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:

= \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.