What is the ratio in which two sugar solutions of concentrations 15% and 40% are to be mixed to get a solution of concentration 30% ?

Answer

To solve the problem of mixing two sugar solutions of concentrations 15% and 40% to obtain a solution of concentration 30%, we can follow these steps:

### Step-by-Step Solution:

1. **Define Variables**:
Let the amount of the 15% sugar solution be \( x \) and the amount of the 40% sugar solution be \( 1 - x \) (considering the total amount of the mixture as 1 unit).

2. **Set Up the Equation**:
The total concentration of the final mixture can be expressed as:
\[ \text{Concentration of 15% solution} \times \text{Amount of 15% solution} + \text{Concentration of 40% solution} \times \text{Amount of 40% solution} = \text{Final concentration} \times \text{Total amount} \]
This can be written as:
\[ 0.15x + 0.4(1 - x) = 0.3 \]

3. **Expand and Simplify the Equation**:
Expanding the equation gives:
\[ 0.15x + 0.4 - 0.4x = 0.3 \]
Combining like terms results in:
\[ -0.25x + 0.4 = 0.3 \]

4. **Isolate \( x \)**:
Subtract 0.4 from both sides:
\[ -0.25x = 0.3 - 0.4 \]
\[ -0.25x = -0.1 \]
Dividing both sides by -0.25 gives:
\[ x = \frac{-0.1}{-0.25} = 0.4 \]

5. **Find the Amount of Each Solution**:
Since \( x = 0.4 \), the amount of the 15% solution is 0.4 and the amount of the 40% solution is:
\[ 1 - x = 1 - 0.4 = 0.6 \]

6. **Calculate the Ratio**:
The ratio of the two solutions is:
\[ \text{Ratio} = \frac{0.4}{0.6} = \frac{4}{6} = \frac{2}{3} \]

### Final Answer:
The ratio in which the two sugar solutions of concentrations 15% and 40% are to be mixed to get a solution of concentration 30% is **2:3**.

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