Application problems including rational equations

Set up a rational equation and then solve the following problems.

1.      A positive integer is twice another. The sum of the reciprocals of the two positive integers is. Find the two integers.

let one integer = x

other integer = 2x

sum of reciprocal = 1/x + 1/2x=3/14

=> 3x/3=21/3

=> x = 7

one no. = 7

other no. = 14.

2.      A positive integer is twice another. The difference between the reciprocals of the two positive integers is  Find the two integers.

Let one of the integers be x

The other is 2x.

 1/x-1/2x=1/18.

Multiply through by18x:

18-9=x so x=9.

 The two integers are 9 and 18.

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?

:

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.

4.      A bus​ averages 2 miles per hour faster than a motorcycle. If the bus travels 165 miles at the same time it takes the motorcycle​ to travel 155 miles, then what is the speed of each?

Let the speed of the motorcycle be x

Speed of the bus = x+2

165/(x+2)=155/x

165x=155x+310

X=31

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

Speed of the bus is x+2=31+2 = 33 Miles/Hour.

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?

Let the time taken by the associate alone be x.

Therefore;

1/7+1/x=1/3

Multiply all through by  21x

3x+21=7x

21=4x

X=21/4hrs