In an unknown fraction the denominator is one more than twice the numerator. When 2 1/10 is added to this fraction, the result is equal to the reciprocal of the unknown fraction.
Find the unknown fraction.
(Hint; the the numerator of the fraction to be the x)
using their hint:
let the numerator of original fraction be x
then the denominator is 2x+1
so the original fraction is x/(2x+1)
condition stated:
x/(2x+1) + 2 1/10 = (2x+1)/x
x(2x+1) + 21/10 = (2x+1)/x
multiply each term by 10x(2x+1) , the LCD
10x^2 + 21x(2x+1) = 10(2x+1)^2
10x^2 + 42x^2 + 21x = 40x^2 + 40x + 10
12x^2 - 19x - 10 = 0
(x-2)(12x + 5) = 0
x = 2 or x = -5/12
if x = 2, the original fraction was 2/5
check:
2/5 + 21/10 = 5/2, that works
could it work for x = -5/12 ?
the fraction would be
(-5/12) / (-10/12 + 1)
= -5/2
what if we add 21/10 to -5/2 ??
-5/2 + 21/10 = -2/5
so the original fraction could have been
2/5 or -5/2
(I verified that either one will work)
Let's break down the problem step by step to find the unknown fraction.
Step 1: Define the fraction
Let's assume the numerator of the unknown fraction is x. According to the problem, the denominator is one more than twice the numerator, which can be expressed as (2x + 1).
Therefore, the fraction is x / (2x + 1).
Step 2: Add 2 1/10 to the fraction
To add 2 1/10 to the fraction, we need to find the decimal equivalent of 2 1/10, which is 2.1. Now, we add this to the fraction:
x / (2x + 1) + 2.1
Step 3: Set up the equation
According to the problem, the result of adding 2.1 to the fraction should be equal to the reciprocal of the unknown fraction. The reciprocal of x / (2x + 1) is (2x + 1) / x.
Now, let's set up the equation:
x / (2x + 1) + 2.1 = (2x + 1) / x
Step 4: Solve the equation
To solve the equation, we need to get rid of the fractions by cross-multiplying.
First, let's multiply both sides of the equation by x to get rid of the fraction:
x * (x / (2x + 1)) + 2.1x = 2x + 1
x^2 / (2x + 1) + 2.1x = 2x + 1
Now, let's multiply both sides of the equation by (2x + 1) to eliminate another fraction:
x^2 + 2.1x(2x + 1) = (2x + 1)(2x + 1)
x^2 + 4.2x^2 + 2.1x = 4x^2 + 4x + 1
Combine like terms:
5.2x^2 + 2.1x - 4x^2 - 4x - 1 = 0
x^2 - 1.9x - 1 = 0
Now, we can solve this quadratic equation using factoring, completing the square, or the quadratic formula.
Step 5: Solve for x
Factoring the equation x^2 - 1.9x - 1 = 0 may not be immediately obvious, so let's use the quadratic formula:
x = (-b ± sqrt(b^2 - 4ac)) / 2a
For our equation, a = 1, b = -1.9, and c = -1.
Plugging in these values into the quadratic formula, we get:
x = (-(-1.9) ± sqrt((-1.9)^2 - 4(1)(-1))) / 2(1)
Simplifying further:
x = (1.9 ± sqrt(3.61 + 4)) / 2
x = (1.9 ± sqrt(7.61)) / 2
x ≈ (1.9 ± 2.76) / 2
So, there are two possible values for x:
x ≈ 4.66 / 2 ≈ 2.33
and
x ≈ -0.86 / 2 ≈ -0.43
Since fractions cannot have negative numerators, the only valid value for x is approximately 2.33.
Therefore, the unknown fraction is 2.33 / (2 * 2.33 + 1) = 2.33 / 5.66.