GRE. Rate problem and weighted average isn't intuitive.

Question

Shaundra drove the same route to work each morning, Monday through Friday, in a particular week. On Monday and Tuesday she averaged 20 miles per hour, and on her three remaining work days she averaged 30 miles per hour.

Quantity A Shaundra’s average speed for all five morning commutes

Quantity B 26 miles per hour

• Quantity A is greater.
• Quantity B is greater.
• The two quantities are equal.
• The relationship cannot be determined from the information given.

So the strangest thing to me is that the answer is 25 even though it seems like she drives a greater % of the distance at 30 mph. How do I rectify this intuition?

So let's say distance is d. So she drives 5d.

The time that she drives on Mon and Tues is d/20. The time that she drives on Wed, Thurs, Fri is d/30.

Total time = $2 * \frac{d}{20} + 3 * \frac{d}{30}$ = d/5

So average rate = $\frac{5d}{d/5}$ = 25

What sorcery is this? I can't rectify this answer with the idea in my head that we drove longer at 30mph so the weighted rate should be closer to 30 and not dead in the middle?

Answer 1

Here is the right intuition: she drives a greater distance at 30 mph, yes. But when you are going 30 mph, you cover the same amount of distance in less time than if you are going 20 mph.

Alternatively: if she had gone 30 mph for two days and 20 mph for two days, but the same distance all days, she would have spent more time going slower, and we would expect the average speed to be closer to 20. However, she went 30 mph for three days, so this means she is spending longer going faster, which would increase the average speed. (The fact that they happen to meet dead in the middle of 20 and 30 is a happy accident.)

Answer 2

There's no sorcery here. Simply you found that the average speed is not the average (arithmetic mean) of the 5 speeds. Trust the formulas and the logic, intuition may be misleading in some cases.

Also, she didn't drive longer time at 30 mph, she drove $d/10$ time i.e. exactly the same time which she drove at 20 mph. So she drove longer route, but not longer time at the speed of 30 mph. I guess this makes it clear and intuitive (after all if you drive N hours at 20 mph, and N hours at 30 mph, your average speed is 25 mph).