A device needs 4 batteries to run. Each battery runs for 2 days. If there are a total of 6 batteries available, what is the maximum number of days for which the device can be run by strategically replacing the batteries till all the batteries are completely drained of power?
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Battery Maximum Runtime Calculation
The problem asks for the maximum number of days a device can run given a limited supply of batteries, each with a specific lifespan, and a requirement for a certain number of batteries to operate the device simultaneously. We have 6 batteries in total, each lasting 2 days, and the device needs 4 batteries at any given time.
Understanding Battery-Days
A useful way to think about the total power available is in terms of "battery-days". One battery-day is equivalent to one battery running for one day. The total power available from all batteries is the sum of the life of each battery.
Total number of batteries available = 6
Lifespan of each battery = 2 days
Total battery-days available = Number of batteries × Lifespan per battery
Total battery-days available = \(6 \times 2 = 12\) battery-days
Calculating Daily Power Consumption
The device requires 4 batteries to run at any given time. This means that every day the device is running, it consumes the power equivalent of 4 batteries running for 1 day.
Number of batteries required by device = 4
Battery-days consumed per day = 4 battery-days
Determining Maximum Running Days
To find the maximum number of days the device can run, we divide the total available power (in battery-days) by the amount of power consumed per day (in battery-days per day). This assumes we can strategically replace batteries to utilize all available power.
Maximum running days = Total battery-days available / Battery-days consumed per day
\[ \text{Maximum running days} = \frac{12 \text{ battery-days}}{4 \text{ battery-days/day}} = 3 \text{ days} \]This calculation shows that with 12 total battery-days of power and consuming 4 battery-days per day, the device can run for a maximum of 3 days.
Strategic Battery Usage Explained
Here is one way to achieve 3 days of runtime:
- Day 1: Use 4 batteries (let's call them B1, B2, B3, B4). At the end of Day 1, each of these 4 batteries has 1 day of life remaining. Batteries B5 and B6 are unused (have 2 days of life remaining).
- Day 2: We need 4 batteries. Let's use 2 partially used batteries (B1, B2) and 2 fresh batteries (B5, B6).
- B1, B2: Used 1 day on Day 1 + 1 day on Day 2 = 2 days total. Drained.
- B3, B4: Used 1 day on Day 1. Have 1 day life remaining.
- B5, B6: Used 1 day on Day 2. Have 1 day life remaining.
- Day 3: We need 4 batteries. We use the remaining 4 batteries (B3, B4, B5, B6), each of which has 1 day of life left.
- B3, B4: Used 1 day on Day 1 + 1 day on Day 3 = 2 days total. Drained.
- B5, B6: Used 1 day on Day 2 + 1 day on Day 3 = 2 days total. Drained.
This strategy successfully runs the device for 3 full days using all available battery life.
| Day | Batteries Used | Life Used Today (per battery) | Status End of Day |
|---|---|---|---|
| 1 | B1, B2, B3, B4 | 1 day | B1-B4: 1 day life left; B5-B6: 2 days life left |
| 2 | B1, B2, B5, B6 | 1 day | B1-B2: Drained; B3-B4: 1 day life left; B5-B6: 1 day life left |
| 3 | B3, B4, B5, B6 | 1 day | B3-B4: Drained; B5-B6: Drained. All batteries drained. |
Therefore, the maximum number of days the device can be run is 3.
Important Questions from Fundamental Principles of Counting
What is the number of four digit decimal number (<1) in which no digit is repeated?
Let S = {2, 3, 4, 5, 6, 7, 9}. How many different 3-digit numbers (with all digits different) from S can be made which are less than 500?
Consider the digits 3, 5, 7, 9. What is the number of 5-digit numbers formed by these digits in which each of these four digits appears?
3-digit numbers are formed using the digits 1, 3, 7 without repetition of digits. A number is randomly selected. What is the probability that the number is divisible by 3?
Consider the following paragraph:
THE ABILITY TO REASON ACCURATELY IS VERY IMPORTANT, AS IS THE ABILITY TO COUNT. AS AN EXERCISE IN BOTH, LET US COUNT HOW MANY TIMES THE LETTER "E" OCCURS IN THIS PARAGRAPH. THE CORRECT COUNT IS ________.
Which option when put in the blank in the above paragraph will make the final sentence accurate?
