How many positive integer x are there such that 3x has 3 digits and 4x has four digits ?

Answer

To solve the problem of how many positive integers \( x \) satisfy the conditions that \( 3x \) has 3 digits and \( 4x \) has 4 digits, we can break it down into steps.

### Step 1: Determine the range for \( 3x \)
We know that \( 3x \) must be a 3-digit number. The smallest 3-digit number is 100, and the largest is 999. Therefore, we can set up the inequality:
\[ 100 \leq 3x \leq 999 \]

### Step 2: Solve for \( x \)
To find the range for \( x \), we divide the entire inequality by 3:
\[ \frac{100}{3} \leq x \leq \frac{999}{3} \]
Calculating the values:
\[ \frac{100}{3} \approx 33.33 \quad \text{and} \quad \frac{999}{3} = 333 \]
Thus, we have:
\[ 34 \leq x \leq 333 \]
(We take the ceiling of \( 33.33 \) since \( x \) must be a positive integer.)

### Step 3: Determine the range for \( 4x \)
Next, we need \( 4x \) to be a 4-digit number. The smallest 4-digit number is 1000, and the largest is 9999. Therefore, we set up another inequality:
\[ 1000 \leq 4x \leq 9999 \]

### Step 4: Solve for \( x \) again
Dividing the entire inequality by 4 gives us:
\[ \frac{1000}{4} \leq x \leq \frac{9999}{4} \]
Calculating the values:
\[ \frac{1000}{4} = 250 \quad \text{and} \quad \frac{9999}{4} \approx 2499.75 \]
Thus, we have:
\[ 250 \leq x \leq 2499 \]

### Step 5: Find the intersection of the two ranges
Now we need to find the common values of \( x \) that satisfy both inequalities:
1. From \( 3x \): \( 34 \leq x \leq 333 \)
2. From \( 4x \): \( 250 \leq x \leq 2499 \)

The intersection of these two ranges is:
\[ 250 \leq x \leq 333 \]

### Step 6: Count the integers in the intersection
To find the number of integers in the range \( 250 \) to \( 333 \), we can use the formula for counting integers in a range:
\[ \text{Count} = \text{Last} - \text{First} + 1 \]
Substituting the values:
\[ \text{Count} = 333 - 250 + 1 = 84 \]

### Final Answer
Thus, the number of positive integers \( x \) such that \( 3x \) has 3 digits and \( 4x \) has 4 digits is \( \boxed{84} \).