A cup of milk containing sugar and pure milk in the ratio 3: 2 respectively is mixed with another cup of milk of same volume as the earlier and containing 25% of sugar. Determine the ratio of total quantity of sugar to the total quantity of pure milk.

Let's break down the problem and solve it step-by-step.

### Given:
1. **First cup of milk:**
   - The ratio of sugar to pure milk is 3:2.
   - So, for every 3 parts of sugar, there are 2 parts of pure milk.
   - Let the total volume of this cup be \( V \). Therefore, the sugar in the first cup is \( \frac{3}{5} \times V \) and the pure milk is \( \frac{2}{5} \times V \).

2. **Second cup of milk:**
   - The total volume is the same as the first cup, i.e., \( V \).
   - This cup contains 25% sugar, meaning \( \frac{25}{100} = 0.25 \) of the volume is sugar and \( 0.75 \) of the volume is pure milk.
   - So, sugar in the second cup is \( 0.25 \times V \) and pure milk is \( 0.75 \times V \).

### Total quantities of sugar and pure milk:
- Total sugar:
  \[
  \text{Sugar} = \left( \frac{3}{5} \times V \right) + \left( 0.25 \times V \right)
  = \frac{3}{5}V + 0.25V = \frac{3}{5}V + \frac{1}{4}V
  \]
  To add these fractions, find the common denominator (which is 20):
  \[
  \text{Sugar} = \frac{12}{20}V + \frac{5}{20}V = \frac{17}{20}V
  \]

- Total pure milk:
  \[
  \text{Pure milk} = \left( \frac{2}{5} \times V \right) + \left( 0.75 \times V \right)
  = \frac{2}{5}V + 0.75V = \frac{2}{5}V + \frac{3}{4}V
  \]
  Again, find the common denominator (which is 20):
  \[
  \text{Pure milk} = \frac{8}{20}V + \frac{15}{20}V = \frac{23}{20}V
  \]

### Ratio of total sugar to total pure milk:
The ratio of total sugar to total pure milk is:
\[
\text{Ratio} = \frac{\frac{17}{20}V}{\frac{23}{20}V} = \frac{17}{23}
\]

Thus, the ratio of the total quantity of sugar to the total quantity of pure milk is **17:23**.

Would you like more details, or do you have any questions?

Here are 5 related questions for further practice:
1. If the volume of each cup was 10 liters, how much sugar is there in total?
2. What would the ratio be if the second cup had 30% sugar instead?
3. How does changing the ratio of sugar to milk in the first cup affect the final ratio?
4. If both cups had different volumes, how would that change the final ratio?
5. Can you find the percentage of sugar in the final mixture?

**Tip:** When solving ratio problems, always try to express each component in terms of a common variable to simplify calculations.

A cup of milk containing sugar and pure milk in the ratio 3:2 respectively is mixed with another cup of milk of same volume as the earlier and containing 25% of sugar. Determine the ratio of total quantity of sugar to the total quantity of pure milk.

Learn how to calculate the ratio of total sugar to total pure milk when mixing two cups of milk with different sugar contents. This step-by-step solution covers ratio concepts, fraction addition, and percentage calculations. Suitable for students in Grades 8-10.

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