The proportion of acid and water in three samples is 2:1, 3:2 and 5:3. A mixture containing equal quantities of all three samples is made. The ratio of water and acid in the mixture is:

Options

A. `120:133`
B. `227:133`
C. `227:120`
D. `133:227`

Answer

To solve the problem of finding the ratio of water and acid in a mixture made from three samples with given proportions of acid and water, we can follow these steps:

### Step 1: Understand the Ratios
The ratios of acid to water in the three samples are:
- Sample 1: Acid:Water = 2:1
- Sample 2: Acid:Water = 3:2
- Sample 3: Acid:Water = 5:3

### Step 2: Express the Ratios in Parts
From the ratios, we can express the amounts of acid and water in terms of parts:
- Sample 1: 2 parts acid and 1 part water
- Sample 2: 3 parts acid and 2 parts water
- Sample 3: 5 parts acid and 3 parts water

### Step 3: Calculate Total Parts for Each Sample
Now, we calculate the total parts for each sample:
- Sample 1: Total = 2 + 1 = 3 parts
- Sample 2: Total = 3 + 2 = 5 parts
- Sample 3: Total = 5 + 3 = 8 parts

### Step 4: Find the LCM of Total Parts
To mix equal quantities of all three samples, we need to find the least common multiple (LCM) of the total parts (3, 5, and 8):
- LCM(3, 5, 8) = 120

### Step 5: Determine the Quantity of Each Sample
Next, we determine how many times we need to multiply each sample to get 120 parts:
- Sample 1: \( \frac{120}{3} = 40 \) (multiply by 40)
- Sample 2: \( \frac{120}{5} = 24 \) (multiply by 24)
- Sample 3: \( \frac{120}{8} = 15 \) (multiply by 15)

### Step 6: Calculate the Amount of Acid and Water in Each Sample
Now we calculate the total amount of acid and water from each sample:
- Sample 1:
- Acid = \( 2 \times 40 = 80 \)
- Water = \( 1 \times 40 = 40 \)

- Sample 2:
- Acid = \( 3 \times 24 = 72 \)
- Water = \( 2 \times 24 = 48 \)

- Sample 3:
- Acid = \( 5 \times 15 = 75 \)
- Water = \( 3 \times 15 = 45 \)

### Step 7: Total Acid and Water in the Mixture
Now, we sum the amounts of acid and water from all samples:
- Total Acid = \( 80 + 72 + 75 = 227 \)
- Total Water = \( 40 + 48 + 45 = 133 \)

### Step 8: Find the Ratio of Water to Acid
Finally, the ratio of water to acid in the mixture is:
- Ratio = Water:Acid = \( 133:227 \)

### Final Answer
The ratio of water to acid in the mixture is **133:227**.

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