A certain sum of money was divided among A , B , C in
a certain way . B got one tenth as much as A and C together got
A got one ninth of what B and C together got. Find ratio of
share of A: B: C ?

Let's denote the shares of A, B, and C as \( a \), \( b \), and \( c \) respectively. We are given two conditions:

1. **B got one tenth as much as A and C together**:
   \[
   b = \frac{1}{10}(a + c)
   \]

2. **A got one ninth of what B and C together got**:
   \[
   a = \frac{1}{9}(b + c)
   \]

### Step 1: Expressing \( b \) and \( c \)
From the first condition, we can express \( b \) as:
\[
b = \frac{1}{10}(a + c)
\]

From the second condition, we can express \( a \) as:
\[
a = \frac{1}{9}(b + c)
\]

### Step 2: Substituting \( b \) in terms of \( a \) and \( c \)
Substitute the expression for \( b \) from the first equation into the second equation:
\[
a = \frac{1}{9}\left(\frac{1}{10}(a + c) + c\right)
\]

### Step 3: Simplifying the equation
Now simplify the right-hand side:
\[
a = \frac{1}{9}\left(\frac{1}{10}(a + c) + c\right)
= \frac{1}{9}\left(\frac{1}{10}a + \frac{1}{10}c + c\right)
= \frac{1}{9}\left(\frac{1}{10}a + \frac{11}{10}c\right)
\]

Multiply both sides by 9 to eliminate the denominator:
\[
9a = \frac{1}{10}a + \frac{11}{10}c
\]

Multiply through by 10 to get rid of the fractions:
\[
90a = a + 11c
\]

### Step 4: Solving for \( a \) in terms of \( c \)
Rearrange the equation:
\[
90a - a = 11c
\]
\[
89a = 11c
\]
\[
a = \frac{11}{89}c
\]

### Step 5: Finding \( b \) in terms of \( c \)
Now substitute \( a = \frac{11}{89}c \) into the first condition:
\[
b = \frac{1}{10}(a + c) = \frac{1}{10}\left(\frac{11}{89}c + c\right)
= \frac{1}{10}\left(\frac{11}{89}c + \frac{89}{89}c\right)
= \frac{1}{10}\left(\frac{100}{89}c\right)
= \frac{10}{89}c
\]

### Step 6: Final ratio of \( A : B : C \)
Now we have:
\[
a = \frac{11}{89}c, \quad b = \frac{10}{89}c, \quad c = c
\]

Thus, the ratio \( A : B : C \) is:
\[
A : B : C = 11 : 10 : 89
\]

### Conclusion:
The ratio of the shares of A, B, and C is \( 11:10:89 \).

---

Do you need any further explanation or have additional questions? Here are some related problems you might find interesting:

1. If the total amount was 1000 units, how much did each person receive?
2. How would the ratio change if B received one-fifth of what A and C together got?
3. What if A received twice as much as B and C together? How would that affect the ratio?
4. How can we use the method of ratios to solve other real-world problems?
5. How would the ratio change if instead of sums, fractions of the total were given?

**Tip:** When solving ratio problems, always aim to express variables in terms of one quantity to simplify solving multiple equations.

A certain sum of money was divided among A, B, C in a certain way. B got one tenth as much as A and C together. A got one ninth of what B and C together got. Find the ratio of the share of A: B: C?

Solve a problem where a sum of money is divided among A, B, and C using algebraic expressions and ratios. B receives one tenth as much as A and C together, while A receives one ninth of what B and C together get. The final ratio is calculated as 11:10:89, suitable for Grades 8-10 students studying ratios and linear equations.

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