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

Answer:

The answer is 15

Explanation: you divide 20 by 3/4

Answer 2

Answer:

$15$

Step-by-step explanation:

$1$ problem = $\frac{3}{4}$ 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 $\frac{3}{4}$ 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 $\frac{3}{4}$ to itself $20$ times to get our resultant.
2. Multiply $\frac{3}{4}$ 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 $\frac{3}{4}$ a page, doing $20$ problems is the same as doing $\frac{3}{4}$ a page $20$ times, so let's do just that by adding $\frac{3}{4}$ to itself $20$ times.

$\frac{3}{4}+\frac{3}{4}+\frac{3}{4}+\frac{3}{4}+\frac{3}{4}+\frac{3}{4}+\frac{3}{4}+\frac{3}{4}+\frac{3}{4}+\frac{3}{4}+\frac{3}{4}+\frac{3}{4}+\frac{3}{4}+\frac{3}{4}+\frac{3}{4}+\frac{3}{4}+\frac{3}{4}+\frac{3}{4}+\frac{3}{4}+\frac{3}{4}$

• Yep, that's going to be a lot of work, but I'll show that you can add $\frac{3}{4}$ 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 $\frac{3}{4}$s and adding them together.

$\frac{3}{4}+\frac{3}{4}+\frac{3}{4}+\frac{3}{4}+\frac{3}{4}+\frac{3}{4}+\frac{3}{4}+\frac{3}{4}+\frac{3}{4}+\frac{3}{4}+\frac{3}{4}+\frac{3}{4}+\frac{3}{4}+\frac{3}{4}+\frac{3}{4}+\frac{3}{4}+\frac{3}{4}+\frac{3}{4}+\frac{3}{4}+\frac{3}{4}\\\frac{6}{4}+\frac{6}{4}+\frac{6}{4}+\frac{6}{4}+\frac{6}{4}+\frac{6}{4}+\frac{6}{4}+\frac{6}{4}+\frac{6}{4}+\frac{6}{4}$

• 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.

$\frac{6}{4}+\frac{6}{4}+\frac{6}{4}+\frac{6}{4}+\frac{6}{4}+\frac{6}{4}+\frac{6}{4}+\frac{6}{4}+\frac{6}{4}+\frac{6}{4}\\\frac{12}{4}+\frac{12}{4}+\frac{12}{4}+\frac{12}{4}+\frac{12}{4}$

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.

$\frac{12}{4}+\frac{12}{4}+\frac{12}{4}+\frac{12}{4}+\frac{12}{4}\\\frac{24}{4}+\frac{24}{4}+\frac{12}{4}\\\frac{48}{4}+\frac{12}{4}\\\frac{60}{4}\\ 15$      

or      

$\frac{12}{4}+\frac{12}{4}+\frac{12}{4}+\frac{12}{4}+\frac{12}{4}\\3+3+3+3+3\\15$

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, $\frac{3}{4}$ a paper, being done $20$ times, so we are taking the value $\frac{3}{4}$ and adding it to itself $20$ times; we are multiplying $\frac{3}{4}$ 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.

$\frac{3}{4}$ × $20$

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

$\frac{3}{4}$ × $20$ ⇒ $\frac{3}{4}$ × $\frac{20}{1}$

• 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:

$\frac{3}{4}$ × $\frac{20}{1}$

$\frac{3(20)}{4(1)}$

$\frac{60}{4}$

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

$\frac{60}{4}=15$