One number is 5 times another If the sum of their reciprocals is 2/15 find the two numbers Write a sentence that defines the variable you use and check your solution
n is one number
on it another number
n= 5*on
1/n + 1/on= 2/15
1/5on + 1/on= 2/15
multiply both sides by 5on
1 + 5= 10 on/15 solve for on, then n.
To solve the problem, we will define the variable "n" as one of the numbers and "on" as the other number. Therefore, the two numbers can be represented as "n" and "5n" since it is given that one number is 5 times the other.
Now, we can write the equation based on the sum of their reciprocals, which is 1/n + 1/on = 2/15. Substituting the values, we get 1/n + 1/(5n) = 2/15.
To simplify the equation, we need to find a common denominator by multiplying the denominators. In this case, the common denominator is 5n. So, we multiply both fractions by the appropriate factors to get the following equation:
(1/n)(5n/5n) + (1/(5n))(n/n) = (2/15)(5n).
This simplifies to 5/5n + 1/5n = 2n/15.
Now, we can combine the fractions on the left side of the equation to get (5 + 1)/5n = 2n/15.
Simplifying further, we have 6/5n = 2n/15.
To eliminate the fractions, we can cross-multiply and solve for "on":
6 * 15 = 2n * 5n,
90 = 10n^2,
9 = n^2.
Taking the square root of both sides, we get:
n = ±3.
Now, we can substitute this value of "n" back into the equation to find the other number:
on = 5 * n,
on = 5 * 3,
on = 15.
Thus, the two numbers are 3 and 15.
To check our solution, let's substitute the values of "n" and "on" back into the equation 1/n + 1/on = 2/15:
1/3 + 1/15 = 2/15.
Finding a common denominator of 15, we have:
5/15 + 1/15 = 2/15,
6/15 = 2/15.
Both sides of the equation are equal, so our solution is correct.