To find the amount of money shared by [tex]\(A\)[/tex], [tex]\(Q\)[/tex], [tex]\(R\)[/tex], and [tex]\(S\)[/tex], we need to follow these steps:
1. Understand the given ratios and the difference in amounts:
- The ratios given are [tex]\(3: 4: 6: 7\)[/tex].
- [tex]\(S\)[/tex] receives [tex]$6,000 more than \(Q\).
2. Set up the ratio equations:
- Let the common ratio be \(x\).
- Therefore, the amounts each person receives can be expressed in terms of \(x\) as follows:
- \(A = 3x\)
- \(Q = 4x\)
- \(R = 6x\)
- \(S = 7x\)
3. Utilize the given information about \(S\) and \(Q\):
- \(S\) receives $[/tex]6,000 more than [tex]\(Q\)[/tex], so:
[tex]\[
S - Q = 6,000
\][/tex]
Substituting the expressions for [tex]\(S\)[/tex] and [tex]\(Q\)[/tex]:
[tex]\[
7x - 4x = 6,000
\][/tex]
Simplifying gives:
[tex]\[
3x = 6,000
\][/tex]
Solving for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{6,000}{3} = 2,000
\][/tex]
4. Calculate the specific amounts for each person:
- Substitute [tex]\(x = 2,000\)[/tex] back into the expressions for each person's amount:
- [tex]\(A = 3x = 3 \times 2,000 = 6,000\)[/tex]
- [tex]\(Q = 4x = 4 \times 2,000 = 8,000\)[/tex]
- [tex]\(R = 6x = 6 \times 2,000 = 12,000\)[/tex]
- [tex]\(S = 7x = 7 \times 2,000 = 14,000\)[/tex]
5. Sum the amounts to find the total money shared:
[tex]\[
\text{Total amount} = A + Q + R + S
= 6,000 + 8,000 + 12,000 + 14,000
= 40,000
\][/tex]
So, the total amount of money shared is [tex]\( \$40,000 \)[/tex].
Therefore, the answer to the question is: [tex]\( \boxed{40,000} \)[/tex].
