P and Q are two alloys which are prepared by mixing tin and lead in the ratio of 12 : 5 and 4 : 3 respectively. If equal quantities of alloys are melted to form a third alloy R, then what is the ratio of tin and lead in alloy R?

Options

A. 84 : 65
B. 57 : 35
C. 76 : 43
D. 78 : 47

Explanation

To solve the problem of finding the ratio of tin and lead in alloy R formed by melting equal quantities of alloys P and Q, we can follow these steps:

### Step 1: Understand the Ratios of Alloys P and Q
Alloy P is made of tin and lead in the ratio of 12:5. This means for every 12 parts of tin, there are 5 parts of lead.

Alloy Q is made of tin and lead in the ratio of 4:3. This means for every 4 parts of tin, there are 3 parts of lead.

### Step 2: Define Equal Quantities for Mixing
Let’s assume we take 1 unit of each alloy for mixing. This will help us maintain equal quantities of both alloys.

### Step 3: Calculate the Amount of Tin and Lead in Alloy P
For alloy P (12:5):
- Total parts = 12 + 5 = 17 parts
- Amount of tin = (12/17) * 1 unit = 12/17 units
- Amount of lead = (5/17) * 1 unit = 5/17 units

### Step 4: Calculate the Amount of Tin and Lead in Alloy Q
For alloy Q (4:3):
- Total parts = 4 + 3 = 7 parts
- Amount of tin = (4/7) * 1 unit = 4/7 units
- Amount of lead = (3/7) * 1 unit = 3/7 units

### Step 5: Find a Common Denominator
To add the amounts of tin and lead from both alloys, we need a common denominator. The least common multiple of 17 and 7 is 119.

### Step 6: Convert Amounts to Common Denominator
Convert the amounts of tin and lead to have a common denominator of 119:
- For alloy P:
- Tin: (12/17) = (12 * 7)/(17 * 7) = 84/119 units
- Lead: (5/17) = (5 * 7)/(17 * 7) = 35/119 units

- For alloy Q:
- Tin: (4/7) = (4 * 17)/(7 * 17) = 68/119 units
- Lead: (3/7) = (3 * 17)/(7 * 17) = 51/119 units

### Step 7: Add the Amounts of Tin and Lead
Now, we can add the amounts of tin and lead from both alloys:
- Total tin = 84/119 + 68/119 = (84 + 68)/119 = 152/119 units
- Total lead = 35/119 + 51/119 = (35 + 51)/119 = 86/119 units

### Step 8: Form the Ratio of Tin to Lead in Alloy R
Now, we can form the ratio of tin to lead in alloy R:
- Ratio of tin to lead = Total tin : Total lead = 152 : 86

### Step 9: Simplify the Ratio
To simplify the ratio, we can divide both sides by 2:
- 152 ÷ 2 = 76
- 86 ÷ 2 = 43

Thus, the ratio of tin to lead in alloy R is **76 : 43**.