Question

Имеются два сосуда, содержащие 48 кг и 42 кг раствора кислоты различной концентрации. Если их слить вместе, то получим раствор, содержащий 42% кислоты. Если же слить равные массы этих растворов, то полученный раствор будет содержать 40% кислоты. Сколько килограммов кислоты содержится во втором растворе? С полным решением пожалуйста

Answer 1

Обозначим (х) кг кислоты содержится в 48 кг первого сосуда, 
                   (у) кг кислоты содержится в 42 кг второго сосуда.
если их слить вместе, получим (48+42=90) кг раствора, 
содержащего (х+у) кг кислоты --это 42%
90 кг ----- 100%
(х+у) -----   42%
х+у = 90*42/100 = 37.8
если слить равные массы (например, по 1 кг ))) растворов, 
получим 2 кг раствора,
содержащего ((х/48)+(у/42)) кг кислоты --это 40%
2 кг ----- 100%
((х/48)+(у/42)) ----- 40%
((х/48)+(у/42)) = 0.8
получили систему уравнений:
х = 37.8 - у
7х+8у = 0.8*6*7*8
-------------------------
7*37.8 + у = 268.8
у = 268.8 - 264.6 = 4.2 (кг) кислоты было во втором растворе.
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ПРОВЕРКА:
х = 37.8-4.2 = 33.6
48 кг ----- 100%
33.6 -----   р%
р = 3360/48 = 70% раствор был в первом сосуде
42 кг ----- 100%
4.2   -----   р%
р = 420/42 = 10% раствор был во втором сосуде
90 кг ----- 100%
37.8 -----  р%
р = 3780/90 = 378/9 = 42%
в 1 кг 70% раствора содержится 1*0.7 кг кислоты
в 1 кг 10% раствора содержится 1*0.1 кг кислоты
2 кг ----- 100%
0.7+0.1 ----- р%
р = 80/2 = 40%

Answer 2

Problem Analysis

We have two vessels containing solutions of acid with different concentrations. When these solutions are mixed together, a solution with a certain acid concentration is obtained. Additionally, if equal masses of these solutions are mixed, a solution with a different acid concentration is obtained. We need to determine the amount of acid in the second solution.

Solution

Let's assume that the first vessel contains x kg of acid solution with a concentration of a% and the second vessel contains y kg of acid solution with a concentration of b%.

According to the problem statement, when the two solutions are mixed together, a solution with a concentration of 42% acid is obtained. This can be expressed as:

(x + y) * 42/100 = x * a/100 + y * b/100

Simplifying the equation, we get:

42(x + y) = ax + by

Similarly, when equal masses of the two solutions are mixed, a solution with a concentration of 40% acid is obtained. This can be expressed as:

(x/2 + y/2) * 40/100 = x * a/100 + y * b/100

Simplifying the equation, we get:

20(x + y) = ax + by

We now have a system of two equations with two variables. We can solve this system to find the values of x and y.

Let's solve the system of equations:

42(x + y) = ax + by (Equation 1) 20(x + y) = ax + by (Equation 2)

To eliminate the variables, we can subtract Equation 2 from Equation 1:

42(x + y) - 20(x + y) = ax + by - (ax + by)

Simplifying, we get:

22(x + y) = 0

Dividing both sides by 22, we get:

x + y = 0

Since the sum of x and y is zero, it means that x = -y.

Now, let's substitute this value in Equation 1:

42(x + y) = ax + by

42(x + (-x)) = ax + b(-x)

42(0) = ax - bx

0 = (a - b)x

Since x cannot be zero (as it represents the mass of the acid solution in the first vessel), we can conclude that a - b = 0.

Therefore, a = b.

This means that the concentrations of the two acid solutions are equal.

Now, let's substitute a = b in Equation 1:

42(x + y) = ax + by

42(x + y) = ax + ax

42(x + y) = 2ax

Dividing both sides by 2x, we get:

21 = a

Since a represents the concentration of the acid solution, we can conclude that the concentration of the acid in both vessels is 21%.

To find the amount of acid in the second solution, we need to determine the mass of the second solution. Since the mass of the first solution is 48 kg, the mass of the second solution is also 48 kg.

Therefore, the amount of acid in the second solution is:

48 kg * 21% = 10.08 kg.

Answer

The second solution contains 10.08 kg of acid.

Verification

To verify the solution, let's check if the given conditions are satisfied.

When the two solutions are mixed together, the resulting solution has a concentration of 42% acid. Let's calculate the concentration of the resulting solution:

(x + y) * 42/100 = x * a/100 + y * b/100

(48 + 48) * 42/100 = 48 * 21/100 + 48 * 21/100

96 * 42/100 = 48 * 21/100 + 48 * 21/100

40.32 = 10.08 + 10.08

40.32 = 20.16

The equation is satisfied.

When equal masses of the two solutions are mixed, the resulting solution has a concentration of 40% acid. Let's calculate the concentration of the resulting solution:

(x/2 + y/2) * 40/100 = x * a/100 + y * b/100

(48/2 + 48/2) * 40/100 = 48 * 21/100 + 48 * 21/100

24 * 40/100 = 48 * 21/100 + 48 * 21/100

9.6 = 4.032 + 4.032

9.6 = 8.064

The equation is satisfied.

Therefore, the solution is verified.