1. A positive integer is twice another. The sum of the reciprocals of the two positive integers is . Find the two integers.
2. A positive integer is twice another. The difference of the reciprocals of the two positive integers is Find the two integers.
3. John can jog twice as fast as he can walk. He was able to jog the first 5 miles to his grandmother’s house, but then he tired and walked the remaining 2 miles. If the total trip took 0.9 hours, then what was his average jogging speed?
4. A bus averages 2 miles per hour faster than a motorcycle. If the bus travels 165 miles in the same time it takes the motorcycle to travel 155 miles, then what is the speed of each?
5. Jane can paint the office by herself in 7 hours. Working with an associate, she can paint the office in 3 hours. How long would it take her associate to do it working alone?
Expert Solution Preview
1. Let’s first assign variables to represent the two positive integers. Let the first positive integer be x and the second positive integer be 2x (since one is twice the other).
According to the given information, the sum of the reciprocals of these two positive integers is equal to . We can express this mathematically as:
1/x + 1/(2x) =
To solve this equation, we need to find a common denominator and combine the fractions:
(2 + 1)/(2x) =
Simplifying the expression gives:
3/(2x) =
Now, we can cross-multiply to solve for x:
3 = 2x
Dividing both sides by 2 gives:
x =
Therefore, the two positive integers are x = and 2x = .
2. Similar to the previous question, let’s assign variables to represent the two positive integers. Let the first positive integer be x and the second positive integer be 2x (since one is twice the other).
According to the given information, the difference of the reciprocals of these two positive integers is equal to . We can express this mathematically as:
1/x – 1/(2x) =
To solve this equation, we need to find a common denominator and combine the fractions:
(2 – 1)/(2x) =
Simplifying the expression gives:
1/(2x) =
Now, we can cross-multiply to solve for x:
1 = 2x
Dividing both sides by 2 gives:
x =
Therefore, the two positive integers are x = and 2x = .
3. Let’s assume the walking speed of John is w miles/hour.
Since John can jog twice as fast as he can walk, his jogging speed is 2w miles/hour.
The distance traveled while jogging is 5 miles and the distance traveled while walking is 2 miles. The total time taken for the trip is 0.9 hours.
Using the formula Time = Distance/Speed, we can calculate the time taken for jogging and walking:
Time taken for jogging = Distance jogged / Jogging speed = 5 / 2w hours
Time taken for walking = Distance walked / Walking speed = 2 / w hours
The total time taken is the sum of these two times:
0.9 = (5 / 2w) + (2 / w)
To solve this equation, we find a common denominator and simplify:
0.9 = (10 + 4w) / (2w^2)
(10 + 4w) / (2w^2) = 0.9
Now we cross-multiply to solve for w:
10 + 4w = 0.9 * 2w^2
10 + 4w = 1.8w^2
Moving all terms to one side gives us a quadratic equation:
1.8w^2 – 4w – 10 = 0
To solve this quadratic equation, we can either factor it or use the quadratic formula. After solving for w, we can find the jogging speed (2w).
4. Let’s assume the speed of the motorcycle is x miles/hour.
Since the bus travels 2 miles per hour faster than the motorcycle, the speed of the bus is (x + 2) miles/hour.
The distance traveled by the motorcycle is 155 miles and the distance traveled by the bus is 165 miles. The time taken for both journeys is the same.
Using the formula Time = Distance/Speed, we can calculate the time taken for the motorcycle and bus:
Time taken for motorcycle = Distance traveled by motorcycle / Speed of motorcycle = 155 / x hours
Time taken for bus = Distance traveled by bus / Speed of bus = 165 / (x + 2) hours
Since the time taken for both journeys is equal, we have:
155 / x = 165 / (x + 2)
To solve this equation, we can cross-multiply and simplify:
155(x + 2) = 165x
155x + 310 = 165x
Moving all terms to one side gives us a linear equation:
10x = 310
Solving for x gives:
x = 31
Therefore, the speed of the motorcycle is 31 miles/hour, and the speed of the bus is 31 + 2 = 33 miles/hour.
5. Let’s assume Jane’s painting speed is p square feet per hour.
Working alone, Jane can paint the office in 7 hours, covering the entire area.
Working with an associate, they can paint the office in 3 hours, covering the entire area.
Using the formula Rate × Time = Work, we know that the work done is the same in both cases.
Jane’s work alone: p × 7 = Total area of the office
Combined work: (p + a) × 3 = Total area of the office
Since the total area of the office is the same in both cases, we can equate the two equations:
p × 7 = (p + a) × 3
Expanding the equation gives:
7p = 3p + 3a
Simplifying the equation gives:
4p = 3a
To find the time it would take her associate to paint the office alone, we need to solve for a:
a = (4/3)p
Therefore, it would take her associate (4/3) times the amount of time Jane takes alone to paint the office.
