Question

Olivia is doing her math homework. For each problem she uses 3/4 sheet of paper. How many sheets of paper will Olivia need to complete 20 math problems?

Answer 1

The answer is 15

Explanation: you divide 20 by 3/4

Answer 2

`15`

Explanation:

`1` problem =
`(3)/(4)` a page

`20` problems = __ pages

- First, we need to figure out what this problem is asking us. This problem is asking for how many pieces of paper we need to do
`20` math questions. Well, the first step is knowing how many pieces of paper are needed for a single question. Luckily the question tells us that we need
`(3)/(4)` a paper for
`1` problem. So I have shown above what we know, and what we need to know. How do we figure out how many pages are needed for
`20` math problems?

• There are a couple ways to solve this problem:

1. Add
   `(3)/(4)` to itself
   `20` times to get our resultant.
2. Multiply
   `(3)/(4)` by
   `20`.

(I listed them in order of time it takes to solve; #1 will take the longest, and #2 the shortest)

• I'll do both methods and you can decide which you are most comfortable with.

Addition Method:

• If we need to do
  `20` problems, and
  `1` problem is
  `(3)/(4)` a page, doing
  `20` problems is the same as doing
  `(3)/(4)` a page
  `20` times, so let's do just that by adding
  `(3)/(4)` to itself
  `20` times.

`(3)/(4)+(3)/(4)+(3)/(4)+(3)/(4)+(3)/(4)+(3)/(4)+(3)/(4)+(3)/(4)+(3)/(4)+(3)/(4)+(3)/(4)+(3)/(4)+(3)/(4)+(3)/(4)+(3)/(4)+(3)/(4)+(3)/(4)+(3)/(4)+(3)/(4)+(3)/(4)`

• Yep, that's going to be a lot of work, but I'll show that you can add
  `(3)/(4)` to itself this many times and get the same answer as the
  `2` step solution seen with multiplication. Here's my work.

Note: (in case you struggle with fractions) Fractions when added together do NOT have a change in denominator (bottom number), only the numerator (top number) is added. Treat this as the same thing as addition, but you've got a number on the bottom that we'll deal with later.

• In this step, I'm taking every pair of
  `(3)/(4)`s and adding them together.

`(3)/(4)+(3)/(4)+(3)/(4)+(3)/(4)+(3)/(4)+(3)/(4)+(3)/(4)+(3)/(4)+(3)/(4)+(3)/(4)+(3)/(4)+(3)/(4)+(3)/(4)+(3)/(4)+(3)/(4)+(3)/(4)+(3)/(4)+(3)/(4)+(3)/(4)+(3)/(4)\\(6)/(4)+(6)/(4)+(6)/(4)+(6)/(4)+(6)/(4)+(6)/(4)+(6)/(4)+(6)/(4)+(6)/(4)+(6)/(4)`

• Now, I'm again going to add my pairs together. Using this method saves time adding long chains of numbers and keeps your work minimized.

`(6)/(4)+(6)/(4)+(6)/(4)+(6)/(4)+(6)/(4)+(6)/(4)+(6)/(4)+(6)/(4)+(6)/(4)+(6)/(4)\\(12)/(4)+(12)/(4)+(12)/(4)+(12)/(4)+(12)/(4)`

Note: at this point you can actually convert your fractions to the whole number
`3`. Four goes into twelve
`3` times, so you'll end up with a clean number as your answer without any difficult conversions. Not converting now won't change your end result if you chose not to, but it will make things easier.

`(12)/(4)+(12)/(4)+(12)/(4)+(12)/(4)+(12)/(4)\\(24)/(4)+(24)/(4)+(12)/(4)\\(48)/(4)+(12)/(4)\\(60)/(4)\\ 15`

or

`(12)/(4)+(12)/(4)+(12)/(4)+(12)/(4)+(12)/(4)\\3+3+3+3+3\\15`

Multiplication Method:

This method is much, much less time consuming compared to doing the addition, so I recommend using this method in most word problems like this one.

Note: (read if you're having trouble with multiplication.) We have
`1` problem,
`(3)/(4)` a paper, being done
`20` times, so we are taking the value
`(3)/(4)` and adding it to itself
`20` times; we are multiplying
`(3)/(4)` by
`20`. Adding a number to itself
`n` times and multiplying a certain number by the value
`n` holds no difference except in how it's written and how many steps we have to use to solve it.

`(3)/(4)` ×
`20`

• If you're used to seeing only whole numbers being multiplied, this may help you grasp it:

`(3)/(4)` ×
`20` ⇒
`(3)/(4)` ×
`(20)/(1)`

• Remember that not only are both the numerators and denominators being multiplied, but separately. Here's the multiplications I did for both top and bottom.

numorater:
`3` ×
`20=60` (look up "long-multiplication if you are confused by this answer.)

denominator:
`1` ×
`4=1+1+1+1=4`

• Once we have multiplied the numerators and denominators together, this is your answer:

`(3)/(4)` ×
`(20)/(1)`

`(3(20))/(4(1))`

`(60)/(4)`

• Look familiar? This is exactly what our unsimplified answer when adding our fractions together was.

`(60)/(4)=15`