Badki can bake 45 cakes in 9 hours when working alone. Badki and Chutki together can bake 80 cakes in 10 hours. If only Chutki works, how many cakes can she bake in 40 hours at the same constant rate?

Introduction / Context:
This problem deals with individual and combined productivity. You are given how many cakes Badki bakes alone and how many cakes both Badki and Chutki bake together in a given time. Using this information, you need to determine how many cakes Chutki alone can bake in a longer period, assuming all rates are constant.

Given Data / Assumptions:

Badki alone bakes 45 cakes in 9 hours.
Badki and Chutki together bake 80 cakes in 10 hours.
We assume both Badki and Chutki work at constant individual rates throughout.
We are asked for the number of cakes Chutki bakes in 40 hours when working alone.

Concept / Approach:
First, we compute Badki's individual baking rate by dividing total cakes by time. Next, we compute the combined rate of Badki and Chutki together. Subtracting Badki's rate from the combined rate gives Chutki's individual baking rate. Finally, we multiply Chutki's rate by 40 hours to find how many cakes she can bake alone in that time.

Step-by-Step Solution:
Badki bakes 45 cakes in 9 hours, so Badki's rate = 45 / 9 = 5 cakes per hour. Badki and Chutki together bake 80 cakes in 10 hours, so combined rate = 80 / 10 = 8 cakes per hour. Let Chutki's rate be c cakes per hour. Then 5 + c = 8, because Badki and Chutki together bake 8 cakes per hour. So c = 8 - 5 = 3 cakes per hour. This means Chutki alone bakes 3 cakes every hour. In 40 hours, total cakes baked by Chutki = 3 * 40 = 120 cakes.

Verification / Alternative check:
As a check, if Chutki bakes 3 cakes per hour and Badki bakes 5 cakes per hour, then together they bake 8 cakes per hour. Over 10 hours that would give 80 cakes, which matches the given data. Badki alone at 5 cakes per hour over 9 hours gives 45 cakes, again agreeing with the information. Therefore, our calculation of 3 cakes per hour for Chutki and 120 cakes in 40 hours is consistent.

Why Other Options Are Wrong:
Option 125 cakes does not correspond to any integer rate when divided by 40 hours and is inconsistent with the derived rate of 3 cakes per hour.
Option 10 cakes in 40 hours would imply Chutki bakes only 0.25 cakes per hour, which is far too slow compared to the combined rate with Badki.
Option 20 cakes would mean a rate of 0.5 cakes per hour, again inconsistent with the combined productivity.
Option 160 cakes would imply a rate of 4 cakes per hour for Chutki, which would make the combined rate 9 cakes per hour and contradict the given data of 80 cakes in 10 hours.

Common Pitfalls:
Some learners mistakenly average the total cakes or times instead of working with rates. Another common error is to assume that each person contributes equally to the joint result without calculating individual contributions. Always compute individual and combined rates carefully, and remember that combined rate is the sum of individual rates, not an average of times.

Final Answer:
Therefore, Chutki alone can bake 120 cakes in 40 hours, so the correct option is 120 cakes.