A positive integer is 2 less than another. If the sum of the reciprocal of the smaller and twice the reciprocal of the larger is 5/12, then find the two integers.
algebraic expression is:
1/n-2 + (2)1/n = 5/12
small = s and b = big
s = b-2
1/s + 2/b = 5/12
so
1/(b-2) + 2/b = 5/12 agreed
multiply by 12 b (b-2)
12 b + 24(b-2) = 5 b(b-2)
12 b + 24 b - 48 = 5 b^2 - 10 b
0 = 5 b^2 - 46 b + 48
0 = (b-8)(5b-6)
so use b = 8
s = b-2 = 6
To find the two integers, let's solve the equation step by step.
The given equation is:
1/n - 2 + 2/n = 5/12
Let's simplify the equation by combining like terms:
(1 + 2)/n - 2 = 5/12
3/n - 2 = 5/12
Now let's get rid of the fraction by multiplying both sides of the equation by 12:
12 * (3/n - 2) = 12 * (5/12)
36/n - 24 = 5
Next, let's isolate the term with n by adding 24 to both sides of the equation:
36/n - 24 + 24 = 5 + 24
36/n = 29
To eliminate the fraction, we can cross-multiply:
36 * n = 29 * 1
36n = 29
Now, divide both sides of the equation by 36 to solve for n:
36n/36 = 29/36
n = 29/36
So, the value of n is 29/36.
Since we are given that the positive integer is 2 less than another, let's find the other integer by subtracting 2 from n:
n - 2 = 29/36 - 2
To subtract fractions, let's find a common denominator of 36:
n - 2 = 29/36 - 2 * 36/36
n - 2 = 29/36 - 72/36
Now, let's subtract the fractions:
n - 2 = (29 - 72)/36
n - 2 = -43/36
Therefore, the two integers are 29/36 and -43/36.
To solve the equation 1/n-2 + 2/n = 5/12, we need to find the values for n that satisfy this equation. Let's go step by step to solve for n.
Step 1: Simplify the left side of the equation.
To do this, we first need to find a common denominator for the two fractions. The common denominator for n-2 and n is n(n-2). Rewriting the equation with the common denominator, we have:
[(1)(n) + 2(n-2)] / (n)(n-2) = 5/12
Simplifying further, we get:
(n + 2n - 4) / (n)(n-2) = 5/12
(3n - 4) / (n)(n-2) = 5/12
Step 2: Cross-multiply and simplify.
Cross-multiplying the fractions, we get:
12(3n - 4) = 5(n)(n-2)
Expanding both sides of the equation:
36n - 48 = 5n^2 - 10n
Rearranging the equation to set it equal to zero:
5n^2 - 46n + 48 = 0
Step 3: Solve the quadratic equation.
To find the value(s) of n that satisfy the equation, we can either factorize the quadratic equation or use the quadratic formula.
Using the quadratic formula:
n = (-b ± √(b^2 - 4ac)) / 2a
For our equation, a = 5, b = -46, and c = 48. Substituting these values into the quadratic formula:
n = (-(-46) ± √((-46)^2 - 4(5)(48))) / (2(5))
n = (46 ± √(2116 - 960)) / 10
n = (46 ± √1156) / 10
n = (46 ± 34) / 10
This gives us two possible values for n:
n = (46 + 34) / 10 = 80 / 10 = 8
n = (46 - 34) / 10 = 12 / 10 = 1.2
Step 4: Find the corresponding values for the two integers.
Since n is a positive integer and is 2 less than the other positive integer, we can conclude that one of the integers is n + 2.
For n = 8, the other integer is 8 + 2 = 10.
For n = 1.2, the other integer is not a positive integer, so it does not satisfy the condition.
Therefore, the two integers are 8 and 10.