A solution of sugar syrup has `15%` sugar. Another solution has `5%` sugar.How many litre of the second solution must be added to 20 litres of the first solution to make a solution of `10%` sugar ?

Options

A. 10
B. 5
C. 15
D. 20

Answer

To solve the problem step by step, we need to determine how many liters of the second solution (5% sugar) must be added to 20 liters of the first solution (15% sugar) to create a new solution that has a sugar concentration of 10%.

### Step 1: Define Variables
Let \( A \) be the amount of the second solution (5% sugar) that we need to add in liters.

### Step 2: Calculate the Amount of Sugar in Each Solution
1. **Sugar in the first solution (15% sugar)**:
- Amount of sugar in 20 liters of the first solution:
\[ \text{Sugar from first solution} = 15\% \text{ of } 20 = \frac{15}{100} \times 20 = 3 \text{ liters} \]

2. **Sugar in the second solution (5% sugar)**:
- Amount of sugar in \( A \) liters of the second solution:
\[ \text{Sugar from second solution} = 5\% \text{ of } A = \frac{5}{100} \times A = 0.05A \text{ liters} \]

### Step 3: Set Up the Equation for the Final Solution
The total amount of sugar in the final mixture (which has a total volume of \( 20 + A \) liters) should equal 10% of that total volume:
\[ \text{Total sugar} = \text{Sugar from first solution} + \text{Sugar from second solution} \]
\[ \text{Total sugar} = 3 + 0.05A \]
The concentration of sugar in the final solution should be:
\[ 10\% \text{ of } (20 + A) = \frac{10}{100} \times (20 + A) = 0.1(20 + A) \]

### Step 4: Set Up the Equation
Now, we can set up the equation:
\[ 3 + 0.05A = 0.1(20 + A) \]

### Step 5: Simplify the Equation
Expanding the right side:
\[ 3 + 0.05A = 2 + 0.1A \]

### Step 6: Rearrange the Equation
Rearranging gives:
\[ 3 - 2 = 0.1A - 0.05A \]
\[ 1 = 0.05A \]

### Step 7: Solve for \( A \)
Now, divide both sides by 0.05:
\[ A = \frac{1}{0.05} = 20 \]

### Conclusion
Thus, the amount of the second solution that must be added is \( A = 20 \) liters.

### Final Answer
**20 liters of the second solution must be added.**

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