the river flow is less than the peed of the boat in still water
Find a number greater than ! such that the sum of the number and its reciprocal is
The dilterence of the speed of a faster dur and a slower car is 200 Ader | the slower war
Answers
First, let us explain the meaning of "upstream" and "downstream."
When a boat travels in the same direction as the current, we say that it is traveling downstream.
downstream
Thus if b is the speed of the boat in still water, and c is the speed of the current, then its total speed is
Downstream speed = b + c
When a boat travels against the current, it travels upstream.
upstream
In this case, its total speed is
Upstream speed = b − c
Problem. The speed of a boat in still water is 30 mph. It takes the same time for the boat to travel 5 miles upstream as it does to travel 10 miles downstream. Find the speed of the current.
Solution. The key to this type of problem is same time. That will give the equation,
Time upstream = Time downstream
Now, speed, or velocity, is distance divided by time -- so many miles per hour:
v = d
t
Therefore,
t = d
v
The equation will be
Time upstream = Time downstream
Distance upstream
Speed upstream = Distance downstream
Speed downstream
Let x be the speed of the current. Then according to the problem:
_5_
30 − x = 10
30 + x
Therefore,
5(30 + x) = 10(30 − x)
150 + 5x = 300 − 10x
5x + 10x = 300 − 150
15x = 150
x = 10 mph
Problem 7. The speed of a boat in still water is 15 mi/hr. If the boat travels 8 miles downstream in the same time it takes to travel 4 miles upstream, what is the speed of the current?
To see the equation, pass your mouse over the colored area.
To cover the answer again, click "Refresh" ("Reload").
But do the problem yourself first!
Time upstream = Time downstream
Distance upstream
Speed upstream = Distance downstream
Speed downstream
Let x be the speed of the current. Then according to the problem:
_4_
15 − x = 8
15 + x
4(15 + x) = 8(15 − x)
60 + 4x = 120 − 8x
4x + 8x = 120 − 60
12x = 60
x = 5 mph
Problem 8. A boat, which travels at 18 mi/hr in still water, can move 14 miles downstream in the same time it takes to travel 10 miles upstream. Find the speed of the current.
Distance upstream
Speed upstream = Distance downstream
Speed downstream
_10_
18 − x = 14
18 + x
10(18 + x) = 14(18 − x)
180 + 10x = 252 − 14x
10x + 14x = 252 − 180
24x = 72
x = 3 mph
Problem 9. Train A has a speed 15 mi/hr greater than train B. If train A travels 150 miles in the same time train B travels 120 miles, what are the speeds of the two trains?
Solution. Let x be the speed of train A. Then the speed of train B is x − 15.
Train A's time = Train B's time
A's distance
A's speed = B's distance
B's speed
150
x = 120
x − 15
150(x − 15) = 120x
150x − 150· 15 = 120x
30x = 150· 15
x = 150· 15
30
= 150
2
x = 75 mph
x − 15 = 60 mph
Problem 10. A train travels 30 mi/hr faster than a car. If the train covers 120 miles in the same time the car covers 80 miles, what is the speed of each of them?
Solution. Let x be the speed of the train. Then the speed of the car is x − 30.
Train's time = Car's time
Train's distance
Train's speed = Car's distance
Car's speed
120
x = 80
x − 30
120(x − 30) = 80x
120x − 120· 30 = 80x
40x = 120· 30
x = 120· 30
40
= 3· 30
x = 90 mph
x − 30 = 60 mph
Example 5. Total time problem. Katrina drove her car to Boston at a speed of 100 kph (kilometers per hour). She drove back at 75 kph. The total driving time was 7 hours. How far away was Boston?
Solution. Let x be the distance to Boston. Then
Time going + Time returning = Total time.
Again, time is
time problem
time problem = 7.
time problem = 7. x is a common factor.
word problem = 7. Lesson 24 of Arithmetic
x = word problem Lesson 9, Example 6.
= 300 km.
Problem 11. You have exactly h hours at your disposal. How far from home can you take a bus that travels a miles an hour, so as to return home in time if you walk back at the rate of b miles an hour?
Solution. Let x = that distance. Then
Time going + Time returning = Total time
x
a + x
b = h
bx + ax = hab
(a + b)x = hab
x = _hab_
a + b
Example 6. Job problem. Raymond can do a job in 3 hours, while it takes Robert 2 hours. How long will it take them if they work together?
Solution. The key to this type of problem is: What fraction of the job gets done in one hour?
For example, if a job takes 3 hours, then in one hour, 1
3 will get done.
In general, if a job takes x hours, then in one hour, 1
x will get done.
So, let x answer the question. Let x be how long will it take them if
they work together. Then 1
x is that fraction of the job that gets done in
one hour.
We have,
1
x = 1
3 + 1
2
For, in one hour, Raymond does 1
3 of the job, and Robert, 1
2 .
Since x, or its reciprocal, is already isolated on the left, simply add the fractions on the right:
1
x = 2 + 3
6 = 5
6
Therefore, on taking reciprocals,
x = 6
5 = 1 1
5 hours.
Problem 12. Carlos can do a certain job in three days, while it takes Alec six days. If they work together, how long will it take them?
Let x be that time. Here is the equation:
1
x = 1
3 + 1
6
= 6 + 3
18
= 9
18
1
x = 1
2
Therefore,
x = 2 days