First, let us define the total sum of money as [tex]\( T \)[/tex].
### Step 1: Determine Ali's Share
Ali received 45% of the total sum of money, [tex]\( T \)[/tex]. Therefore, Ali's share can be expressed as:
[tex]\[ \text{Ali's share} = 0.45T \][/tex]
### Step 2: Set Up the Ratio between Ali's and Bala's Shares
We know the ratio of Ali's share to Bala's share is 9:7. Therefore, if we let Ali's share be [tex]\( 9x \)[/tex], then Bala's share would be [tex]\( 7x \)[/tex].
Since Ali's share is also given by [tex]\( 0.45T \)[/tex], we have:
[tex]\[ 9x = 0.45T \][/tex]
[tex]\[ x = \frac{0.45T}{9} \][/tex]
[tex]\[ x = \frac{0.05T}{1} \][/tex]
Now, Bala's share can be given as:
[tex]\[ \text{Bala's share} = 7x = 7 \left(\frac{0.05T}{1}\right) = 0.35T \][/tex]
### Step 3: Express Catherine's Share
Catherine's share is given directly as [tex]$400. Therefore:
\[ \text{Catherine's share} = 400 \]
### Step 4: Total Sum of Money Equation
The total sum of money \( T \) is the sum of the shares of Ali, Bala, and Catherine:
\[ \text{Total sum of money} = \text{Ali's share} + \text{Bala's share} + \text{Catherine's share} \]
\[ T = 0.45T + 0.35T + 400 \]
Combine the shares:
\[ T = 0.80T + 400 \]
Solve for \( T \):
\[ T - 0.80T = 400 \]
\[ 0.20T = 400 \]
\[ T = \frac{400}{0.20} \]
\[ T = 2000 \]
### Step 5: Determine Ali's Share
Now that we know the total sum of money \( T = 2000 \), we can calculate Ali's share:
\[ \text{Ali's share} = 0.45 \times 2000 = 900 \]
### Step 6: Calculate the Difference between Ali and Catherine's Shares
The difference between the amount received by Ali and Catherine is:
\[ \text{Difference} = 900 - 400 = 500 \]
Therefore, the difference between the amount received by Ali and Catherine is \( \$[/tex]500 \). The correct answer is:
[tex]\[ \boxed{500} \][/tex]
