Application problems including rational equations

Question

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?

Show work

Answer 1

Q3. Jog time + walk time = 0.9 hrs. Let x = walk speed; then jog speed is 2x.

Jog time = dist/speed = 5/2x; Walk time = 2/x

5/2x + 2/x = 0.9 -> 5/2x + 4/2x = 9/2x = 0.9

9 = 1.8x -> x = 9/1.8 = 5 mph walk speed

Then jog avg speed is 10 mph.

Check: Jog 5mi/10 mph = 0.5 hrs

Walk 2mi/5mph = 0.4 hrs

Total 0.9 hrs.

Answer 2

Q.3

Explanation:

Let x = his jogging speed



2x = his walking speed



write a time equation, time = dist/speed



jog time + walk time = 0.9 hrs



5/x+2/2x=9/10



50+10=9x



x = 6.67



John’s jogging speed is therefore 6.67 miles/hr.

Q.4

Explanation:

Let the speed of motorcycle be x



Speed of the bus = x+2

then we have

165/(x+2)=155/x



165x=155x+310

x=31



Hence: the speed of the motorcycle is 31 Miles/Hour

Q.5

Explanation:

Let the time taken by associate alone be x.

Therefore;

1/7+1/x=1/3

Multiply all through by 21x

3x+21=7x



21=4x

X=21/4hrs