I follow a well-known physics professor who will, from time to time post a "contest" for people to solve physics problems. The problems are not too difficult but are often counter intuitive. That means the answer goes against our feelings or intuition.
It is interesting to me how many people are quick to post because they 'feel' they have the right answer. Those quick answers are usually wrong.
I confess, I'm not great with math. I nearly always get the answer right. How can both of those statements be true?
Most will get that question wrong! Most will think my idea of being bad at math is different from their idea. If I get the answer right most of the time, I must be good at math! Wrong.
Good at math means a person can set up a formula, run through the steps and arrive at the right answer. I was never good at the setting up the formula part. So in reality, I'm not good at math. But I can break the problem down to parts. I calculate the parts and then compile the answers. Then I check to see if my answer is reasonable.
Here's an example. (A very easy one!) Let's say four kids are carrying apples. Sally, Jane, Joe, and Jim each are carrying apples. Sally has 3. Jane has 2. Joe has 6. Jim has only 1. If they put all the apples into a basket, how many apples will be in the basket?
OK, this is very easy. But if I am not good at math, I might make a model. Instead of names, which might confuse me, or even bias my answer (can boys carry more apples?), and because the names don't matter, I simplify.
a=3 b=2 c=6 d=1
Now, again, because I am not good at math, I might add 3+2=5 and 6+1=7 Now I can see that 5+7=12
Are there any tricks to the question? Does it mention eating, dropping or sharing apples? No. Could 4 kids carry 12 apples? Sure. So my answer makes sense.
This is a very simple example but I wanted to give you and idea of how to break a problem down into parts. You can solve the problem even if you are not good at math.
To test whether my model is valid, I might run a test using 'easy' numbers. What if each of our example kids had 1 apple? Will this method lead to an answer of 4? If not, my model may not work. Adding up a and b and then c and d might not be a valid method. So I test it with easy numbers. (Round numbers) It works.
Here is a real world example of where doing a little math gives an answer that might not agree with your feelings.
We have several salespeople. The top seller (Mary) produces $270,000 per year in sales. Out least productive seller (Kristy) sells only $50,000 per year. For the sake of simplicity, we will not consider the other salespeople.
We have money in the budget for sales training but only enough for one person. Mary already knows most of what it takes to be a great seller. And, she is after all, our best producer. If she attends the training, we could expect her sales to improve by only about 10%.
If we send Kristy, her sales should increase by 50%! Wow. That would really be great and well worth the cost.
But if we do a little math we see something that might change our mind.
If Mary increases her sales by only 10%, we gain $27,000 in sales. If Kristy increases her sales by 50%, we gain $25,000. The money (this is almost always the case!) is better spent on the top producer than on the bottom. This is a huge thing to consider---the government spends billions on the least productive. Businesses throw billions annually at the least productive employees. Improvements in the best performers generally yields the best returns.
In this case, we should play with the model because the net gain to the company is fairly close either way. It might be better to send Kristy because a slight improvement of projected 50% to 55% would change the total outcome. And, maybe a 15% improvement in one of our 'middle' place sellers might even produce more than sending either Mary or Kristy.
The point is this. Don't go with your feelings. Run some "what if" models and get a good sense of where the best return is.
Make a model to solve this.
John bought two items at the store. He bought an apple and a pint of milk. He spent $1.50. The milk was one dollar more than the apple. How much was the apple?
Quickly! Write down your "gut" answer.
Now make a little model and see what you get. I would get a $1 bill and enough change to make 50 cents and two items to represent milk and an apple. (I'm not good at math, remember?!) You'll soon see that your model makes a great calculator!