If x beakers of 100 ml containing 1: 4 acid-water solution are mixed with y beakers of 200 ml. containing 3:17 acid-water solution then the ratio of acid to water in the resulting mixture be comes 19:91. Find x: y

Options

A. `5:3`
B. `3:5`
C. `7:13`
D. `13:7`

Answer

To solve the problem step by step, we will follow the logic presented in the transcript and break it down into clear steps.

### Given:
- **Beaker 1**: Contains 100 ml of a 1:4 acid-water solution.
- **Beaker 2**: Contains 200 ml of a 3:17 acid-water solution.
- The resulting mixture has a ratio of acid to water of 19:91.

### Step 1: Calculate the amount of acid and water in Beaker 1
- The total volume of Beaker 1 is 100 ml.
- The ratio of acid to water is 1:4, which means:
- Acid = \( \frac{1}{1+4} \times 100 = \frac{1}{5} \times 100 = 20 \) ml
- Water = \( \frac{4}{1+4} \times 100 = \frac{4}{5} \times 100 = 80 \) ml

For **x** beakers:
- Total Acid from x beakers = \( 20x \) ml
- Total Water from x beakers = \( 80x \) ml

### Step 2: Calculate the amount of acid and water in Beaker 2
- The total volume of Beaker 2 is 200 ml.
- The ratio of acid to water is 3:17, which means:
- Acid = \( \frac{3}{3+17} \times 200 = \frac{3}{20} \times 200 = 30 \) ml
- Water = \( \frac{17}{3+17} \times 200 = \frac{17}{20} \times 200 = 170 \) ml

For **y** beakers:
- Total Acid from y beakers = \( 30y \) ml
- Total Water from y beakers = \( 170y \) ml

### Step 3: Calculate the total acid and water in the mixture
- Total Acid = \( 20x + 30y \) ml
- Total Water = \( 80x + 170y \) ml

### Step 4: Set up the ratio of acid to water
According to the problem, the ratio of acid to water in the resulting mixture is given as 19:91. Therefore, we can set up the equation:
\[ \frac{20x + 30y}{80x + 170y} = \frac{19}{91} \]

### Step 5: Cross-multiply to eliminate the fraction
Cross-multiplying gives us:
\[ 91(20x + 30y) = 19(80x + 170y) \]

### Step 6: Expand both sides
Expanding both sides results in:
\[ 1820x + 2730y = 1520x + 3230y \]

### Step 7: Rearrange the equation
Rearranging the equation to isolate terms involving x and y:
\[ 1820x - 1520x = 3230y - 2730y \]
\[ 300x = 500y \]

### Step 8: Simplify to find the ratio x:y
Dividing both sides by 100 gives:
\[ 3x = 5y \quad \Rightarrow \quad \frac{x}{y} = \frac{5}{3} \]

Thus, the ratio \( x:y \) is \( 5:3 \).

### Final Answer
The ratio of x to y is \( 5:3 \).

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