A fraction's numerator is 3 less than its denominator. Its reciprocal is 4 times its value. It is greater than 0. What is the fraction?
(D-3)/D = fraction=1/4fraction
fraction^2=1/4
fraction=1/2
fraction=3/6
let the denominator be x
then the numerator is x-3
and the fraction is (x-3)/x
the reciprocal is 4 times the original fraction
---> x/(x-3) = 4(x-3)/x
cross-multiplying and simplifying I got
x^2 - 8x + 12 = 0
(x-2)(x-6) = 0
x = 2 or x = 6
but x=2 produces a negative value, which was not allowed
so x = 6
so the fraction was 3/6 = 1/2
reciprocal is 2/1 = 2
is that 4(1/2) ? yes!
Can you help me with a math problem
To solve this problem, let's break it down step by step:
1. Let's assume the denominator of the fraction is represented by the variable 'x'.
So, the numerator would be 'x - 3' because it is 3 less than the denominator.
2. The reciprocal of a fraction is obtained by flipping the fraction, so if the original fraction is 'a/b', its reciprocal would be 'b/a'.
3. According to the problem, the reciprocal of the fraction is 4 times the value of the fraction itself. We can write this as an equation:
(x - 3)/x = 4 * (x/(x - 3))
4. Now, we need to solve this equation to find the value of 'x'. To do that, we'll multiply both sides of the equation by 'x * (x - 3)' to eliminate the denominators:
(x - 3)(x * (x - 3))/x = 4 * (x * (x - 3))/(x - 3)
5. Simplifying this equation will give:
(x - 3)(x - 3) = 4 * x
6. Expanding the left side of the equation:
(x^2 - 6x + 9) = 4x
7. Rearranging the equation to one side:
x^2 - 10x + 9 = 0
8. To solve this quadratic equation, we can either factor it or use the quadratic formula. In this case, the equation can be factored as:
(x - 1)(x - 9) = 0
9. So, we have two possible solutions: x = 1 or x = 9.
10. However, we need to remember that the fraction must be greater than 0. Since the denominator cannot be 0 (to avoid division by zero), the valid solution in this case is x = 9.
11. Therefore, the fraction is:
numerator = x - 3 = 9 - 3 = 6
denominator = x = 9
So, the fraction is 6/9, which can be further simplified to 2/3.