A natural number decreased by 30 times its reciprocal is 1. Find the natural number.
the number --- x
its reciprocal --- 1/x
x - 30/x = 1
times x
x^2 - 30 = x
x^2 - x - 30 = 0
(x - 6)(x + 5) = 0
x = 6 or x = -5, but a natural number is positive, so
x = 6
check:
6 - 30(1/6)
= 6 - 5
= 1
looking good.
Let's assume the natural number is "x".
Given that the natural number decreased by 30 times its reciprocal is 1, we can write the equation:
x - 30(1/x) = 1
To solve this equation, we can simplify it by getting rid of the fraction:
Multiply every term by "x" to eliminate the fraction:
x(x) - 30(1) = 1(x)
x^2 - 30 = x
Now, rearrange the equation to get all the terms on one side:
x^2 - x - 30 = 0
To solve this quadratic equation, we can factor it:
(x - 6)(x + 5) = 0
Set each factor equal to zero and solve for "x":
x - 6 = 0 or x + 5 = 0
x = 6 or x = -5
Since we're looking for a natural number, we can discard the negative solution.
Therefore, the natural number is x = 6.
To find the natural number, let's set up an equation based on the given information.
Let the natural number be represented by a variable, say "x."
According to the problem, the natural number decreased by 30 times its reciprocal is 1. Mathematically, we can represent this as:
x - 30(1/x) = 1
To solve this equation, we first multiply both sides by "x" to eliminate the fraction:
x^2 - 30 = x
Next, move all terms to one side to have a quadratic equation:
x^2 - x - 30 = 0
Now, we can factor the quadratic equation or use the quadratic formula to find the values of "x." In this case, let's factorize it:
(x - 6)(x + 5) = 0
By setting each factor equal to zero, we get:
x - 6 = 0 or x + 5 = 0
Solving each equation gives us:
x = 6 or x = -5
Since we are looking for a natural number, we disregard the negative value. Therefore, the natural number is:
x = 6
So, the natural number is 6.