Question

За 2 футбольних і 4 волейбольних м'ячі заплатили 2000 грн. Після того як футбольний м'яч подешевшав на 20 %, а волейбольний подорожчав на 10 %, то за один футбольний і один волейбольний м'ячі заплатили 650 грн. Якою була початкова ціна кожного м'яча (у гривнях)?

Answer 1

Х - початкова цена футбольного мяча
у - початкова цена волейбольного мяча
20% = 0,2 (20:100)
10% = 0,1 (10:100)
1 - 0,2 = 0,8
1 + 0,1 = 1,1
0,8х - новая цена футбольного мяча
1,1у - новая цена волейбольного мяча
Составим систему уравнений:
2х + 4у = 2000 грн
0,8х + 1,1у = 650 грн
Из первого уравнения 2х = 2000 - 4у; и х = (2000-4у)/2=1000 - 2у
Подставляем во второе ур-ие
0,8(1000-2у) + 1,1у = 650, тогда
800 - 1,6у + 1,1у = 650, отсюда
800-650=1,6у-1,1у, то есть
0,5у = 150, значит у = 150*2 = 300 грн (початкова цена волейбольного мяча), отсюда, зная, что х = 1000 - 2у
х = 1000 - 2*300 = 1000 - 600 = 400 грн (початкова цена футб. мяча)
Ответ: начальная цена футбольного мяча 400 грн
             начальная цена волейбольного мяча 300 грн

Answer 2

Problem Analysis

We are given that 2 footballs and 4 volleyballs were purchased for a total of 2000 UAH. After the price of the football decreased by 20% and the price of the volleyball increased by 10%, the total cost of 1 football and 1 volleyball was 650 UAH. We need to find the initial price of each ball.

Solution

Let's assume the initial price of a football is x UAH and the initial price of a volleyball is y UAH.

According to the given information, we can set up the following equations:

Equation 1: 2x + 4y = 2000 (total cost of 2 footballs and 4 volleyballs is 2000 UAH)

Equation 2: (0.8x) + (1.1y) = 650 (cost of 1 football after a 20% decrease and 1 volleyball after a 10% increase is 650 UAH)

We can solve this system of equations to find the values of x and y.

Solving the Equations

Let's solve the system of equations using the substitution method.

From Equation 1, we can express x in terms of y:

2x = 2000 - 4y

x = (2000 - 4y) / 2

Substituting this value of x into Equation 2, we get:

(0.8 * ((2000 - 4y) / 2)) + (1.1y) = 650

Simplifying the equation:

(1600 - 3.2y) + 1.1y = 650

1600 - 3.2y + 1.1y = 650

-2.1y = 650 - 1600

-2.1y = -950

y = -950 / -2.1

y ≈ 452.38

Substituting this value of y back into Equation 1, we can find x:

2x + 4(452.38) = 2000

2x + 1809.52 = 2000

2x = 2000 - 1809.52

2x = 190.48

x = 190.48 / 2

x ≈ 95.24

Answer

The initial price of each football was approximately 95.24 UAH and the initial price of each volleyball was approximately 452.38 UAH.