Mr Wong gave a sum of money to his 3 sons, Alfred, Ben and Clive. Alfred's share was 1/6 of the total sum of money received by Ben and CLive. Ben's share was 1/2 of the total sum received by Alfred and CLive. CLive received $1024 more than Alfred. How much was the sum of money given to the 3 boys?



Let's define the variables and formulate the equations based on the given information:

- Let $ A $ be the amount of money Alfred received.
- Let $ B $ be the amount of money Ben received.
- Let $ C $ be the amount of money Clive received.

### Step 1: Set Up Equations

1. **Alfred's Share Equation:**

   Alfred's share is $ \frac{1}{6} $ of the total sum received by Ben and Clive. This means:

   \[
   A = \frac{1}{6} (B + C)
   \]

2. **Ben's Share Equation:**

   Ben's share is $ \frac{1}{2} $ of the total sum received by Alfred and Clive. This means:

   \[
   B = \frac{1}{2} (A + C)
   \]

3. **Clive's Additional Amount Equation:**

   Clive received $1024 more than Alfred. This means:

   \[
   C = A + 1024
   \]

### Step 2: Solve the Equations

Let's solve these equations step-by-step to find the values of $ A $, $ B $, and $ C $.

First, substitute $ C = A + 1024 $ into the first equation:

\[
A = \frac{1}{6} (B + (A + 1024))
\]

Simplify this equation:

\[
A = \frac{1}{6} (B + A + 1024)
\]

Multiply both sides by 6:

\[
6A = B + A + 1024
\]

Rearrange:

\[
6A - A = B + 1024
\]

\[
5A = B + 1024
\]

\[
B = 5A - 1024
\]

Now, substitute $ B = 5A - 1024 $ and $ C = A + 1024 $ into the second equation:

\[
B = \frac{1}{2} (A + C)
\]

\[
5A - 1024 = \frac{1}{2} (A + (A + 1024))
\]

Simplify inside the parentheses:

\[
5A - 1024 = \frac{1}{2} (2A + 1024)
\]

\[
5A - 1024 = A + 512
\]

Subtract $ A $ from both sides:

\[
4A - 1024 = 512
\]

Add 1024 to both sides:

\[
4A = 1536
\]

Divide by 4:

\[
A = 384
\]

### Step 3: Find $ B $ and $ C $

Now that we know $ A = 384 $, let's find $ B $ and $ C $:

\[
C = A + 1024 = 384 + 1024 = 1408
\]

\[
B = 5A - 1024 = 5(384) - 1024 = 1920 - 1024 = 896
\]

### Step 4: Calculate the Total Sum

The total sum of money given to the three sons is:

\[
A + B + C = 384 + 896 + 1408 = 2688
\]

### Conclusion

The total sum of money given to the three boys is **$2688**.

Would you like more details on any step, or do you have any questions?

### Related Questions

1. What if the conditions for Alfred's or Ben's share change? How would that affect the total sum?
2. How can we solve this problem using a different method, such as matrices or graphical representation?
3. What if Clive received a different amount more than Alfred? How would that affect the result?
4. How would the problem change if there were more than three sons?
5. Can this problem be generalized for any number of sons with different share rules?

### Tip

When solving word problems involving multiple variables, always carefully translate the words into algebraic equations step-by-step to avoid errors.

Mr Wong gave a sum of money to his 3 sons, Alfred, Ben and Clive. Alfred's share was 1/6 of the total sum of money received by Ben and Clive. Ben's share was 1/2 of the total sum received by Alfred and Clive. Clive received $1024 more than Alfred. How much was the sum of money given to the 3 boys?

Solve a complex algebraic word problem where Mr. Wong divides his money between three sons, Alfred, Ben, and Clive. Learn how to use linear equations to determine each son's share and calculate the total sum of $2688. Suitable for Grades 9-11, with step-by-step explanations.

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