Hardly Anyone Remembers How to do Fractions

[Whateverist]

I own a business where employees need to now how to work out fractions and long multiplication to do their job efficiently. Here are my questions that I placed in their interview exam to determine if they know basic maths: 
1. 1/4 + 1/3 =
2. 1/4 x 1/3 =
3. 132 x 64 =

Only about 3 applicants over dozens that have sat the exam have answered all questions correctly. Most of them got the last one right, but they used their mobile phones.  I originally thought that the questions were too easy, but apparently they are now redundant? I don't know why teachers today don't teach fractions so that their students are competent in their usage, or is it a case that they aren't considered important enough to warrant any special attention?

It was either in 5th or 6th grade that problems like your #1 caused me to do some algebraic generalizing for the first time. After performing some number of straight forward problems of that sort it occurred to me the sum would always be sum of the denominators over their product, i.e.:  1/A + 1/B = (A + B)/AB.  Of course that just comes from generalizing the algorithm and not any deep insight into the meaning of fractions. 

Problems like you #2 should be trivial to anyone, at least algorithmically.  Unless of course lack of use and lack of insight into the algorithm's construction has just totally deserted you.  If one remembered only that multiplication by a unit fraction is the same as dividing by its numerator it wouldn't be too hard to work out an answer.  So  N • 1/3 = N ÷ 3 asks us to find just one third of N, the amount we started with.  We started with 1/4 so we only want a third of that.  If we imagine a cup measurer marked out in 1/4's with each fourth further divided by smaller lines into three equal sections, we'd wind up with 12 equal sections (of this smaller size) in the entire cup.  So our problem means how much of the entire cup is just one of those smaller sections and the answer becomes 1 of 12 equal parts, 1/12.

Operations with fractions were always taught between 2 and 3 years before students came to me.  Part of the failure of the american education system traditionally has been that we teach a shit load of things every year and then go on reteaching them since we never do any of it in enough depth to stick.  So I almost always had to work in some review too.  With the common core that was implemented just before I retired there is a chance this cycle will finally be broken. But then there are so many other things wrong with the education system as a whole that I wouldn't bet on it.  The only think I am positive about is that if we continually jump between one reform system and another forever students are screwed.