would it take for them to complete the job if they were working together?

Accepted Answer

Let's use the formula:
Time = Work / Rate
Let the amount of work be represented by 1 (the whole job). 
The rate of the painter is 1/7 (finishing 1 job in 7 hours). 
The rate of the assistant is 1/9 (finishing 1 job in 9 hours). 
When working together, their rates will add up so:
1 = (1/7 + 1/9) * Time
1 = (16/63) * Time
Time = 6.3 hours or approximately 6 hours and 18 minutes. 
The closest answer choice to that is C (6 hours). Let x be the time it takes for them to complete the job when working together.
Then, the amount of work done by the painter in one hour is 1/7 and the amount of work done by the assistant in one hour is 1/9.
Working together, they can finish the job in one hour, which means the amount of work done in one hour is 1/x.
So, we can set up the equation:
1/7 + 1/9 = 1/x
Solving for x, we get x = 63/16, which is approximately equal to 3.94 hours or 3 hours and 56 minutes.
Therefore, the best choice is C, which is closest to the calculated value of 3.94 hours.

Question 18

Solve the problem and answer the question.
-A baker can decorate the day's cookie supply four times as fast as his new assistant. If they decorate all the cookies working together in 28 minutes, how long would it take for each of them to decorate the cookies
Working individually?

Accepted Answer

Let's assume that the new assistant takes x minutes to decorate all the cookies alone.
Then the baker can decorate all the cookies in x/4 minutes.
Now, if they work together, they can decorate all the cookies in 28 minutes.
Using the formula: (time taken by A * time taken by B) / (time taken by A + time taken by B) = Total time
we can find x as:
(x * x/4) / (x/4 + x) = 28
Solving this equation, we get x = 112 minutes
So, the baker takes 28 minutes / 4 = 7 minutes to decorate all the cookies alone.
Therefore, the answer would be:
A) 11ebb63d_1b61_da39_b851_5913bd83e1aa_TB8475_11 - Incorrect, as it doesn't match with the calculations.
B) 11ebb63d_1b61_da3a_b851_11e4ed93110a_TB8475_11 - Correct, as the assistant takes 112 minutes and the baker takes 7 minutes to decorate all the cookies alone.
C) 11ebb63d_1b61_da3b_b851_d9239ca75b6e_TB8475_11 - Incorrect, as it doesn't match with the calculations.
D) 11ebb63d_1b62_014c_b851_5fe8afb5255c_TB8475_11 - Incorrect, as it doesn't match with the calculations. Let the time taken by the assistant be x minutes. Then the time taken by the baker will be x/4 minutes. Together they take 28 minutes to decorate all the cookies. So, their combined rate of work is 1/28. Using the formula "rate x time = work", we can write:
(x/4) + x = 1/28
Solving this equation gives x = 40 minutes (approx) for the assistant, and x/4 = 10 minutes (approx) for the baker. Therefore, the answer is B, as 10 minutes is the time taken by the baker to decorate all the cookies working individually. And the assistant takes 40 minutes to decorate all the cookies working individually.

Question 19

Find the quantity indicated.
-y varies directly as z. If y = 110 when z = 11, find y when z = 13.

Accepted Answer

If y varies directly as z, then y = kz, where k is the constant of variation. 
To find k, we can use the given information: 
y = 110 when z = 11 
110 = k(11) 
k = 10

Now we can use this value of k to find y when z = 13: 
y = kz 
y = 10(13) 
y = 130

Therefore, y = 130 when z = 13. The best choice is B. If y varies directly as z, then we can write y = kz for some constant k. 
To find k, we can use the given information that y = 110 when z = 11:
110 = k(11)
k = 10
Now we can use this value of k to find y when z = 13:
y = 10(13) = 130

Question 20

Solve the problem and answer the question.
-One pump can drain a pool in 10 minutes. When a second pump is also used, the pool only takes 5 minutes to drain. How long would it take the second pump to drain the pool if it were the only pump in use?

Accepted Answer

Let x be the time it takes the second pump to drain the pool alone. Then, the first pump can drain 1/10 of the pool in 1 minute, and the second pump can drain 1/x of the pool in 1 minute. Together, they can drain the pool in 5 minutes, so they can drain 1/5 of the pool in 1 minute. This gives us the equation:

1/10 + 1/x = 1/5

Multiplying both sides by 10x, we get:

x + 10 = 2x

Simplifying, we get x = 10. Therefore, the second pump can drain the pool alone in 10 minutes.

Quiz 7: Rational Expressions and Equations

Solve the problem and answer the question. -A barge takes 6 hours to move (at a constant rate) downstream for 60 miles, helped by a current of 2 miles per hour. If the barge's engines are set at the same pace, find the time of its return trip against the current.

Find the quantity indicated. -s varies directly as r. Find k when r = 56 and s = 4.

Find the quantity indicated. -g varies directly as f. Find f when g = 81 and k = 9.

Solve the problem. -The amount of water used to take a shower is directly proportional to the amount of time that the shower is in use. A shower lasting 17 minutes requires 13.6 gallons of water. Find the amount of water used in a shower Lasting 5 minutes.

Find the quantity indicated. -y varies directly as If y = 405 when z = 9, find y when z = 4.

Write the variation. -r varies directly as the square of t.

Solve the problem and answer the question. -A twin-engined aircraft can fly 1600 miles from city A to city B in 5 hours with the wind and make the return trip in 8 hours against the wind. What is the speed of the wind?

Find the quantity indicated. -y varies directly as the square of x. Find y when k = 5 and x = 3.

Solve the problem. -The distance that an object falls when it is dropped is directly proportional to the square of the amount of time since it was dropped. An object falls 39.2 meters in 2 seconds. Find the distance the object falls in 5 seconds.

Solve the problem and answer the question. -Julie and Eric row their boat (at a constant speed) 40 miles downstream for 4 hours, helped by the current. Rowing at the same rate, the trip back against the current takes 10 hours. Find the rate of the current.

Solve the problem. -The amount of gas that a helicopter uses is directly proportional to the number of hours spent flying. The helicopter flies for 2 hours and uses 22 gallons of fuel. Find the number of gallons of fuel that the helicopter uses To fly for 6 hours.

Solve the problem and answer the question. -BJ can overhaul a boat's diesel inboard engine in 15 hours. His apprentice takes 30 hours to do the same job. How long would it take them working together assuming no gain or loss in efficiency?

Write the variation. -c varies directly as b.

Solve the problem and answer the question. -A painter can finish painting a house in 7 hours. Her assistant takes 9 hours to finish the same job. How long would it take for them to complete the job if they were working together?

Solve the problem and answer the question. -A baker can decorate the day's cookie supply four times as fast as his new assistant. If they decorate all the cookies working together in 28 minutes, how long would it take for each of them to decorate the cookies Working individually?

Find the quantity indicated. -y varies directly as z. If y = 110 when z = 11, find y when z = 13.

Solve the problem and answer the question. -One pump can drain a pool in 10 minutes. When a second pump is also used, the pool only takes 5 minutes to drain. How long would it take the second pump to drain the pool if it were the only pump in use?