We have 101 candles. Every day we light a candle. With the surplus of 5 used candles we can make a new one. For how many days are 101 candles enough?

To determine how many days 101 candles will last, including the ones made from surplus, we can set up the problem and solve it step-by-step.

1. **Initial candles:** We start with 101 candles. Each day we light one candle.

2. **Surplus candles:** For every 5 used candles, we can make one new candle from the surplus.

First, let's calculate the number of days we can light candles before we need to start making new ones from the surplus:
\[ 101 \text{ days (lighting one candle each day)} \]

Now, after 101 days, we have used 101 candles, which gives us a surplus of:
\[ \left\lfloor \frac{101}{5} \right\rfloor = 20 \text{ new candles} \]

Next, we continue the process with the 20 new candles:
\[ 20 \text{ days (lighting one candle each day)} \]

After these 20 days, we have used 20 candles, which gives us a surplus of:
\[ \left\lfloor \frac{20}{5} \right\rfloor = 4 \text{ new candles} \]

Continuing with the 4 new candles:
\[ 4 \text{ days (lighting one candle each day)} \]

After these 4 days, we have used 4 candles, which gives us a surplus of:
\[ \left\lfloor \frac{4}{5} \right\rfloor = 0 \text{ new candles} \]

Since we can no longer make new candles, the process stops here. Now, summing up all the days:

\[ 101 + 20 + 4 = 125 \text{ days} \]

Therefore, the 101 candles are enough to last for 125 days.

Would you like more details on any part of this process or have any questions?

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**Follow-up Questions:**
1. How does the process change if we start with a different number of candles?
2. What if it takes 4 used candles to make a new one instead of 5?
3. Can we create a general formula to determine the number of days for any starting number of candles?
4. How would the solution be affected if we could make a new candle from 3 used candles instead?
5. What if we increase the initial number of candles to 200?
6. How does the surplus calculation change if we have a different conversion rate for used candles to new candles?
7. What if we have a continuous daily increase in the initial number of candles by 1 each day?
8. Can we graphically represent the decrease in the number of candles over time including the surplus?

**Tip:** When dealing with a similar problem, always look for patterns or cycles that repeat, as they often simplify the calculations.

Discover how many days 101 candles can last, including surplus candle production from used candles. Step-by-step calculation provides insights into arithmetic and division concepts, applicable for Grades 5-6.

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