Calculate the 3rd proportional to 16 and 32 .

Answer

To find the 3rd proportional to the numbers 16 and 32, we can follow these steps:

### Step 1: Understand the definition of 3rd proportional
The 3rd proportional to two numbers a and b is a number c such that the ratio of a to b is the same as the ratio of b to c. This can be expressed mathematically as:
\[ \frac{a}{b} = \frac{b}{c} \]

### Step 2: Substitute the values
In this case, we have:
- \( a = 16 \)
- \( b = 32 \)

So, we can set up the equation:
\[ \frac{16}{32} = \frac{32}{c} \]

### Step 3: Cross multiply
Cross multiplying gives us:
\[ 16 \cdot c = 32 \cdot 32 \]

### Step 4: Calculate \( 32 \cdot 32 \)
Now, calculate \( 32 \cdot 32 \):
\[ 32 \cdot 32 = 1024 \]
So, we have:
\[ 16c = 1024 \]

### Step 5: Solve for c
Now, divide both sides by 16 to find c:
\[ c = \frac{1024}{16} \]

### Step 6: Calculate \( \frac{1024}{16} \)
Now, perform the division:
\[ c = 64 \]

### Conclusion
Thus, the 3rd proportional to 16 and 32 is:
\[ \boxed{64} \]

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