A 19-liter mixture consists by volume of 1 part juice to 18 parts water. If x liters of juice and y liters of water are added to this mixture to make a 54-liter mixture consisting by volume of 1 part juice to 2 parts water, what is the value of x. I'm not sure, but would it be 27?

Question

Visualize an image depicting an open book on a wooden table. On the left page of the book, show a 15th century-style sketch of a 19-liter jug, with a small portion coloured to indicate juice and the remainder indicating water. On the right page of the book, depict a larger, 54-liter jug with three distinct sections, each one-third of the total capacity, to represent the new composition of juice to water. Make sure no numbers or text are included in the image.

A 19-liter mixture consists by volume of 1 part juice to 18 parts water. If x liters of juice and y liters of water are added to this mixture to make a 54-liter mixture consisting by volume of 1 part juice to 2 parts water, what is the value of x. I'm not sure, but would it be 27?

Answer 1

let x = volume of juice

let y = volume of water

Volume of Mixture
(x + 1) + (y + 18) = 54

solve for y

y = 35 - x

"1 part juice to 2 parts water"
ratio of juice to water = 1/2

(x + 1) / (y + 18) = 1/2

(x + 1) / ([35 - x] + 18) = 1/2

solve for x

x + 1 = 1/2 (53 - x)
2x + 2 = 53 - x
3x + 2 = 53
3x = 51
x = 17

Answer 2

Well, let's see if your guess is juicily correct or just a bit watered down! To find the value of x, we can set up an equation using the information given.

Initially, we have a 19-liter mixture that consists of 1 part juice to 18 parts water. Since the total volume of the mixture is 19 liters, the amount of juice present is (1/19) * 19 = 1 liter.

Now, we want to add x liters of juice and y liters of water to this mixture to make a 54-liter mixture consisting of 1 part juice to 2 parts water. This means that the amount of juice in the final mixture should be (1/3) * 54 = 18 liters.

Since we started with 1 liter of juice and added x liters to it, the total amount of juice in the final mixture will be 1 + x liters.

Setting up an equation, we have:

1 + x = 18

Solving for x, we find:

x = 18 - 1

So, x = 17 liters.

Therefore, my fruity friend, the value of x is 17, not 27. It seems like your guess was just a little squeezed!

Answer 3

To solve this problem, we can set up an equation based on the given information.

First, let's calculate the initial amount of juice and water in the 19-liter mixture.

The ratio of juice to water in the 19-liter mixture is 1:18.

So, the amount of juice in the 19-liter mixture is (1/19) * 19 = 1 liter.

And the amount of water in the 19-liter mixture is (18/19) * 19 = 18 liters.

Now, let's consider the 54-liter mixture.

The ratio of juice to water in the 54-liter mixture is 1:2.

So, the amount of juice in the 54-liter mixture is (1/3) * 54 = 18 liters.

And the amount of water in the 54-liter mixture is (2/3) * 54 = 36 liters.

To find the amount of juice (x) and water (y) added to the 19-liter mixture, we need to find the difference between the amounts of juice and water between the two mixtures.

The change in juice is x - 1 liters.

The change in water is y - 18 liters.

Since the ratio of juice to water in both mixtures is the same, we can set up the following equation:

(x - 1) / (y - 18) = 1 / 18

Now, let's solve this equation to find the value of x.

Cross-multiplying the equation, we get:

18(x - 1) = y - 18

Expanding the equation, we have:

18x - 18 = y - 18

Simplifying further, we obtain:

18x = y

Since we know that the amount of juice in the 54-liter mixture is 18 liters, we can substitute y with 36 in the equation:

18x = 36

Dividing both sides of the equation by 18, we get:

x = 2

Therefore, the value of x is 2 liters, not 27.

Answer 4

To solve this question, we can set up equations based on the given information.

Let's denote the amount of juice added (x liters) and the amount of water added (y liters) to make a 54-liter mixture.

First, let's calculate the amount of juice and water in the initial 19-liter mixture:
- According to the given information, the mixture consists of 1 part juice to 18 parts water. So, the amount of juice in the 19-liter mixture is (1/19) * 19 = 1 liter.
- The amount of water in the 19-liter mixture is 19 - 1 = 18 liters.

Now, let's consider the final 54-liter mixture:
- The mixture consists of 1 part juice to 2 parts water. So, the amount of juice in the 54-liter mixture is (1/3) * 54 = 18 liters, since the ratio of juice to water is 1:2.
- The amount of water in the 54-liter mixture is 54 - 18 = 36 liters.

Since we know that the amount of juice in the final mixture is obtained by adding x liters of juice to the initial mixture, we can write the following equation:
1 + x = 18.

Solving this equation, we find that x = 18 - 1, so x = 17.

Therefore, the value of x (the amount of juice added) is 17 liters.

Note: Your initial guess of 27 liters is not correct, but the correct answer is 17 liters.