one positive number is 14 less than another positive number. if the reciprocal of the smaller number is added to five times to the reciprocal of the larger number the sum is 1/4. what are the two numbers
To solve this problem, let's assign variables to the two unknown numbers. Let's call the larger number x and the smaller number y.
According to the problem, "one positive number is 14 less than another positive number," so we can write the equation:
x = y + 14 (Equation 1)
The problem also states that "the reciprocal of the smaller number is added to five times the reciprocal of the larger number, and the sum is 1/4." We can write this as an equation as well:
1/y + 5(1/x) = 1/4 (Equation 2)
Now we have a system of two equations with two unknowns (x and y). We can solve this system of equations to find the values of x and y.
The first step is to substitute the value of x from Equation 1 into Equation 2:
1/y + 5(1/(y + 14)) = 1/4
Next, we multiply both sides of the equation by 4y(y + 14) to eliminate the denominators:
4(y + 14) + 20y = y(y + 14)
Expanding the equation:
4y + 56 + 20y = y^2 + 14y
Rearranging the equation to set it equal to zero:
y^2 + 14y - 4y - 56 - 20y = 0
Combining like terms:
y^2 + 10y - 56 = 0
Now we can solve this quadratic equation by factoring or using the quadratic formula. By factoring, we find:
(y + 14)(y - 4) = 0
Setting each factor equal to zero, we get two possible values for y:
y + 14 = 0 -> y = -14 (rejected since it is given that the numbers are positive)
y - 4 = 0 -> y = 4
Since y represents the smaller number, we can substitute this value back into Equation 1 to find x:
x = y + 14 = 4 + 14 = 18
Therefore, the two numbers are 4 and 18.
Let's assign variables to the two positive numbers. Let x represent the smaller number and y represent the larger number.
Given that one positive number is 14 less than the other, we can write the following equation:
y = x + 14
We are also given that the reciprocal of the smaller number (1/x) added to five times the reciprocal of the larger number (5/y) equals 1/4. So we can write the following equation:
1/x + 5/y = 1/4
Now we have a system of two equations. To solve this system, we will eliminate one variable. Let's eliminate y by substituting its value from the first equation into the second equation:
1/x + 5/(x + 14) = 1/4
To simplify this equation, we can find a common denominator, which is 4x(x + 14):
4(x + 14) + 20x = x(x + 14)
Simplifying further:
4x + 56 + 20x = x^2 + 14x
Collecting like terms:
x^2 + 30x - 4x - 14x - 56 = 0
x^2 + 16x - 56 = 0
Now we have a quadratic equation. Factoring or using the quadratic formula:
(x + 4)(x + 14) = 0
This gives us two possible values for x: -4 and -14. However, since the numbers are positive, we can disregard the negative solution.
Therefore, x = -4 is not a valid solution.
The only valid solution is:
x + 14 = 0
x = -14
So the smaller number, x, is -14.
To find the larger number, y, we can substitute this value back into the first equation:
y = x + 14
y = -14 + 14
y = 0
The larger number, y, is 0.
Therefore, the two positive numbers that satisfy the given conditions are -14 and 0.