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
If x < y, then we have
(x+y)/(y+1) = 3/2
(y-x)/x = 1/2
Now just solve for x and y
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
Hmm. I got 6 and 9
Better read the problem again. You got your division backwards: the greater divides the sum.
To solve this problem, let's assume that the two numbers are x and y, where x is the greater number and y is the smaller number.
Based on the given information, we can form two equations:
1) (x + 1)/(x + y) = 3/2
2) (x - y)/y = 1/2
We can simplify these equations by cross-multiplying:
1) 2(x + 1) = 3(x + y)
2x + 2 = 3x + 3y
2) 2(x - y) = y
2x - 2y = y
Now, let's solve these equations simultaneously.
Rewriting equation 1) as 3y - 2x = 2, we can multiply equation 2) by -3 to eliminate y:
-6x + 6y = -3y
Adding this new equation to equation 1) will cancel out the x terms:
0x + 9y = 2 - 3y
9y + 3y = 2
12y = 2
y = 2/12
y = 1/6
Substitute the value of y back into equation 2) to find x:
2x - 2(1/6) = 1/6
2x - 1/3 = 1/6
2x = 1/6 + 1/3
2x = 1/6 + 2/6
2x = 3/6
x = (3/6) / 2
x = 1/4
Therefore, the two numbers are x = 1/4 and y = 1/6.