A and B start moving from a place X to Y and Y to X , respectively , at the same time on the same day . After crossing each other ,they complete their remaining joumeys in `5(4)/(9)` hours and x hours ,respectively .If the ratio of the speeds (in km/h) of A and B is `9 :7` ,then what is the value of x ?

Options

A. `8(1)/(2)`
B. 9
C. 7
D. `7(1)/(2)`

Answer

To solve the problem step by step, we will use the relationship between speed, time, and distance, particularly focusing on the time taken by A and B after they cross each other.

### Step 1: Understand the Given Information
- A and B start from points X and Y respectively and move towards each other.
- After they cross each other, A takes \(5 \frac{4}{9}\) hours to reach Y, and B takes \(x\) hours to reach X.
- The ratio of their speeds is given as \(9:7\).

### Step 2: Convert Mixed Fraction to Improper Fraction
Convert \(5 \frac{4}{9}\) into an improper fraction:
\[ 5 \frac{4}{9} = \frac{5 \times 9 + 4}{9} = \frac{45 + 4}{9} = \frac{49}{9} \]

### Step 3: Set Up the Ratio of Speeds and Times
Using the formula that relates speeds and times:
\[ \frac{v_A}{v_B} = \frac{\sqrt{t_B}}{\sqrt{t_A}} \]
Where:
- \(v_A\) and \(v_B\) are the speeds of A and B respectively.
- \(t_A = \frac{49}{9}\) (time taken by A after crossing).
- \(t_B = x\) (time taken by B after crossing).

Given the ratio of speeds:
\[ \frac{9}{7} = \frac{\sqrt{x}}{\sqrt{\frac{49}{9}}} \]

### Step 4: Simplify the Right Side
Calculate \(\sqrt{\frac{49}{9}}\):
\[ \sqrt{\frac{49}{9}} = \frac{7}{3} \]

Now substitute this back into the equation:
\[ \frac{9}{7} = \frac{\sqrt{x}}{\frac{7}{3}} \]

### Step 5: Cross Multiply to Solve for \(\sqrt{x}\)
Cross multiplying gives:
\[ 9 \cdot \frac{7}{3} = 7 \cdot \sqrt{x} \]
\[ \frac{63}{3} = 7 \cdot \sqrt{x} \]
\[ 21 = 7 \cdot \sqrt{x} \]

### Step 6: Isolate \(\sqrt{x}\)
Divide both sides by 7:
\[ \sqrt{x} = \frac{21}{7} = 3 \]

### Step 7: Square Both Sides to Find \(x\)
\[ x = 3^2 = 9 \]

### Final Answer
The value of \(x\) is \(9\).

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