Let's solve the problem step by step.

Step 1: Define the rates of work

Let:

• A's rate of work = $A$ (work per day)
• B's rate of work = $B$
• C's rate of work = $C$

From the given information, we can write the following equations:

• A and B together can complete the work in 30 days, so: A + B = \frac{1}{30} \quad \text{(This means A and B together can complete \frac{1}{30} of the work in 1 day.)}
• B and C together can complete the work in 24 days, so: $B + C = \frac{1}{24}$
• C and A together can complete the work in 20 days, so: $C + A = \frac{1}{20}$

Step 2: Find individual work rates

We have three equations:

1. $A + B = \frac{1}{30}$
2. $B + C = \frac{1}{24}$
3. $C + A = \frac{1}{20}$

To find $A$, $B$, and $C$, let's add all three equations: $(A + B) + (B + C) + (C + A) = \frac{1}{30} + \frac{1}{24} + \frac{1}{20}$ This simplifies to: $2(A + B + C) = \frac{1}{30} + \frac{1}{24} + \frac{1}{20}$

Now, find the common denominator of 30, 24, and 20. The least common denominator (LCD) is 120, so: $\frac{1}{30} = \frac{4}{120}, \quad \frac{1}{24} = \frac{5}{120}, \quad \frac{1}{20} = \frac{6}{120}$ Thus, $2(A + B + C) = \frac{4 + 5 + 6}{120} = \frac{15}{120} = \frac{1}{8}$ Dividing both sides by 2: $A + B + C = \frac{1}{16}$

Step 3: Solve for individual rates

Now, use the value $A + B + C = \frac{1}{16}$ in the original equations to find each person's rate:

1. $A + B = \frac{1}{30}$
2. $B + C = \frac{1}{24}$
3. $C + A = \frac{1}{20}$

From $A + B + C = \frac{1}{16}$, we know: $A = (A + B + C) - (B + C) = \frac{1}{16} - \frac{1}{24} = \frac{3}{48} - \frac{2}{48} = \frac{1}{48}$ So, $A = \frac{1}{48}$ (A completes $\frac{1}{48}$ of the work in 1 day).

Now, use this to find $B$ and $C$:

• From $A + B = \frac{1}{30}$, we get: $B = \frac{1}{30} - \frac{1}{48} = \frac{8 - 5}{240} = \frac{3}{240} = \frac{1}{80}$
• From $C + A = \frac{1}{20}$, we get: $C = \frac{1}{20} - \frac{1}{48} = \frac{12 - 5}{240} = \frac{7}{240} = \frac{7}{240}$

Thus, $A = \frac{1}{48}, B = \frac{1}{80}, C = \frac{1}{80}$.