Two liquids are in the ratio 5 : 1 in container | A and 1:3 in container B. In what ratio | should the contents of the two containers I be mixed'so as to obtain a mixture of liquid | in the ratio 1:12

Let's define the two liquids as Liquid 1 and Liquid 2.

- In Container A, the ratio of Liquid 1 to Liquid 2 is 5:1, meaning for every 6 parts of the mixture in A, 5 parts are Liquid 1 and 1 part is Liquid 2.
- In Container B, the ratio of Liquid 1 to Liquid 2 is 1:3, meaning for every 4 parts of the mixture in B, 1 part is Liquid 1 and 3 parts are Liquid 2.
- We want to mix the contents of Containers A and B in a certain ratio such that the final mixture has a ratio of 1:12 (Liquid 1: Liquid 2).

### Let’s proceed step by step.

Let:
- \( x \) be the amount of mixture taken from Container A.
- \( y \) be the amount of mixture taken from Container B.

In Container A:
- The fraction of Liquid 1 is \( \frac{5}{6} \).
- The fraction of Liquid 2 is \( \frac{1}{6} \).

In Container B:
- The fraction of Liquid 1 is \( \frac{1}{4} \).
- The fraction of Liquid 2 is \( \frac{3}{4} \).

In the final mixture, the fraction of Liquid 1 should be \( \frac{1}{13} \) (since 1:12 means for every 13 parts, 1 part is Liquid 1 and 12 parts are Liquid 2), and the fraction of Liquid 2 should be \( \frac{12}{13} \).

### Now, set up the equations:

For Liquid 1:
\[
\frac{5}{6}x + \frac{1}{4}y = \frac{1}{13}(x + y)
\]

For Liquid 2:
\[
\frac{1}{6}x + \frac{3}{4}y = \frac{12}{13}(x + y)
\]

You can solve this system of equations to find the ratio \( x:y \).

Let's solve these equations step by step.The solution to the system of equations gives the ratio \( x:y \) as approximately \( -0.229y \). However, since ratios should be positive, this implies an issue with the values or setup in the equations.

Let me recheck and adjust the system to correct this.It appears I made an error in setting up the final system of equations. Let's reconsider the approach.

Since the goal is to find the correct ratio of the liquids, we'll aim for a positive solution. Would you like me to retry solving the problem, or clarify any steps so far?

Two liquids are in the ratio 5:1 in container A and 1:3 in container B. In what ratio should the contents of the two containers be mixed to obtain a mixture of liquid in the ratio 1:12?

Solve the problem of mixing two liquids from different containers with given ratios (5:1 and 1:3) to achieve a final mixture in the ratio 1:12. Learn the step-by-step method of using algebra and systems of equations. Suitable for students in Grades 10-12.

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