Find out a number which will satisfy both the blank spaces?
`(21)/(?)=(?)/(2(1)/(3))`

Options

A. `(4)/(3)`
B. `7(1)/(2)`
C. 7
D. `7(2)/(3)`

Explanation

To solve the equation \( \frac{21}{?} = \frac{?}{2 \frac{1}{3}} \), we will denote the unknown number as \( a \).

### Step-by-Step Solution:

1. **Set up the equation**:
We can rewrite the equation as:
\[ \frac{21}{a} = \frac{a}{2 \frac{1}{3}} \]

2. **Convert the mixed number to an improper fraction**:
The mixed number \( 2 \frac{1}{3} \) can be converted to an improper fraction:
\[ 2 \frac{1}{3} = 2 + \frac{1}{3} = \frac{6}{3} + \frac{1}{3} = \frac{7}{3} \]
Now, substitute this back into the equation:
\[ \frac{21}{a} = \frac{a}{\frac{7}{3}} \]

3. **Simplify the right side**:
To simplify \( \frac{a}{\frac{7}{3}} \), we multiply by the reciprocal:
\[ \frac{a}{\frac{7}{3}} = a \cdot \frac{3}{7} = \frac{3a}{7} \]
Now the equation looks like:
\[ \frac{21}{a} = \frac{3a}{7} \]

4. **Cross-multiply**:
Cross-multiplying gives us:
\[ 21 \cdot 7 = 3a \cdot a \]
This simplifies to:
\[ 147 = 3a^2 \]

5. **Solve for \( a^2 \)**:
Divide both sides by 3:
\[ a^2 = \frac{147}{3} = 49 \]

6. **Take the square root**:
Now, take the square root of both sides:
\[ a = \sqrt{49} = 7 \]

Thus, the number that satisfies both blank spaces is \( 7 \).