Respuesta :
Answer:
[tex]15[/tex]
Step-by-step explanation:
[tex]1[/tex] problem = [tex]\frac{3}{4}[/tex] a page
[tex]20[/tex] 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 [tex]20[/tex] 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 [tex]\frac{3}{4}[/tex] a paper for [tex]1[/tex] 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 [tex]20[/tex] math problems?
- There are a couple ways to solve this problem:
- Add [tex]\frac{3}{4}[/tex] to itself [tex]20[/tex] times to get our resultant.
- Multiply [tex]\frac{3}{4}[/tex] by [tex]20[/tex].
(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 [tex]20[/tex] problems, and [tex]1[/tex] problem is [tex]\frac{3}{4}[/tex] a page, doing [tex]20[/tex] problems is the same as doing [tex]\frac{3}{4}[/tex] a page [tex]20[/tex] times, so let's do just that by adding [tex]\frac{3}{4}[/tex] to itself [tex]20[/tex] times.
[tex]\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}[/tex]
- Yep, that's going to be a lot of work, but I'll show that you can add [tex]\frac{3}{4}[/tex] to itself this many times and get the same answer as the [tex]2[/tex] 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 [tex]\frac{3}{4}[/tex]s and adding them together.
[tex]\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}[/tex]
- 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.
[tex]\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}[/tex]
Note: at this point you can actually convert your fractions to the whole number [tex]3[/tex]. Four goes into twelve [tex]3[/tex] 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.
[tex]\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[/tex]
or
[tex]\frac{12}{4}+\frac{12}{4}+\frac{12}{4}+\frac{12}{4}+\frac{12}{4}\\3+3+3+3+3\\15[/tex]
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 [tex]1[/tex] problem, [tex]\frac{3}{4}[/tex] a paper, being done [tex]20[/tex] times, so we are taking the value [tex]\frac{3}{4}[/tex] and adding it to itself [tex]20[/tex] times; we are multiplying [tex]\frac{3}{4}[/tex] by [tex]20[/tex]. Adding a number to itself [tex]n[/tex] times and multiplying a certain number by the value [tex]n[/tex] holds no difference except in how it's written and how many steps we have to use to solve it.
[tex]\frac{3}{4}[/tex] × [tex]20[/tex]
- If you're used to seeing only whole numbers being multiplied, this may help you grasp it:
[tex]\frac{3}{4}[/tex] × [tex]20[/tex] ⇒ [tex]\frac{3}{4}[/tex] × [tex]\frac{20}{1}[/tex]
- 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: [tex]3[/tex] × [tex]20=60[/tex] (look up "long-multiplication if you are confused by this answer.)
denominator: [tex]1[/tex] × [tex]4=1+1+1+1=4[/tex]
- Once we have multiplied the numerators and denominators together, this is your answer:
[tex]\frac{3}{4}[/tex] × [tex]\frac{20}{1}[/tex]
[tex]\frac{3(20)}{4(1)}[/tex]
[tex]\frac{60}{4}[/tex]
- Look familiar? This is exactly what our unsimplified answer when adding our fractions together was.
[tex]\frac{60}{4}=15[/tex]
