Question

Алеша, Боря и Вася покупали блокноты и трехкопеечные карандаши. Алеша купил 4 карандаша и 2 блокнота, Боря купил 6 карандашей и 1 блокнот, Вася купил 3 карандаша и 1 блокнот. Известно, что суммы денег, заплаченные Алешей, Борей и Васей, образуют соответственно первый, второй и третий члены геометрической прогрессии. Сколько стоит блокнот?

Answer 1

Пусть 1 блокнот стоит х копеек.
Тогда:
Алеша заплатил: 12+2х копеек
Боря заплатил: 18+х копеек
Вася заплатил: 9+х копеек.
Говорят: 12+2х;18+х;9+х- это геометрическая прогрессия. 18+х- это средний член геометрической прогрессии, значит он равен среднему геометрическому(произведению под корнем) 12+2х и 9+х. Остается только найти х, решив уравнение:
$18+x= \sqrt{(12+2x)(9+x)}$
(18+x)²=(12+2x)(9+x)
x²+36x+324=108+30x+2x²
x²-6x-216=0
x₁×x₂=-216
x₁+x₂=6
x₁=-12(не удовлетворяет, так как стоимость отрицательной быть не может)
x₂=18→ответ.
Ответ: блокнот стоит 18 копеек.

Answer 2

To find the cost of a notebook, we need to solve the problem using the given information. Let's break it down step by step.

Step 1: Assigning Variables

Let's assign variables to the quantities and prices mentioned in the problem: - Let x be the cost of a notebook (in rubles). - Let y be the cost of a three-kopeck pencil (in rubles).

Step 2: Analyzing Alesha's Purchase

According to the problem, Alesha bought 4 pencils and 2 notebooks. Therefore, the total amount of money Alesha spent can be calculated as follows: - For pencils: 4 * y = 4y - For notebooks: 2 * x = 2x

Step 3: Analyzing Borya's Purchase

Borya bought 6 pencils and 1 notebook. Therefore, the total amount of money Borya spent can be calculated as follows: - For pencils: 6 * y = 6y - For notebooks: 1 * x = x

Step 4: Analyzing Vasya's Purchase

Vasya bought 3 pencils and 1 notebook. Therefore, the total amount of money Vasya spent can be calculated as follows: - For pencils: 3 * y = 3y - For notebooks: 1 * x = x

Step 5: Forming a Geometric Progression

The problem states that the amounts of money spent by Alesha, Borya, and Vasya form a geometric progression. In a geometric progression, each term is obtained by multiplying the previous term by a constant ratio.

Let's assume the common ratio is r.

According to the problem, the amounts of money spent by Alesha, Borya, and Vasya form a geometric progression in that order. Therefore, we can write the following equations:

- Borya's amount / Alesha's amount = r - Vasya's amount / Borya's amount = r

Substituting the amounts we calculated earlier, we get:

- (6y + x) / (4y + 2x) = r - (3y + x) / (6y + x) = r

Step 6: Solving the Equations

We have two equations with two unknowns (x and y). We can solve these equations to find the values of x and y.

Simplifying the equations, we get:

- (6y + x) / (4y + 2x) = r - Cross-multiplying: (6y + x) * (4y + 2x) = r * (4y + 2x) - Expanding: 24y^2 + 12xy + 4x^2 = 4ry + 2rx - Rearranging: 24y^2 + (12x - 4ry)y + (4x^2 - 2rx) = 0 [[1]]

- (3y + x) / (6y + x) = r - Cross-multiplying: (3y + x) * (6y + x) = r * (6y + x) - Expanding: 18y^2 + 9xy + 6yx + x^2 = 6ry + rx - Rearranging: 18y^2 + (15x - 6ry)y + (x^2 - rx) = 0 [[2]]

Now, we have a system of equations [[1]] and [[2]] that we can solve to find the values of x and y.

Unfortunately, I am unable to solve this system of equations directly. However, you can use various methods such as substitution or elimination to find the values of x and y. Once you have the values of x and y, you can determine the cost of a notebook (x).

I hope this helps you solve the problem! Let me know if you have any further questions.