The sum of two numbers is 4000. 10% of one number is `6 (2)/(3)` % of the other. The difference of the

Answer

To solve the problem step by step, we will follow the information given in the question carefully.

### Step 1: Define the Variables
Let the first number be \( x \). Since the sum of the two numbers is 4000, the second number will be:
\[ 4000 - x \]

**Hint:** Define variables for the unknowns in the problem to make calculations easier.

### Step 2: Set Up the Equation
According to the problem, 10% of the first number is equal to \( 6 \frac{2}{3} \% \) of the second number. We can express this mathematically as:
\[ 0.1x = \left(6 \frac{2}{3}\% \text{ of } (4000 - x)\right) \]
First, convert \( 6 \frac{2}{3} \% \) into a decimal:
\[ 6 \frac{2}{3} = \frac{20}{3} \quad \text{so} \quad \frac{20}{3}\% = \frac{20}{3 \times 100} = \frac{20}{300} = \frac{1}{15} \]
Thus, the equation becomes:
\[ 0.1x = \frac{1}{15}(4000 - x) \]

**Hint:** Convert percentages to decimals or fractions to simplify calculations.

### Step 3: Clear the Equation
Multiply both sides of the equation by 15 to eliminate the fraction:
\[ 15 \times 0.1x = 4000 - x \]
This simplifies to:
\[ 1.5x = 4000 - x \]

**Hint:** Multiplying through by a common denominator can help eliminate fractions.

### Step 4: Solve for \( x \)
Now, add \( x \) to both sides:
\[ 1.5x + x = 4000 \]
This simplifies to:
\[ 2.5x = 4000 \]
Now, divide both sides by 2.5:
\[ x = \frac{4000}{2.5} = 1600 \]

**Hint:** When solving for a variable, isolate it by performing inverse operations.

### Step 5: Find the Second Number
Now that we have \( x \), we can find the second number:
\[ 4000 - x = 4000 - 1600 = 2400 \]

**Hint:** Use the relationship between the two numbers to find the second variable.

### Step 6: Calculate the Difference
Finally, we need to find the difference between the two numbers:
\[ 2400 - 1600 = 800 \]

**Hint:** The difference can be found by subtracting the smaller number from the larger number.

### Final Answer
The difference between the two numbers is \( 800 \).