If x < y, then we have
(x+y)/(y+1) = 3/2
(y-x)/x = 1/2
Now just solve for x and y
When the greater of the two numbers increased by 1 divides the sum of the numbers the result is3/2 .when the difference of these numbers is divide by the smaller ,the result is 1/2 .find the numbers
3 answers
a = the greater number
b = the smaller number
When the greater of the two numbers increased by 1 divides the sum of the numbers the result is 3 / 2 mean :
( a + 1 ) / ( a + b ) = 3 / 2
When the difference of these numbers is divide by the smaller ,the result is 1 / 2 mean:
( a - b ) / b = 1 / 2
Now you must solve system:
( a + 1 ) / ( a + b ) = 3 / 2
( a - b ) / b = 1 / 2
( a - b ) / b = 1 / 2 Multiply both sides by b
a - b = b / 2 Add b to both sides
a - b + b = b / 2 + b
a = b / 2 + b
a = b / 2 + 2 b / 2
a = 3 b / 2
Replace this value in equation:
( a + 1 ) / ( a + b ) = 3 / 2
( 3 b / 2 + 1 ) / ( 3 b / 2 + b ) = 3 / 2
( 3 b / 2 + 2 / 2 ) / ( 3 b / 2 + 2 b / 2 ) = 3 / 2
[ ( 3 b + 2 ) / 2 ] / ( 5 b / 2 ) = 3 / 2
2 * ( 3 b + 2 ) / ( 2 * 5 b ) = 3 / 2
( 3 b + 2 ) / 5 b = 3 / 2 Multiply both sides by 5 b
3 b + 2 = 3 * 5 b / 2
3 b + 2 = 15 b / 2 Multiply both sides by 2
2 * ( 3 b + 2 ) = 15 b
6 b + 4 = 15 b Subtract 6 b to both sides
6 b + 4 - 6 b = 15 b - 6 b
4 = 9 b Divide both sides by 9
4 / 9 = b
b = 4 / 9
Replace this value in equation:
a = 3 b / 2
a = ( 3 * 4 / 9 ) / 2
a = 12 / ( 2 * 9 )
a = 12 / 18
a = 6 * 2 / 6 * 3
a = 2 / 3
The solutions are:
a = 2 / 3 , b = 4 / 9
Proof:
( a + 1 ) / ( a + b ) = 3 / 2
( 2 / 3 + 1 ) / ( 2 / 3 + 4 / 9 ) =
( 2 / 3 + 3 / 3 ) / ( 3 * 2 / 3 * 3 + 4 / 9 ) =
( 5 / 3 ) / ( 6 / 9 + 4 / 9 ) =
( 5 / 3 ) / (10 / 9 ) =
5 * 9 / 3 * 10 =
5 * 3 * 3 / 3 * 5 * 2 =
3 / 2
( a - b ) / b = 1 / 2
( 2 / 3 - 4 / 9 ) / ( 4 / 9 ) =
( 2 * 3 / 3 * 3 - 4 / 9 ) / ( 4 / 9 ) =
( 6 / 9 - 4 / 9 ) / ( 4 / 9 ) =
( 2 / 9 ) / ( 4 / 9 ) =
2 * 9 / 4 * 9 =
2 / 4 = 1 * 2 / 2 * 2 = 1 / 2
b = the smaller number
When the greater of the two numbers increased by 1 divides the sum of the numbers the result is 3 / 2 mean :
( a + 1 ) / ( a + b ) = 3 / 2
When the difference of these numbers is divide by the smaller ,the result is 1 / 2 mean:
( a - b ) / b = 1 / 2
Now you must solve system:
( a + 1 ) / ( a + b ) = 3 / 2
( a - b ) / b = 1 / 2
( a - b ) / b = 1 / 2 Multiply both sides by b
a - b = b / 2 Add b to both sides
a - b + b = b / 2 + b
a = b / 2 + b
a = b / 2 + 2 b / 2
a = 3 b / 2
Replace this value in equation:
( a + 1 ) / ( a + b ) = 3 / 2
( 3 b / 2 + 1 ) / ( 3 b / 2 + b ) = 3 / 2
( 3 b / 2 + 2 / 2 ) / ( 3 b / 2 + 2 b / 2 ) = 3 / 2
[ ( 3 b + 2 ) / 2 ] / ( 5 b / 2 ) = 3 / 2
2 * ( 3 b + 2 ) / ( 2 * 5 b ) = 3 / 2
( 3 b + 2 ) / 5 b = 3 / 2 Multiply both sides by 5 b
3 b + 2 = 3 * 5 b / 2
3 b + 2 = 15 b / 2 Multiply both sides by 2
2 * ( 3 b + 2 ) = 15 b
6 b + 4 = 15 b Subtract 6 b to both sides
6 b + 4 - 6 b = 15 b - 6 b
4 = 9 b Divide both sides by 9
4 / 9 = b
b = 4 / 9
Replace this value in equation:
a = 3 b / 2
a = ( 3 * 4 / 9 ) / 2
a = 12 / ( 2 * 9 )
a = 12 / 18
a = 6 * 2 / 6 * 3
a = 2 / 3
The solutions are:
a = 2 / 3 , b = 4 / 9
Proof:
( a + 1 ) / ( a + b ) = 3 / 2
( 2 / 3 + 1 ) / ( 2 / 3 + 4 / 9 ) =
( 2 / 3 + 3 / 3 ) / ( 3 * 2 / 3 * 3 + 4 / 9 ) =
( 5 / 3 ) / ( 6 / 9 + 4 / 9 ) =
( 5 / 3 ) / (10 / 9 ) =
5 * 9 / 3 * 10 =
5 * 3 * 3 / 3 * 5 * 2 =
3 / 2
( a - b ) / b = 1 / 2
( 2 / 3 - 4 / 9 ) / ( 4 / 9 ) =
( 2 * 3 / 3 * 3 - 4 / 9 ) / ( 4 / 9 ) =
( 6 / 9 - 4 / 9 ) / ( 4 / 9 ) =
( 2 / 9 ) / ( 4 / 9 ) =
2 * 9 / 4 * 9 =
2 / 4 = 1 * 2 / 2 * 2 = 1 / 2
Hmm. I got 6 and 9
Better read the problem again. You got your division backwards: the greater divides the sum.
Better read the problem again. You got your division backwards: the greater divides the sum.
1 answer