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The problem in the image involves finding the total sum of money that three people, Aminah, Getha, and Muthu, share, based on given ratios and additional information.

### Let's break down the problem:

1. **Ratio of Aminah's money to the total money of Getha and Muthu:**
   - This ratio is given as $1 : 3$. 
   - If Aminah's amount of money is $A$, then Getha's and Muthu's combined amount is $3A$.

2. **Ratio of Getha's money to the combined amount of Aminah and Muthu:**
   - This ratio is $5 : 11$.
   - If Getha's amount of money is $G$, then the combined amount of Aminah and Muthu is $\frac{11}{5}G$.

3. **Additional information:**
   - Muthu has $51 more than Aminah, meaning $M = A + 51$.

### Solution Approach:

We need to establish a system of equations to find the values of $A$, $G$, and $M$ (Aminah's, Getha's, and Muthu's amounts, respectively).

### Step 1: Express Getha's money in terms of $A$.

From the first ratio $A : (G + M) = 1 : 3$:

\[
A = \frac{G + M}{3}
\]

### Step 2: Express Muthu's money in terms of $A$.

From the additional information $M = A + 51$.

### Step 3: Express $A$ and $M$ in terms of $G$.

Using the second ratio $G : (A + M) = 5 : 11$:

\[
\frac{G}{A + M} = \frac{5}{11} \implies 11G = 5(A + M)
\]

Substitute $M = A + 51$ into the equation:

\[
11G = 5(A + A + 51) = 5(2A + 51) \implies 11G = 10A + 255
\]

### Step 4: Solve for $G$, $A$, and $M$.

Now, let's solve these equations to find $A$, $G$, and $M$, and then calculate the total sum of money.

I will solve these equations.It appears that the algebraic setup led to a zero value for the total sum of money, which isn't possible in a real-world scenario. This suggests that there might have been an error in the equation setup or the assumptions made. Let's review the problem and correct the approach:

1. **Corrected setup:**
   - We know that the ratio of Aminah's money to the combined money of Getha and Muthu is $1:3$.
   - We know the ratio of Getha's money to the combined money of Aminah and Muthu is $5:11$.
   - Muthu has $51 more than Aminah.

2. **Corrected equations:**
   - We should directly use the ratios and additional condition given to correctly set up and solve the system of equations.

I'll revise the approach and recalculate the sum.It seems there was an issue in how the solutions were accessed. Let me correct that and find the correct values for $A$, $G$, and $M$, and then calculate the total sum of money again.It seems there was an issue when attempting to extract the solutions. Instead of proceeding step by step, I'll take a more straightforward approach to re-solve the system of equations directly and correctly identify the values of $A$, $G$, and $M$. Let's start fresh with the equation and obtain the solutions in a simplified manner.It seems like I can’t do more advanced data analysis right now. Please try again later. If you'd like, you can describe another problem, or if you have any additional questions about the math problem, I'd be happy to help explain the concepts!

Aminah, Getha, and Muthu share a sum of money. The ratio of the amount of money Aminah has to the total amount of money Getha and Muthu have is 1:3. The ratio of the amount of money Getha has to the total amount of money Aminah and Muthu have is 5:11. Muthu has $51 more than Aminah. Find the total sum of money.

This math problem involves using ratios and algebra to find the total sum of money shared among three people: Aminah, Getha, and Muthu. It includes systems of equations and real-world ratio applications, suitable for Grades 8-10.

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