We need to calculate how long it will take for the digital clock, which gains 5 seconds every 2 hours, to accumulate a 2-minute lead over the actual time.

Step 1: Convert 2 minutes to seconds

Since 1 minute = 60 seconds,
$2 \, \text{minutes} = 2 \times 60 = 120 \, \text{seconds}.$

The clock needs to gain 120 seconds.

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Step 2: Determine the time it takes to gain 5 seconds

The clock gains 5 seconds every 2 hours.

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Step 3: Calculate how many 2-hour intervals are needed to gain 120 seconds

To find the number of intervals required, divide 120 seconds by 5 seconds:
$\text{Number of intervals} = \frac{120}{5} = 24.$

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Step 4: Calculate the total time required

Each interval is 2 hours long. For 24 intervals:
$\text{Total time} = 24 \times 2 = 48 \, \text{hours}.$

Convert this to days:
$\text{Total time in days} = \frac{48}{24} = 2 \, \text{days}.$

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Final Answer:

It will take 2 days for the clock to be 2 minutes ahead of the actual time.

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Would you like further clarification or have other questions?

Related Questions:

1. How would the calculation change if the clock gains 6 seconds every 2 hours?
2. What if the clock loses 5 seconds every 2 hours? How would that affect the time?
3. Can you calculate how long it will take for the clock to be 5 minutes ahead instead?
4. If the clock gains time every 1 hour instead of 2 hours, what changes in the calculation?
5. How can this type of problem be modeled using a formula?

Tip: Always convert time measurements (minutes, hours, days) to the same unit for consistent calculations.