Mr. P’s Parent Maths Problem of the Week

At the end of Term 2, I posed a particularly tricky problem. Here’s a quick recap:

An old clattery car is to drive a 2-mile stretch, up and down a hill. Because it is so old, it cannot drive the first mile, the ascent, faster than with an average speed of 15 miles per hour.
Question: How fast does it have to drive the second mile, on going down (it can, of course, go a lot faster) to obtain an average speed for the whole distance of 30 miles an hour?

Conceptually, this problem is a real mind-twister. But this is how it works: First, ask yourself how long it takes the old car to reach the top of the hill? The uphill stretch is one mile. At a speed of 15 miles per hour, the car takes four minutes to get there (since one hour divided by fifteen equals four minutes). 

Secondly, ask yourself how long it would take to complete the full journey up and back down, with an average speed of 30 miles per hour. The total distance is two miles, and at 30 miles per hour, that corresponds to two miles in four minutes. So the entire trip would need to take four minutes. However, those four minutes have already been spent just reaching the top! Therefore, to achieve an average of 30 miles per hour for the whole trip, the car would have to cover the remaining mile instantaneously!

Problem for Week 1 Term 3

The following problem was used in this week’s National Junior Maths Challenge.

Ada likes adding numbers! On Monday, she added all the three-digit numbers, each of whose digits is either 0 or 1. On Tuesday, she added all the two-digit numbers each of whose digits is either 2 or 3. On Wednesday, she added her answers from Monday and Tuesday. 

What was the result of Ada’s final addition?

A) 310                 B) 421                 C) 532                 D) 643                 E) 754