## Historically Multiplying and Dividing Fractions by a Number

Posted by mark schwartz on April 27, 2016

__Introduction__

I was trolling through some of my old texts (1850-1875) and I noticed some consistency in their explaining and demonstrating multiplying and dividing fractions by a number. This method is not in today’s text because, once you’ve played with the examples in this article, you’ll realize that today’s rules are easier to apply. However, in some cases, at least knowing what was done in the 1850-1875 era might prove useful … you’ll find out as you read.

__The Story__

In today’s texts, there are methods for teaching how to multiply and divide a fraction by a number, with perhaps some slight variation by instructor but the core “rule” is the same. Basically, write the whole number as a fraction with a denominator of 1. For multiplication, multiply the numerators together and multiply the denominators together. For division, multiply the reciprocal of the whole number by the other term.

In Loomis’s 1872 *Treatise on Algebra *(page 62) he states (italics and bold mine):

“** In order to multiply a fraction by any number, we just multiply its numerator or divide its denominator by that number**”. For division: “

**”.**

*In order to divide a fraction by any number, we must divide its numerator or multiply its denominator by that number*The following carefully crafted problem demonstrates how his rules apply. Take your time reading each step and refer to his statements above as needed.

Multiply: (12/15)(3) = (12•3)/15 = 36/15 = 12/5 or (12/15)(3) = 12/(15 ÷ 3) = 12/5.

Divide: (12/15) ÷ 3 = (12/3)/15 = 4/15 or (12/15) ÷ 3 = 12/(15•3) = 12/45 = 4/15

I chose the above problems to demonstrate his rules because, __depending on the numbers in the problem__, his rules can lead to fussy and messy solutions. Note the second example.

Division: (3/4) ÷5 = (3/5) ÷ 4 = 3/(5•4) = 3/20 or (3/4) ÷5 = 3/(4•5) = 3/20

Multiplication: (3/4)(5) = (3•5)/4 = 15/4 or (3/4)(5) = 3/(4/ 5) = ?

The question mark is there because … what is to be done next using his rules? We’re not dividing a fraction by a number. This is where Loomis’s idea gets complicated and why today’s rules are easier to apply, although I believe playing with his rules gives some background for today’s rules. If he ever ran into this he didn’t say so.

Now, about the problem which ended with a question mark. Here’s how it can be finished. Step one is to __take the reciprocal of the problem__, which changes the problem to dividing a fraction by a whole number, which allows the use of the Loomis rule. Taking the reciprocal is one of those sneaky math things we occasionally employ and later in the process, we take the reciprocal of the answer. It’s unusual but it works.

The reciprocal of 3/(4/5) is (4/5)/3 = 4/(5•3) = 4/15, the reciprocal of which is the correct answer of 15/4. I told you that his rule would give the correct answer but it’s a process requiring the reciprocal of the reciprocal!

In applying Loomis’s rules it becomes clear that using the division option when either multiplying or dividing creates problems that can be solved but, as you saw, in one case it’s a complicated “trick” (double reciprocal) that makes it work. *It seems then that unless you can see that the division will give a whole number quotient, use only the multiplication option* … which is exactly what the current rule does … “in order to divide, multiply by the reciprocal”.

Loomis, I believe, provided a combined rules. Their application sometimes depended on the numbers in the problem and further, as I discovered, sometimes depended on the double-reciprocal trick. And one more thing. I was wondering, as you might be, if his rule worked when both terms are fractions. I played with it and it doesn’t; his rules are only for a fraction and a whole number.

It was fun exploring his ideas but again, the current rules for multiplication and division are easier to apply … stick with them!

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