A group of data entrants A, B, and C start working at 8:30 am. A takes 10 minutes per learner, B takes 15 minutes, and C takes 10 minutes. When will they all get a break together and how many learners

Data entrant A takes 10 minutes per learner.
Data entrant B takes 15 minutes per learner.
Data entrant C takes 10 minutes per learner.
To find when they all finish at the same time, we need to find the Least Common Multiple (LCM) of their individual times.

$$\text{LCM}(10, 15, 10) = \text{LCM}(10, 15)$$

Prime factorization of 10 is $2 \times 5$.
Prime factorization of 15 is $3 \times 5$.

$$\text{LCM}(10, 15) = 2 \times 3 \times 5 = 30 \text{ minutes}$$

So, they will all finish a batch of learners simultaneously every 30 minutes.

a) (i) At what time will all the data entrants get a break together?
They started working at 8:30 am. The first break will occur after 30 minutes.

$$8:30 \text{ am} + 30 \text{ minutes} = 9:00 \text{ am}$$

The data entrants will get a break together at $\boxed{\text{9:00 am}}$.

a) (ii) Obtain the number of learners whose data was captured before the first break.
The duration before the first break is 30 minutes.
Number of learners captured by A: $\frac{30 \text{ minutes}}{10 \text{ minutes/learner}} = 3 \text{ learners}$
Number of learners captured by B: $\frac{30 \text{ minutes}}{15 \text{ minutes/learner}} = 2 \text{ learners}$
Number of learners captured by C: $\frac{30 \text{ minutes}}{10 \text{ minutes/learner}} = 3 \text{ learners}$
Total number of learners captured:

$$3 + 2 + 3 = 8 \text{ learners}$$

The total number of learners whose data was captured before the first break is $\boxed{8 \text{ learners}}$.

Task (b):

School fees per learner: Ugx. 900,000.
Cost for refreshments for guests per learner: Ugx. 5,000.
Percentage of remaining fees for food: 40%.

b) (i) Determine how much money the school spent on refreshments if it was based on the learners whose data was captured before the first break.
From part (a)(ii), 8 learners had their data captured before the first break.
Cost for refreshments per learner: Ugx. 5,000.
Total money spent on refreshments:

$$8 \text{ learners} \times \text{Ugx. } 5,000/\text{learner} = \text{Ugx. } 40,000$$

The school spent $\boxed{\text{Ugx. } 40,000}$ on refreshments.

b) (ii) How much money did each learner pay on food.
First, calculate the remaining fees per learner after deducting the guest refreshments.
Remaining fees per learner = Total fees per learner - Guest refreshments per learner

$$\text{Ugx. } 900,000 - \text{Ugx. } 5,000 = \text{Ugx. } 895,000$$

Next, calculate 40% of these remaining fees for food.
Money for food per learner = $40\% \times \text{Ugx. } 895,000$

$$0.40 \times \text{Ugx. } 895,000 = \text{Ugx. } 358,000$$

Each learner paid $\boxed{\text{Ugx. } 358,000}$ on food.