Question

Знаменатель дроби на 3 больше числителя. Если числитель увеличить на 4, а знаменатель на 30, то дробь уменьшится на 1/5 . Найти исходную дробь.

Answer 1

Если х=-9, то дробь -9/12, но по условию дробь несократима
Если х=2, то дробь 2/5, что удовлетворяет условию
Ответ: 2/5 

Answer 2

Problem Analysis

We are given a fraction where the denominator is 3 more than the numerator. If we increase the numerator by 4 and the denominator by 30, the fraction decreases by 1/5. We need to find the original fraction.

Solution

Let's assume the original fraction is x/y.

From the given information, we can form the following equations:

1. The denominator is 3 more than the numerator: y = x + 3. 2. If we increase the numerator by 4 and the denominator by 30, the fraction decreases by 1/5: (x + 4)/(y + 30) = x/y - 1/5.

To solve this system of equations, we can substitute the value of y from the first equation into the second equation and solve for x.

Detailed Solution

Let's substitute y = x + 3 into the second equation:

(x + 4)/(x + 3 + 30) = x/y - 1/5

Simplifying the equation:

(x + 4)/(x + 33) = x/(x + 3) - 1/5

To eliminate the fractions, we can multiply both sides of the equation by the least common denominator, which is 5(x + 33)(x + 3):

5(x + 4)(x + 3) = (x + 33)(x) - (x + 3)(x + 33)

Expanding and simplifying:

5(x^2 + 7x + 12) = x^2 + 33x - x^2 - 33x - 3x - 99

5x^2 + 35x + 60 = x^2 + 33x - x^2 - 33x - 3x - 99

Combining like terms:

4x^2 + 5x + 60 = -99

Rearranging the equation:

4x^2 + 5x + 159 = 0

Now we can solve this quadratic equation using the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

For our equation, a = 4, b = 5, and c = 159. Substituting these values into the quadratic formula:

x = (-5 ± √(5^2 - 4 * 4 * 159)) / (2 * 4)

Simplifying:

x = (-5 ± √(25 - 2544)) / 8

x = (-5 ± √(-2519)) / 8

Since the discriminant is negative, there are no real solutions for x. Therefore, there is no solution for the original fraction.

Answer

There is no solution for the original fraction.