Question
2. The sum of the reciprocals of two consecutive positive integers is 17/72. Write an equation that can be used to find the two integers. What are the integers?
Steve helped me yesterday and gave me the hint 8+9=17. Then I thought about it and saw 8*9=72, but I am confused about how to put it into an equation.
Steve helped me yesterday and gave me the hint 8+9=17. Then I thought about it and saw 8*9=72, but I am confused about how to put it into an equation.
Answers
x = first number
x + 1 = second number
1 / x + 1 / ( x + 1 ) = 17 / 72
[ 1 * ( x + 1 ) + 1 * x ] / [ x * ( x + 1 ) ] = 17 / 72
[ ( x + 1 ) + x ] / [ x * x + x * 1 ) ] = 17 / 72
( 2 x + 1 ) / ( x ^ 2 + x ) = 17 / 72 Multiply both sides by 72
72 * ( 2 x + 1 ) / ( x ^ 2 + x ) = 17
( 72 * 2 x + 72 * 1 ) / ( x ^ 2 + x ) = 17
( 144 x + 72 ) / ( x ^ 2 + x ) = 17 Multiply both sides by x ^ 2 + x
144 x + 72 = 17 * ( x ^ 2 + x )
144 x + 72 = 17 * x ^ 2 + 17 * x
144 x + 72 = 17 x ^ 2 + 17 x Subtract 144 x to both sides
144 x + 72 - 144 x = 17 x ^ 2 + 17 x - 144 x
72 = 17 x ^ 2 - 127 x Subtract 72 to both sides
72 - 72 = 17 x ^ 2 - 127 x - 72
0 = 17 x ^ 2 - 127 x - 72
17 x ^ 2 - 127 x - 72 = 0
The solutions are:
x = - 9 / 17 and x = 8
- 9 / 17 isn't positive integers so x = 8
first number = 8
second number = 8 + 1 = 9
Proof:
1 / x + 1 / ( x + 1 ) =
1 / 8 + 1 / 9 =
( 1 * 9 + 1 * 8 ) / ( 8 * 9 ) =
( 9 + 8 ) / 72 = 17 / 72
x + 1 = second number
1 / x + 1 / ( x + 1 ) = 17 / 72
[ 1 * ( x + 1 ) + 1 * x ] / [ x * ( x + 1 ) ] = 17 / 72
[ ( x + 1 ) + x ] / [ x * x + x * 1 ) ] = 17 / 72
( 2 x + 1 ) / ( x ^ 2 + x ) = 17 / 72 Multiply both sides by 72
72 * ( 2 x + 1 ) / ( x ^ 2 + x ) = 17
( 72 * 2 x + 72 * 1 ) / ( x ^ 2 + x ) = 17
( 144 x + 72 ) / ( x ^ 2 + x ) = 17 Multiply both sides by x ^ 2 + x
144 x + 72 = 17 * ( x ^ 2 + x )
144 x + 72 = 17 * x ^ 2 + 17 * x
144 x + 72 = 17 x ^ 2 + 17 x Subtract 144 x to both sides
144 x + 72 - 144 x = 17 x ^ 2 + 17 x - 144 x
72 = 17 x ^ 2 - 127 x Subtract 72 to both sides
72 - 72 = 17 x ^ 2 - 127 x - 72
0 = 17 x ^ 2 - 127 x - 72
17 x ^ 2 - 127 x - 72 = 0
The solutions are:
x = - 9 / 17 and x = 8
- 9 / 17 isn't positive integers so x = 8
first number = 8
second number = 8 + 1 = 9
Proof:
1 / x + 1 / ( x + 1 ) =
1 / 8 + 1 / 9 =
( 1 * 9 + 1 * 8 ) / ( 8 * 9 ) =
( 9 + 8 ) / 72 = 17 / 72
bruh why is that so long
cope
Related Questions
The sum of the reciprocals of three consecutive positive integers is equal to 47 divided by the prod...
The sum of the reciprocals of two consecutive positive integers is 17/12. Write an equation that can...
Enter your answer and show all the steps that you use to solve this problem in the space provided....