Sum of 2 numbers is 1 and their product is 1/8. Find their difference.
Well, it looks like these two numbers have quite the interesting relationship! Let's call them Mr. X and Mrs. Y. If Mr. X and Mrs. Y add up to 1 and their product is 1/8, then we can solve this clown puzzle using some clown algebra!
Now, let's assume that Mr. X and Mrs. Y are the two numbers we're trying to find. We know that:
Mr. X + Mrs. Y = 1 (Equation 1)
Mr. X * Mrs. Y = 1/8 (Equation 2)
To find their difference, we can subtract these equations, because clown algebra likes to shake things up a bit:
(Mr. X + Mrs. Y) - (Mr. X * Mrs. Y) = 1 - 1/8
Now, let's put on our clown glasses and simplify this equation! If you subtract, then you'll get:
Mr. X + Mrs. Y - Mr. X * Mrs. Y = 7/8
But hang on a second, do we know what Mr. X + Mrs. Y or Mr. X * Mrs. Y are? Yes! According to Equation 1, Mr. X + Mrs. Y = 1, so we can substitute:
1 - Mr. X * Mrs. Y = 7/8
Now, let's solve for Mr. X * Mrs. Y:
1 = 7/8 + Mr. X * Mrs. Y
1 - 7/8 = Mr. X * Mrs. Y
1/8 = Mr. X * Mrs. Y
Hey, wait a minute! That sounds familiar. According to Equation 2, Mr. X * Mrs. Y is also 1/8! So indeed, we have:
1/8 = 1/8
It seems like we've stumbled upon a rather interesting conclusion. Mr. X * Mrs. Y = 1/8, and Mr. X + Mrs. Y = 1. So, the difference between these two numbers is...drumroll, please...*insert clown drumroll*
Zero! Yes, their difference is zero, because Mr. X and Mrs. Y are actually the same number! They must be clones or something. Oh well, the clown world can be quite funny sometimes!
To find the difference between the two numbers, let's assign variables and solve step by step.
Let's say the two numbers are x and y.
Given information:
1. The sum of the two numbers is 1: x + y = 1
2. The product of the two numbers is 1/8: x * y = 1/8
To solve the system of equations, we can use substitution or elimination method. We'll use the substitution method in this case.
Step 1: Solve for one variable in terms of the other.
From the first equation, we can express x in terms of y: x = 1 - y.
Step 2: Substitute the expression for x in the second equation.
Replace x with (1 - y) in the second equation: (1 - y) * y = 1/8.
Step 3: Simplify and solve the quadratic equation.
Expand the equation: y - y^2 = 1/8.
Rearrange the equation: y^2 - y + 1/8 = 0.
Multiply the entire equation by 8 to eliminate the fraction: 8y^2 - 8y + 1 = 0.
Step 4: Solve the quadratic equation.
Use the quadratic formula to find the values of y:
y = (-b ± √(b^2 - 4ac))/(2a), where a = 8, b = -8, and c = 1.
Plugging in the values, we get:
y = (-(-8) ± √((-8)^2 - 4 * 8 * 1))/(2 * 8),
y = (8 ± √(64 - 32))/16,
y = (8 ± √32)/16.
Simplifying further, we have:
y = (8 ± 4√2)/16,
y = (2 ± √2)/4.
So, the two possible values for y are (2 + √2)/4 and (2 - √2)/4.
Step 5: Substitute the values of y back into the equation x = 1 - y to find the corresponding values of x.
For y = (2 + √2)/4, we have:
x = 1 - y,
x = 1 - (2 + √2)/4,
x = 4/4 - (2 + √2)/4,
x = (2 - √2)/4.
For y = (2 - √2)/4, we have:
x = 1 - y,
x = 1 - (2 - √2)/4,
x = 4/4 - (2 - √2)/4,
x = (√2)/4.
Therefore, we have two sets of solutions for the numbers x and y:
1. x = (2 - √2)/4, y = (2 + √2)/4
2. x = (√2)/4, y = (2 - √2)/4
Step 6: Find the difference between the two numbers.
