Question

Помогите срочно :знаменатель дроби на 2 больше её числителя.Если числитель уменьшить на 2 , а знаменатель увеличить на 9 то дробь уменьшиться на 1/2.Найдите данную дробь обозначив буквой X знаменатель дроби.СПАСАЙТЕ СРОЧНО

Answer 1

Обозначим, как задано, дробь у/х,
 по условию    х=у+2  (1)
Уменьшив числитель и увеличив знаменатель, получим дробь: (у-2)/(х+9); по условию: (у-2)/(х+9) = у/х - 1/2  (2);
подставив (1) в (2), получим: 
(у-2)/(у+11) = у/(у+2) - 1/2 ; Приведем все члены уравнения к общему знаменателю 2(у+11)(у+2) и избавимся от него,  :
 2(у-2)(у+2) = 2у(у+11) - (у+2)(у+11);
2у² - 8 - 2у² -22у + у² + 2у + 11у + 22 = 0;
у² - 9у + 14 = 0; у₁ = (9+√(81-56))/2 = 7; у₂ = (9-5)/4 = 2;
из( 1) найдем х:   х₁=7+2 = 9; х₂ =2+2 = 4
то есть наша дробь 7/9 или 2/4 
Проверка: 7/9 - 5/18 = 9/18 = 1/2;  
2/4 - 0/13=2/4=1/2, если дробь сократить 1/2 - (-1)/11= 11/22 + 2/22 = 13/22; 13/22≠1/2. Проверка в этом случае не пройдена.
 

Answer 2

Problem Analysis

We are given a fraction where the denominator is 2 more than the numerator. If we decrease the numerator by 2 and increase the denominator by 9, the fraction decreases by 1/2. We need to find the original fraction by representing the denominator with the letter X.

Solution

Let's assume the original fraction is a/b, where b = a + 2.

According to the problem, if we decrease the numerator by 2 and increase the denominator by 9, the fraction decreases by 1/2. This can be represented as:

(a - 2) / (b + 9) = a / b - 1/2

Simplifying the equation:

(a - 2) / (a + 2 + 9) = a / (a + 2) - 1/2

(a - 2) / (a + 11) = a / (a + 2) - 1/2

To solve this equation, we can cross-multiply:

(a - 2)(a + 2) = (a + 11)(a / 2 - 1/2)

Simplifying further:

a^2 - 4 = (a + 11)(a / 2 - 1/2)

a^2 - 4 = (a^2 + 11a) / 2 - (a + 11) / 2

Multiplying both sides by 2 to eliminate the fraction:

2a^2 - 8 = a^2 + 11a - a - 11

2a^2 - 8 = a^2 + 10a - 11

Rearranging the equation:

a^2 - 10a + 3 = 0

Now we can solve this quadratic equation to find the value of a.

Solving the Quadratic Equation

Using the quadratic formula, we have:

a = (-b ± √(b^2 - 4ac)) / (2a)

For our equation a^2 - 10a + 3 = 0, the coefficients are: - a = 1 - b = -10 - c = 3

Substituting these values into the quadratic formula:

a = (-(-10) ± √((-10)^2 - 4 * 1 * 3)) / (2 * 1)

Simplifying:

a = (10 ± √(100 - 12)) / 2

a = (10 ± √88) / 2

a = (10 ± 2√22) / 2

a = 5 ± √22

So, the possible values for a are 5 + √22 and 5 - √22.

Finding the Original Fraction

Now that we have the possible values for a, we can find the corresponding values for b.

If a = 5 + √22, then b = (5 + √22) + 2 = 7 + √22.

If a = 5 - √22, then b = (5 - √22) + 2 = 7 - √22.

Therefore, the original fraction can be represented as (5 + √22) / (7 + √22) or (5 - √22) / (7 - √22), depending on the value of X.

Conclusion

The original fraction, represented as a/b, can be expressed as (5 + √22) / (7 + √22) or (5 - √22) / (7 - √22), depending on the value of X.

Please note that the solution provided is based on the given information and the mathematical calculations.