Why does this question use multiplication instead of subtraction?

Question

Ok so I'm reading a book right now that tackles pre-algebra and I stumbled on this example problem that goes like this.

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?

The authors solution is to solve it like this.

First, subtract 1/5 from 1/1 (pie) which equals 4/5, and then multiply 1/6 to 4/5 to get how much Doreen ate from the whole pie, 4/30.

My way obviously is to solve it like this.

First subtract 1/1 by 1/5 (the same with the author), but then I subtract 4/5 with 1/6, which gives me a totally different answer, 19/30.

So my questions are these,

1.) Why does subtracting one fraction to another not work in this scenario. (Because I have a different result)

2.) Obviously (for me) I should subtract based on the question, but why does the author use multiplication instead?

3.) And why does multiplying instead of subtraction work?

Answer 1

Here, I recommend we use variables.

Let $p$ represent the pie that is being eaten (somewhat timidly).

Jerry eats $1/5$ of it, so $1-1/5 = 4/5$ of the original pie is left: $4p/5$.

The his wife comes and eats $1/6$ of what is left of the pie.

Take the remaining pie ($4p/5$), and take away $1/6$ of this new pie stuffs, not the original pie.

$(4p/5)(1-1/6) = 4p/5 \cdot 5/6 = 2p/3$. So two-thirds of the pie remains.

1. You are subtracting two fractions of the entire pie. The question asks you first about the enitre pie, then the remainder.

2. 'Of' implies multiplication: $10\%\ \text{of}\ 500 = .1 \cdot 500 = 50$.

3. Multiplication is the only way it will work, because subtraction is the incorrect way to think about the wording of the problem.

Cheers!

Answer 2

If you eat a fraction $q$ of something then what is left is $1-q$ of that something, that is to say you are left with the earlier amount of that something multiplied by $(1-q)$

In effect you subtract from $1$ and then multiply

So

• They start off with $1$ pie and Jerry eats $\dfrac15$ of it, leaving $1 \times \left(1-\dfrac15\right) =\dfrac45$ of a pie

• They then have $\dfrac45$ of a pie and Doreen eats $\dfrac16$ of that, leaving $\dfrac45 \times \left(1-\dfrac16\right)=\dfrac23$ of a pie