John walked 10 miles on Saturday. He walked twice as fast on the second 5 miles of his walk than he walked on the first 5 miles of his walk. Which expression represents the time he spent walking? Let x = John's speed on the first half of his walk.
A. fifteen over x
B. five over three x
C. fifteen over two x
D. ten over three x
Let's say John's speed on the first 5 miles was x. Then his speed on the second 5 miles was 2x, since he walked twice as fast.
To find the time he spent walking, we need to divide the distance by the speed.
For the first 5 miles: Time = Distance / Speed = 5 / x
For the second 5 miles: Time = Distance / Speed = 5 / 2x
So the total time John spent walking is:
Total Time = Time for First 5 Miles + Time for Second 5 Miles
Total Time = 5/x + 5/2x
We can simplify this by finding a common denominator:
Total Time = (10 + 5) / 2x
Total Time = 15 / 2x
So the answer is C. fifteen over two x.
Edie can paint a wall in 3 hours. Dan can paint the same wall in 6 hours. If they work together, how many hours will it take Edie and Dan to paint the wall?
A. 1 hour
B. 1.5 hours
C. 2 hours
D. 4.5 hours
Let's let x be the number of hours it takes Edie and Dan to paint the wall together.
In 1 hour, Edie can complete 1/3 of the wall (since she can paint the whole wall in 3 hours). Similarly, in 1 hour, Dan can complete 1/6 of the wall.
Working together, in 1 hour they can complete:
1/3 (the fraction of the wall Edie can do) + 1/6 (the fraction of the wall Dan can do)
= 1/2 of the wall.
So we can set up the equation:
1/2 = 1/x
To solve for x, we can cross-multiply:
2x = 1
x = 1/2
So it will take Edie and Dan 1/2 hour, or 30 minutes, to paint the wall together.
The answer is not one of the choices given, so either the answer choices are incorrect or there was a mistake in the problem.
To find the expression that represents the time John spent walking, we need to determine his speed on the second 5 miles of his walk.
Let's assume John's speed on the first half of his walk (5 miles) is x.
Since John walks twice as fast on the second half of his walk, his speed on the second half (5 miles) would be 2x.
To calculate the time spent walking, we can use the formula Speed = Distance / Time. Therefore, Time = Distance / Speed.
For the first half of the walk (5 miles), the time spent is: Time₁ = 5 / x.
For the second half of the walk (5 miles), the time spent is: Time₂ = 5 / (2x).
The total time spent walking is the sum of the time spent on the first and second halves of the walk: Total Time = Time₁ + Time₂.
Substituting the values of Time₁ and Time₂, we have: Total Time = 5 / x + 5 / (2x).
To simplify this expression, we need to find a common denominator for the fractions. Let's multiply the first fraction by 2/2 to get the same denominator:
Total Time = (5 * 2) / (2x) + 5 / (2x) = 10 / (2x) + 5 / (2x) = 15 / (2x).
Therefore, the expression that represents the time John spent walking is "fifteen over two x," which is option C.