# The answer is as expected, but this is not proportionality.

A man leaves office everyday at 5 p.m. His driver leaves home everyday at such a time that he can pick up his master from the office at 5 p.m. The driver drives at a constant speed. Now one day, the man left office at 4.40 p.m. and started walking towards home. On that day too the driver had left home on such a time that he could pick up his master at 5 p.m. But since the man had started walking at 4.40 p.m. they met somewhere in between from where the driver took his master home. That day, the man reached home 10 minutes earlier. One week later, a similar thing happened, the man left office at 4.50 p.m. and met the driver on his way home from where the driver took his master. How much earlier did the man reach home the second day?

Solvers, Please show your methods, because answer is 5 minutes, as expected , but how you get it is of interest.

• So why is this not proportionality? Aug 11, 2015 at 11:03
• while doing this problem, it would be wrong if you said, 'if the man leaves office 20 mins earlier, he reaches home 10 minutes earlier, so if he leaves home 10 minutes earlier, then he reaches home 5 minutes earlier,' because the actual calculation is hardly that way, i did the problem in the exact way Arthur did it, actually you have to fix the frame of reference on the driver to get the answer, that's what makes it interesting. you would get the right answer if you said its proportionality but your reasoning in the above case unlike Arthur would be wrong!!! Aug 12, 2015 at 9:00
• I disagree. The result is proportional, you can test this in other cases, and whether saying "because of proportionality" is a valid justification depends on how easy it is for the person it is intended for to see that proportionality must hold. I'd consider proportionality sufficient justification in this case. Aug 12, 2015 at 9:08

The main points are the bullets, a reasoning for each bulleted proposition is right below it.

• The first time he met up with the driver at 4.55.

The reason for this is that the driver's route was ten minutes shorter, which means he must have turned around 5 minutes earlier. Picking up the man five minutes earlier, plus a five minutes shorter trip back home totals 10 minutes.

• The man's walking speed is a third of the driver's driving speed.

What the man walked in 15 minutes (from 4.40 to 4.55), the driver was expected to drive in 5.

• The day he left work at 4.50, he met up with the driver at 4.57.30.

This means he has walked for 7.5 minutes, and the driver meets him two and a half minutes earlier than expected. This is in line with the argument above, that the driver drives three times the man's walking speed.

• The man came home 5 minutes earlier than expected.

The driver picked him up two and a half minutes earlier than planned, and had two and a half minutes shorter drive home, which adds up to five minutes.

• exactly!!!!!!!! Aug 12, 2015 at 8:59