Measuring math ability

I think one of the true signs of mathematical intelligence is the ability to solve a problem you've never seen before with the knowledge you already have. Consider the following:

Bobby and Jenny have a car washing business. If Bobby can wash a car in 8 minutes and Jenny can wash one in 6 minutes, how long will it take if they work together?

If the answer isn't obvious to you, take a few minutes to think about it. The solution requires nothing more than basic algebra. If you're stumped, don't worry. Many American college graduates cannot figure out problems like this. I suspect most people would simply take the average of 6 and 8 to get 7.5 minutes. That is incorrect. 

There are a few key steps to the solution. The first is to identify the unknown. In this case, that is the time to wash one car if they work together. We'll call that amount of time x since that is traditional symbol for an unknown quantity. 

Next, it is important to realize that if they work together, the answer has to be less than 6 minutes and that Bobby and Jenny each work for the same length of time. Those amounts are x/8 and x/6 respectively. And when they work together for x minutes, they wash one car. Putting that all together, we get:

x/8 + x/6 = 1

We now want to get x by itself on the left of the equation. That involves multiplying both sides by 8 and 6, collecting the like terms, and dividing by 14.

6x + 8x = 48

14x = 48

x = 48/14 or about 3.4 minutes. 

Most high school graduates could solve the equation easily, but the hard part is figuring out what equation needs to be solved. 

This next question is one I remember from the Florida math teacher qualification test.

Scientists want to estimate the number of fish in a lake. They catch 100 fish, tag them, and put them back. A week later, they go to the same spot in the lake and catch another 100 fish. This time, they count that 5 of them have tags. About how many fish are there in the lake?

We can assume that the number of tagged fish in the second sample is proportional to the number of fish in the lake. We'll call that number x.

100/x = 5/100

x = 2000

In real life, there could be reasons for this estimate to be wrong, but all such problems require making simplifying assumptions. 

When I was a college student, I remember once reading something in one of my engineering textbooks. It was advice not to look for similar worked example problems when doing homework or taking tests. For learning purposes, it's always best to struggle with unfamiliar problems rather than to search for a similar problem and trying to modify its solution.