Two beakers A and B contain syrup and water in the ratios 1: 2 and 1:3 respectively. The quantity of mixture in both the beakers is same. If 60% and 80% mixtures from beakers A and B respectively are mixed in beaker C, then what is the ratio of quantity of syrup to that of water in beaker C?

Let's solve this step by step.

### Step 1: Determine the amount of syrup and water in Beakers A and B

- **Beaker A:** The ratio of syrup to water is 1:2.
  - Let the quantity of syrup be \( x \).
  - Then, the quantity of water will be \( 2x \).

- **Beaker B:** The ratio of syrup to water is 1:3.
  - Let the quantity of syrup be \( y \).
  - Then, the quantity of water will be \( 3y \).

### Step 2: Equate the total quantities of mixtures in both beakers

Given that the total quantity in both beakers is the same:
\[
x + 2x = y + 3y
\]
\[
3x = 4y
\]
This implies:
\[
\frac{x}{y} = \frac{4}{3}
\]
So, \( x = \frac{4}{3}y \).

### Step 3: Calculate the amount of syrup and water transferred to Beaker C

- **From Beaker A:**
  - 60% of the mixture is taken.
  - The quantity of syrup in Beaker A taken to C = \( 0.6x \)
  - The quantity of water in Beaker A taken to C = \( 0.6 \times 2x = 1.2x \)

- **From Beaker B:**
  - 80% of the mixture is taken.
  - The quantity of syrup in Beaker B taken to C = \( 0.8y \)
  - The quantity of water in Beaker B taken to C = \( 0.8 \times 3y = 2.4y \)

### Step 4: Combine the mixtures in Beaker C

- **Total syrup in Beaker C:**
  \[
  0.6x + 0.8y
  \]
- **Total water in Beaker C:**
  \[
  1.2x + 2.4y
  \]

Substituting \( x = \frac{4}{3}y \) into the above equations:

- **Total syrup in Beaker C:**
  \[
  0.6\left(\frac{4}{3}y\right) + 0.8y = 0.8y + 0.8y = 1.6y
  \]
- **Total water in Beaker C:**
  \[
  1.2\left(\frac{4}{3}y\right) + 2.4y = 1.6y + 2.4y = 4y
  \]

### Step 5: Find the ratio of syrup to water in Beaker C

The ratio of syrup to water in Beaker C is:
\[
\text{Ratio} = \frac{1.6y}{4y} = \frac{1.6}{4} = \frac{2}{5}
\]

### Final Answer

The ratio of the quantity of syrup to that of water in Beaker C is **2:5**.

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Would you like further details or have any questions?

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**Here are 8 follow-up questions:**

1. How would the ratio change if 50% of the mixtures from both beakers were used?
2. What if the ratios in Beakers A and B were reversed?
3. How does the total quantity of mixture in Beaker C compare to the original beakers?
4. Can you find the ratio of syrup to water in Beaker C if 100% of the mixtures from A and B were combined?
5. What if the beakers had different total quantities, say Beaker A had double the quantity of Beaker B?
6. How would the ratio change if Beaker B had a ratio of 2:5 instead of 1:3?
7. What if only syrup was mixed from Beaker A and only water from Beaker B?
8. Can the final ratio be simplified further if different percentages were used?

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**Tip:** When mixing two substances, always pay attention to the initial ratio and the percentage used from each source to calculate the resulting ratio accurately.

Calculate the ratio of syrup to water in Beaker C when mixing 60% of beaker A and 80% of beaker B. Understand the method for solving mixture problems involving different ratios and percentages. Suitable for students in Grades 7-9.

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