# Why is the word "of" equivalent to multiplying in fraction word problems?

I know this is a very easy problem, but I'm having a hard time getting my head around this concept, consider this example from a book.

*Jerry bought a pie and ate 1⁄5 of it. Then his wife Doreen ate 1⁄6 of what was left. How much of the total pie was left?

To solve this problem, begin by jotting down what the first sentence tells you: Jerry =1/5

Doreen ate part of what was left, so write a word equation that tells you how much of the pie was left after Jerry was finished. He started with a whole pie, so subtract his portion from 1:

Pie left after Jerry = 1 - 1/5 = 4/5

Next, Doreen ate 1⁄6 of this amount. Rewrite the word of as multiplication and solve as follows. This answer tells you how much of the whole pie Doreen ate: Doreen = 1/6 * 4/5 = 4/30*

Now what I don't understand is the part where 1/6 is multiplied into 4/5 when it should, the book says that in almost all fraction word problems the word "of" almost always means "multiplication" and that it still means that way even when it comes to dividing fractions.

Can anybody explain to me the fraction rules and how they work in word problems?

• Don't try and think of the word "of" as being a math operation. Your best bet is to try and understand the problem. If Doreen eats $1/6$ of what is left, then we can imagine that if the leftovers were divided into $6$ even portions, she would take just $1$. That is the same as dividing $4/5$ by $6$ or multiplying $4/5$ by $1/6$.- Jun 2, 2014 at 19:08
• But how does it make the answer into 4/30? Jun 2, 2014 at 19:38
• So after Jerry has eaten his piece, there is only $4/5$ of the whole pie left. Doreen eats $1/6$ of what Jerry didn't eat, not $1/6$ of the whole. So Doreen ate $\frac{1}{6}\cdot\frac{4}{5}=\frac{4}{30}$. The amount of pie that was left however would be $1-\text{amount eaten by Jerry}-\text{amount eaten by Doreen}=1-\frac{1}{5}-\frac{4}{30}$. Jun 2, 2014 at 19:42
• Ok thanks I get it now, but why is multiplication the required operation when finding out how much of the whole part did Doreen ate? Jun 2, 2014 at 19:51
• "If Doreen eats 1/6 of what is left, then we can imagine that if the leftovers were divided into 6 even portions, she would take just 1." Her taking just one of the portions is equivalent to multiplication by $\frac{1}{6}$. Jun 2, 2014 at 19:54

"OF" think of it like this.

Bob:"We have six 6's".
Tom:"What does a 6 look like ?"
Bob:"A 6 looks like this [x x x x x x]".
Tom:"What does six OF these 6's look like".
Bob:"six of these 6's look like this".
[x x x x x x]
[x x x x x x]
[x x x x x x]
[x x x x x x]
[x x x x x x]
[x x x x x x]

Let $x = \text{a number}$

Then one half "of" a number $=$ $\dfrac1{2}•x$

and one third "of" a number $=$ $\dfrac1{3}•x$

Now replace $x$ with any another fraction thus

the book says that in almost all fraction word problems the word "of" almost always means "multiplication" and that it still means that way even when it comes to dividing fractions.

Since when it comes to dividing fractions it is defined as "multiplication" by the reciprocal of the fraction.

Also note in percent problems the word "of" means "multiply" and the word "is" means "equals."