#PreAgebra - Which fraction is bigger? Smaller?

Questions:



We'll be doing 3c, 4c, 5c, 6b, 6c

Answers:

See below:

Analysis:

In all of these questions, we're dealing with what appears to be a bunch of random numbers dividing other rather numbers. It's all quite confusing! But if we work with these numbers, they'll tell us which of them is bigger and which is smaller.

There are a bunch of ways to think of fractions and to understand what they are telling us, but one of my favourites is a pizza pie analogy. All of these questions have three fractions, so let's imagine three pizza pies all the same size. We're going to cut those pies into some number of same sized pieces - it's the denominator, or the bottom number, that tells us how many. Keep in mind that as the number of pieces increases, the size of those pieces drops (if I cut a pizza in half, 1 slice is really big. If I cut that same pizza into 100 pieces, the pieces are really really small).

After we've cut the pizza into the number of pieces we want, then we'll pick out some number of those pieces. It's the numerator, or the top number, that tells us that. Let's start working 3C so you can see what I mean:



That first pizza has been cut into 7 pieces and we'll take 3, the second pizza has been cut into 2 pieces and we'll take 1, and the third pizza has been cut into 5 pieces and we get 3. Ok - so which option gives us the most amount of pizza? Right now it's a little hard to tell, isn't it? The slices are all different sizes and we're taking different amounts. What would really help is if the slices were all the same size - if we cut the pizza into slices that are all the same size - then all we'd need to do is pick the fraction with the largest numerator (the top number).

Now, as we cut the pizza into smaller and smaller slices, we still want to get the same amount of pizza! So as we increase the number of slices, we get more of them.

As an example, look at the  and think about a pizza where it's cut into 2 and you get 1 slice. If we cut the pizza into quarters, so 4 pieces, how many pieces of pizza do you get now? 2. Getting 2 slices when it's cut into 4 is the same as when you get 1 slice when the pizza is cut into 2 (image from chilimath.com):



This is where the idea of a Common Denominator comes into play - and all that means is that we want the bottom numbers to all be the same (so that the pizza slices are all the same size).

So how do we get to the Common Denominator? One way is to simply multiply all the denominators together (this may not get you to the Lowest Common Denominator, which you'll need to find in other questions, but it'll work well here):

7 x 2 x 5 = 70

Ok - so our denominator will be 70.

Now - we need to remember a couple of things at this point. The first is that we can always multiply by the number 1 and end up with the same number we started with. 3 x 1 = 3. I can multiply by 1 and not change the value of the fraction.

The other thing we need to remember is that 1 doesn't always look like 1. If I divide a number by itself, what do I get? 1. And so:



So now we can multiply the different fractions by versions of the number 1 so that we get our denominators to all be 70:







Remember, these are all equivalent to the fractions I had before - all I did is multiply by 1.

Ok - which amount of pizza is most? The one with 42 slices in it. And which fraction was that? 

All the other questions can be worked this same way:

4C



The Common Denominator is 8 x 6 x 3 = 144:





The one with 36 pieces is the smallest, and that is 

5C



The common denominator is 7 x 6 x 5 = 210



And now they can be arranged easily from largest to smallest:



6B

This question has a little twist to it - they include the number 1 as a fraction. Let's change it's look slightly so that it really does look like a fraction:



This says I cut the pizza into 1 piece (or in other words, it's the whole pizza) and I get that one piece (again, the whole pizza).

Now let's work this the same we have the others:



The common denominator is 5 x 1 x 8 = 40:



From smallest to largest, it's the one with 20, then 24, then 40, or:



6C



The common denominator is 7 x 9 x 3 = 189



From smallest to largest it's the one with 27, then 63, then 84, or:



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Questions and comments always welcome!

#PreAlgebra - How to multiply fractions...

Question

What's 

Answer

Analysis

The basic equation for multiplication is:



Ok - so why does this work?

Let's look at the question and see why.

We can start with the . If we think of a pizza, we're going to cut it in two and hold onto one piece. Now - if we multiply by a natural number, say like 5, what we're saying is that we're going to add  to itself 5 times, so that looks like:



So what happens when we multiply by a fraction? We split the figure into more bits (the denominator says the number, so in our case, each of our 2 pieces is split into 3 more) and then we'll hold onto more bits as well (the numerator says the number, so in our case, for every 1 we're holding, we're going to hold 2).

Let's talk this out. In the denominator, we had 2 pieces. We multiply by 3, which now gives 6 pieces of pizza. In the numerator, we had 1 piece. We multiply by 2 and so now are holding 2 pieces. Holding 2 pieces of a pizza that is cut into 6 is the same as holding 1 piece when it's cut into 3 pieces:



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Questions and comments always welcome!

#PreAlgebra - How to add fractions (and why do I need a common denominator...)

Question

What's ?

Answer

Analysis

First, some fraction stuff! The top number in a fraction is the numerator and the bottom number is the denominator:



Think of a pizza. The denominator tells us the number of pieces there is in 1 whole pizza. The numerator tells us the number of pieces of that pizza that we're concerned with. Notice that if the numerator and the denominator are the same number, we'll end up with the value 1 (or, in other words, the entire pizza):



What our question is asking us to do is to add one slice from the pizza with 4 pieces and 1 slice from the pizza with 6 slices:



Now - if we were simply adding things up, like doing 1+1, all we'd be saying is that I have one slice of pizza here and one there and together they add up to 2:

1+1=2

But with the fractions, we have a bit more information. We can't simply add the numerators together to see that we have 2 pieces of pizza - we need to put more information into our answer. We need to compare the two pizzas on equal terms. To do that, we want the pizzas to both have the same number of pieces in each one - and that is what the Common Denominator is - making the pizzas have the same number of pieces.

Ideally, we'd like to make as few cuts as possible (that pizza is getting cold and I know we're all hungry for some), so we'd like to find the Lowest Common Denominator (LCD). So how do we do that?

I do it by breaking down the two denominators into its prime factors (those numbers that are prime that multiplied together equal the denominator):

4 = 2 X 2
6 = 2 X 3

So now what do we do? The Lowest Common Denominator is the same as the Lowest Common Multiple - we need to include all the primes from each of the biggest groupings. For instance, there are two 2's in the 4, so we need both of those. And there's a 3 in the 6, so we need that too. So we have:

LCD = 2 X 2 X 3 = 12

And now we can scale up our fractions, using clever forms of the number 1, to find the sum.

First remember that anything times 1 equals itself (so we can multiply by 1 and not change the value of the fraction):



And any fraction that has it's numerator and denominator the same is equal to 1. We want to pick values of "1" that when multiplied with our denominators, gives the LCD:





Let's stop here for a second. Notice that with the pizza that had 4 slices, if I cut the pizza into 12 slices, my having 3 of those slices is the same amount of pizza. The fractions are equal. Same with the pizza that had 6 slices - if I cut those slices in half so that there are 12 slices of pizza, and I have 2 of them, it's the same amount of pizza.

Ok, let's finish up:



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Questions and comments always welcome!