Approximately how many years are there in one billion (1,000,000,000) seconds, assuming a year has 365 days?

Introduction / Context:
This puzzle invites you to connect a very large measure of time in seconds with an everyday measure in years. It develops a sense of scale and your ability to handle unit conversions between seconds, minutes, hours, days, and years. Estimating how long one billion seconds lasts is also a useful mental exercise for understanding large quantities that often appear in science, computing, and population statistics.

Given Data / Assumptions:

- Total time to convert is 1,000,000,000 seconds. - One minute has 60 seconds. - One hour has 60 minutes. - One day has 24 hours. - One year is assumed to have 365 days, ignoring leap years for simplicity.

Concept / Approach:
The standard unit conversion approach is to divide step by step. First convert seconds to minutes by dividing by 60. Then convert minutes to hours by again dividing by 60. Next convert hours to days by dividing by 24. Finally, convert days to years by dividing by 365. At each stage, we keep track of the approximate value. The options are close to one another, so accuracy at each step matters. Using an approximate value for seconds per year gives a compact and efficient method.

Step-by-Step Solution:
Step 1: Calculate the number of seconds in one year with 365 days. Step 2: Seconds in a day are 24 * 60 * 60 = 86,400 seconds. Step 3: Seconds in a 365 day year are 365 * 86,400 = 31,536,000 seconds. Step 4: To find how many years are in one billion seconds, divide 1,000,000,000 by 31,536,000. Step 5: 1,000,000,000 / 31,536,000 is approximately 31.7 years. Step 6: Among the choices, 31.68 years is the closest to 31.7 years. Step 7: Therefore, the best approximation from the options is 31.68 years.

Verification / Alternative check:
We can quickly check reasonableness by using a rounded value for seconds in a year, such as 31,500,000. Then 1,000,000,000 / 31,500,000 is about 31.75 years. This once again suggests a value around 31.7 years, consistent with the more precise calculation. When compared to the options, 31.68 years is clearly the best fit, and none of the higher values like 33.31 years or 34.76 years are close enough to the exact quotient to be plausible.

Why Other Options Are Wrong:
Values such as 34.76 years, 33.31 years, and 32.48 years are all significantly larger than 31.7 years and would correspond to fewer seconds per year than our calculation, which conflicts with the standard conversion from days to seconds. A value lower than 31.68 years would also not match the accurate division of one billion by 31,536,000. Therefore, 31.68 years remains the only option that aligns closely with the computed ratio.

Common Pitfalls:
Some people mistakenly divide by 365 too early or forget one of the 60 factors when converting seconds to minutes or hours. Others mix up whether to multiply or divide during the conversion chain. It is important to remember that to move from a smaller unit to a larger unit, such as seconds to years, you divide step by step by 60, 60, 24, and 365. Keeping track of units at each stage helps prevent errors and makes the final answer far more reliable.

Final Answer:
One billion seconds is approximately equal to 31.68 years when using a 365 day year.