Let's break down this mixture problem step by step to find the final ratio of acid and water in the new mixture.

We are given two bottles containing mixtures of acid and water in different ratios. These two mixtures are then combined in a specific ratio.

Analyzing the Initial Mixtures

We have two bottles with the following acid to water ratios:

• Bottle 1: Acid : Water = 2 : 3
• Bottle 2: Acid : Water = 1 : 2

Let's determine the fraction of acid and water in each bottle.

• Bottle 1:
  • Total parts = 2 (acid) + 3 (water) = 5 parts
  • Fraction of Acid = \(\frac{2}{5}\)
  • Fraction of Water = \(\frac{3}{5}\)
• Bottle 2:
  • Total parts = 1 (acid) + 2 (water) = 3 parts
  • Fraction of Acid = \(\frac{1}{3}\)
  • Fraction of Water = \(\frac{2}{3}\)

The problem states that these two mixtures are mixed in the ratio 1 : 3. This means for every 1 unit quantity taken from Bottle 1, 3 units quantity are taken from Bottle 2 to form the new mixture.

Calculating Acid and Water in the New Mixture

Let's assume we take 1 unit of mixture from Bottle 1 and 3 units of mixture from Bottle 2.

From Bottle 1 (1 unit):

• Amount of Acid = \(1 \times \text{Fraction of Acid in Bottle 1} = 1 \times \frac{2}{5} = \frac{2}{5}\)
• Amount of Water = \(1 \times \text{Fraction of Water in Bottle 1} = 1 \times \frac{3}{5} = \frac{3}{5}\)

From Bottle 2 (3 units):

• Amount of Acid = \(3 \times \text{Fraction of Acid in Bottle 2} = 3 \times \frac{1}{3} = 1\)
• Amount of Water = \(3 \times \text{Fraction of Water in Bottle 2} = 3 \times \frac{2}{3} = 2\)

Now, let's find the total amount of acid and total amount of water in the new mixture by summing the amounts from both bottles.

Total Amount of Acid in the new mixture:

\(\text{Total Acid} = \text{Acid from Bottle 1} + \text{Acid from Bottle 2} = \frac{2}{5} + 1\)

To add these, we find a common denominator:

\(\text{Total Acid} = \frac{2}{5} + \frac{5}{5} = \frac{2+5}{5} = \frac{7}{5}\)

Total Amount of Water in the new mixture:

\(\text{Total Water} = \text{Water from Bottle 1} + \text{Water from Bottle 2} = \frac{3}{5} + 2\)

To add these, we find a common denominator:

\(\text{Total Water} = \frac{3}{5} + \frac{10}{5} = \frac{3+10}{5} = \frac{13}{5}\)

Determining the Final Ratio

The ratio of acid to water in the new mixture is the ratio of the total amount of acid to the total amount of water:

\(\text{Ratio (Acid : Water)} = \text{Total Acid} : \text{Total Water}\)

\(\text{Ratio} = \frac{7}{5} : \frac{13}{5}\)

To express this ratio in simpler whole numbers, we can multiply both sides of the ratio by the common denominator, which is 5:

\(\text{Ratio} = \left(\frac{7}{5} \times 5\right) : \left(\frac{13}{5} \times 5\right)\)

\(\text{Ratio} = 7 : 13\)

So, the ratio of acid and water in the new mixture is 7 : 13.

Final Answer Summary

The ratio of acid to water in the new mixture is 7 : 13.

Revision Table: Ratio and Mixture Concepts

Additional Information on Mixture Problems

Mixture problems often involve combining substances or mixtures with different properties (like concentration or price) to create a new mixture. The key is usually to keep track of the total amount of a specific component or property in the final mixture based on its contribution from each source.

• Weighted Average: Mixture problems can sometimes be solved using weighted averages, where the weights are the proportions in which the original mixtures are combined. For example, the final concentration of acid is a weighted average of the concentrations in Bottle 1 and Bottle 2, weighted by the mixing ratio.
• Unit Approach: Assuming a convenient total quantity for the original mixtures (like 1 unit and 3 units as done above, or a total of 4 units) makes calculations straightforward. The final ratio doesn't depend on the absolute quantities mixed, only their ratio.
• Applications: Mixture problems appear in various contexts, such as chemistry (acid solutions, alloys), finance (investments with different interest rates), and everyday life (mixing drinks, calculating average speeds).