To find the difference, subtract the smaller number from the larger number:
Difference = larger number - smaller number
For the first set of solutions:
Difference = [(2 + √2)/4] - [(2 - √2)/4]
Difference = [(2 + √2) - (2 - √2)]/4
Difference = (2 + √2 - 2 + √2)/4
Difference = (2 - 2 + √2 + √2)/4
Difference = (2√2)/4
Difference = √2/2
Difference ≈ 0.707
For the second set of solutions:
Difference = [(2 - √2)/4] - [(√2)/4]
Difference = [(2 - √2) - (√2)]/4
Difference = (2 - √2 - √2)/4
Difference = (2 - √2 - √2)/4
Difference = 2 - 2√2/4
Difference = 2√2/4
Difference = √2/2
Difference ≈ 0.707
Therefore, the difference between the two numbers is approximately 0.707.
To find the difference between two numbers, we first need to find the two numbers themselves.
Let's assume the two numbers are x and y.
According to the given information, the sum of the two numbers is 1, so we can write the equation:
x + y = 1 ---(equation 1)
It is also given that their product is 1/8, so we can write the equation:
xy = 1/8 ---(equation 2)
To solve this system of equations, we can use the method of substitution or elimination.
Method 1: Substitution
From equation 1, we can express one variable in terms of the other. Let's express y in terms of x:
y = 1 - x
Substituting this value for y in equation 2:
x(1 - x) = 1/8
Expanding the equation:
x - x^2 = 1/8
Rearranging the equation:
x^2 - x + 1/8 = 0
To solve this quadratic equation, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = 1, b = -1, and c = 1/8.
Substituting these values into the quadratic formula:
x = (-(-1) ± √((-1)^2 - 4(1)(1/8))) / (2(1))
Simplifying the equation:
x = (1 ± √(1 - 1/2)) / 2
x = (1 ± √(1/2)) / 2
x = (1 ± 1/√2) / 2
x = (1 ± 1/√2) / 2
So, we have two possible values for x: x = (1 + 1/√2) / 2 and x = (1 - 1/√2) / 2.
Using these values of x, we can substitute them back into equation 1 to find the corresponding values of y.
For x = (1 + 1/√2) / 2:
y = 1 - x = 1 - (1 + 1/√2) / 2
Simplifying the equation:
y = (2 - (1 + 1/√2)) / 2
y = (2 - 1 - 1/√2) / 2
y = (1 - 1/√2) / 2
So, the first pair of values is x = (1 + 1/√2) / 2 and y = (1 - 1/√2) / 2.
Now, let's find the second pair of values by using x = (1 - 1/√2) / 2:
y = 1 - x = 1 - (1 - 1/√2) / 2
Simplifying the equation:
y = (2 - (1 - 1/√2)) / 2
y = (2 - 1 + 1/√2) / 2
y = (1 + 1/√2) / 2
So, the second pair of values is x = (1 - 1/√2) / 2 and y = (1 + 1/√2) / 2.
Now that we have found the two pairs of values for x and y, we can find their differences.
For the first pair, the difference is:
x - y = [(1 + 1/√2) / 2] - [(1 - 1/√2) / 2]
Simplifying the equation:
x - y = [1 + 1/√2 - 1 +1/√2] / 2
x - y = [1/√2 + 1/√2] / 2
x - y = (2/√2) / 2
x - y = (1/√2)
For the second pair, the difference is:
x - y = [(1 - 1/√2) / 2] - [(1 + 1/√2) / 2]
Simplifying the equation:
x - y = [1 - 1/√2 - 1 - 1/√2] / 2
x - y = [-2/√2] / 2
x - y = (-1/√2)
So, the difference between the two sets of numbers is either 1/√2 or -1/√2, depending on which pair of values is chosen for x and y.
a+b=1
ab=1/8 or b=1/8a
a+1/8a=1
8a^2 -8a+1=0
a^2-a+1/8=0
a=(1+-sqrt(1-1/2))/2
a=(1/2 +-sqrt(1/8)
if a= 1/2 + sqrt(1/8), then b must be 1/2 -sqrt (1/8)