What is the inverse rule of 3?

Answer

Dear Sandra,

There are indeed two rules of three: proportions and inverse proportions.

You always have four numbers a,b,c,d for the (direct) rule of three, find d so that the ratio between a and b given by a/b is the same as the ratio between c/d. We then find that: a/b=c/d and so we get that d=bc/a.

This typically fits into questions such as: for 5 loaves I get 10 euros, how many loaves do I have to sell to get 100 euros? a=10 euros b= 5 loaves of bread and c= 100 euros. We therefore find that d=500/10=50. Indeed, because the ratio between yield in euros and loaves of bread is 10/5 = 2 euros/bread.

For the rule of three for inverse proportions, you also start with 4 numbers a,b,c and d, but you are looking for the number d such that the inverse proportionality or the product ab is equal to the product cd A and b are said to be inversely is proportional if there exists a constant K such that a=K/b whereas in the previous a and b was proportional if that constant causes a/b=K. So we get the following:

a=K/b but the same proportionality holds for c and d, so c=K/d which leads to: ab=K=cd We can solve for d from this so that we find that d=ab/c.

This typically fits into issues such as: 10 men dig a well in 4 days, how long does it take 8 men to do this? That way we have a=10 males, b=4 days and c=8 males so we find that d=ab/c=40/8=5 days.

How do you find which rule to use? For the first one you can easily see that more loaves of bread yield more money which indicates a proportionality, in the second case more men have to make the digging time shorter which indicates an inverse proportionality.

What if you are wrong? If you’re wrong, the logic doesn’t make sense anymore. If you look at this as a proportionality in the second situation, you find d=bc/a=3.2 days. However, this cannot mean fewer men getting the job done in a shorter amount of time.

Hopefully this settles the matter.

Greetings,

Kurt.