If 64 is divided into the ratio of 2: 4: x and smallest part is `10 (2)/(3).` Find x.

Options

A. `5 (1)/(3)`
B. `6`
C. `10 (2)/(3)`
D. `20`

Answer

To solve the problem, we need to find the value of \( x \) in the ratio \( 2:4:x \) when the total of 64 is divided into this ratio and the smallest part is \( 10 \frac{2}{3} \).

### Step-by-Step Solution:

1. **Convert the Mixed Number to an Improper Fraction**:
\[ 10 \frac{2}{3} = \frac{10 \times 3 + 2}{3} = \frac{30 + 2}{3} = \frac{32}{3} \]

**Hint**: To convert a mixed number to an improper fraction, multiply the whole number by the denominator and add the numerator, then divide by the denominator.

2. **Set Up the Ratio**:
The ratio is \( 2:4:x \). The sum of the parts in the ratio is:
\[ 2 + 4 + x = 6 + x \]

**Hint**: When dealing with ratios, always sum the parts to understand how they relate to the total.

3. **Express Each Part in Terms of the Total**:
The total amount is 64. Each part of the ratio can be expressed as:
- First part: \( \frac{2}{6+x} \times 64 \)
- Second part: \( \frac{4}{6+x} \times 64 \)
- Third part (which corresponds to \( x \)): \( \frac{x}{6+x} \times 64 \)

**Hint**: To find the value of each part in a ratio, multiply the fraction of the part by the total amount.

4. **Identify the Smallest Part**:
The smallest part is given as \( \frac{32}{3} \). We can set up the equation for the smallest part, which corresponds to the first part:
\[ \frac{2}{6+x} \times 64 = \frac{32}{3} \]

**Hint**: Identify which part of the ratio corresponds to the smallest value.

5. **Cross-Multiply to Solve for \( x \)**:
\[ 2 \times 64 = \frac{32}{3} \times (6+x) \]
Simplifying gives:
\[ 128 = \frac{32(6+x)}{3} \]
Multiply both sides by 3:
\[ 384 = 32(6+x) \]
Divide both sides by 32:
\[ 12 = 6 + x \]
Subtract 6 from both sides:
\[ x = 6 \]

**Hint**: When solving equations, isolate the variable by performing inverse operations.

6. **Conclusion**:
The value of \( x \) is \( 6 \).

### Final Answer:
\[ x = 6 \]