If `4(1)/(a)xxb(2)/(3)=7` find the values of a and b.

Options

A. 1,5
B. 2,3
C. 3,2
D. 5,1

Answer

To solve the equation \( 4\left(\frac{1}{a}\right) \times b\left(\frac{2}{3}\right) = 7 \), we will follow these steps:

### Step 1: Rewrite the equation
First, we rewrite the fractions in a more manageable form:
\[ 4 \cdot \frac{1}{a} \cdot b \cdot \frac{2}{3} = 7 \]
This can be simplified to:
\[ \frac{8b}{3a} = 7 \]

### Step 2: Cross-multiply
Next, we cross-multiply to eliminate the fraction:
\[ 8b = 21a \]

### Step 3: Rearranging the equation
We can rearrange this equation to express \( b \) in terms of \( a \):
\[ b = \frac{21a}{8} \]

### Step 4: Substitute values
Now, we can substitute different values for \( a \) to find corresponding values for \( b \). We will check integer values for \( a \) to find valid pairs.

### Step 5: Testing values
1. Let’s try \( a = 1 \):
\[ b = \frac{21 \cdot 1}{8} = \frac{21}{8} \quad \text{(not an integer)} \]

2. Let’s try \( a = 2 \):
\[ b = \frac{21 \cdot 2}{8} = \frac{42}{8} = \frac{21}{4} \quad \text{(not an integer)} \]

3. Let’s try \( a = 3 \):
\[ b = \frac{21 \cdot 3}{8} = \frac{63}{8} \quad \text{(not an integer)} \]

4. Let’s try \( a = 4 \):
\[ b = \frac{21 \cdot 4}{8} = \frac{84}{8} = \frac{21}{2} \quad \text{(not an integer)} \]

5. Let’s try \( a = 5 \):
\[ b = \frac{21 \cdot 5}{8} = \frac{105}{8} \quad \text{(not an integer)} \]

6. Let’s try \( a = 8 \):
\[ b = \frac{21 \cdot 8}{8} = 21 \quad \text{(integer)} \]

### Step 6: Conclusion
The values we found are:
\[ a = 8, \quad b = 21 \]