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FRACTIONS

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Math is black-and-white, with right-or-wrong answers. It’s hard to color outside the lines in math.

While I often argue with this point, there is some truth to it. Just like grammar, chemistry and baking, math is a pretty precise subject matter. Sure, there are many different ways to add 24 and 37 in your head, but fact is, you can’t just decided that the answer is –19, right?

Rules make math work. And algebra helps us write down these rules. Now, we don’t necessarily need to think of math rules in this way, but believe me, when teaching and writing about math, it sure does help. And there are some real-world situations when an equation can  really help make math easier.

Let’s consider the process for multiplying fractions. Do you remember what it is? Take a look at this problem, and see if you can figure it out:

Of course there are several ways to describe what is happening above, right? You can do it in plain English:

To multiply two fractions, multiply the numerators and multiply the denominators.

Or, you can write this using algebra. This is not as hard as you might think! First, assign a variable to each of the unique numbers on the left side of the equation:

a = first numerator

b = first denominator

c = second numerator

d = second denominator

Then substitute those variables for the numbers themselves:

Now, perform the rule that was described in plain English above: multiply the numerators and multiply the denominators.

How about that! Lickity split, we made like mathematicians and created a rule described algebraically. How hard was that really? 

Now you can use this rule to multiply any fractions of any kind. I don’t care if they’re elementary fractions made up of just numbers or if they’re fancy-schmancy algebraic functions that have — gasp! — variables in them. You don’t even have to think of the abc or d. Instead, think of those variables as place holders. (Hint: this is where your mind can be really flexible, even though the rule is not.)

Because you know this rule, you can solve this problem (even with the x and the y). Just multiply the numerators and then multiply the denominators.

Because of the rule for multiplying fractions — which includes the variables aband — you can see how to multiply any fractions. That’s where the algebra comes in handy.

Now, I know exactly what you’re thinking. When will I ever need to solve a problem like the one above. And here’s my honest answer: for most of you, never. Really and truly. I won’t lie.

However, there are times when creating a rule for a specific real-word problem is very useful. That’s when we might create an equation. Stay tuned, when we’ll talk wedding receptions, guest lists, the price per person and rental fees.

So what do you think of algebra and math rules? Did this example help you understand how algebra is important in developing and stating these rules? Do you disagree with me about why this is important? I can take it — so please do share your thoughts in the comments section.

We’re wrapping up a review of fractions today. If you missed Monday’s or Wednesday’s posts, be sure to look back to refresh your memory on multiplying and dividing fractions.

If you’re the product of a traditional elementary and middle school education, you likely spent many, many months (collectively) learning about adding and subtracting fractions. It is definitely one of the trickiest arithmetic skills to have, but it can also be quite useful. Now that you know how to multiply with fractions, you’re ready to unlock the secret of adding and subtracting them. And it all comes down to multiplying by the lowly, little 1.

This process is really easy — if the fractions in question have one important characteristic. Take a look:

Don’t solve the problem! Just look. What do the fractions have in common? You’re one smart cookie, so I’m sure you recognized that the denominators (the numbers on the bottom of the fractions) are the same — 5. And that’s the key in this process. Whenever you’re adding or subtracting fractions, you need to have common denominators. Then, all you need to do is add the numerators together and keep the same denominator.

If you took a few moments to run this through your brain, you probably wouldn’t have even needed to know this rule. And since we’re grownups, we can use this example: If you have 1 fifth of Jack Daniels and 2 fifths of Johnny Walker, how many fifths of alcohol do you actually have? Well, that would be 3 bottles or 3 fifths. (And believe me, while some of my high school students would have appreciated that example, I don’t think I could have gotten away with using it.)

Same thing is true for subtraction. Let’s say that the fraternity πππ (yeah, I made that up) is having a huge party. They’ve purchased 7 fifths of bourbon. But just before the gig gets started, one of the brothers knocks over the bar and breaks 3 of the fifths of bourbon bottles. How many are left? Well, that would be 4, right? Using this analogy, you can see that because the denominator was the same (5), all you needed to do was subtract the numerators (7 – 3) to get what was left (4).

And here’s where you can break even more rules. As a grownup, you can do these things in your head. If you need to add 1/8 yards of fabric to 1/8 yard of fabric, it’s pretty simple to see that you’re dealing with 2/8 yard (simplified, that’s 1/4 yard).

Yeah, things get a little trickier when you have different denominators. Let’s go back to that pizza example from Monday, shall we? Remember, we were figuring out how many pizzas to order, if we knew how much each person typically eats. Let’s say that you can eat 1/4 of a pizza, your sister can eat 1/3 and your brother can eat 1/2? In other words:

Notice something? Yep — no common denominator. So how do you get one? Well, there’s the short cut and then there’s the longer explanation. In case you’re curious, let’s talk explanation first.

You need a number that all three of these denominators will divide into evenly. That’s called a common multiple. In fact, it’s best if you have the least common multiple. (If you have a really good memory, you might remember that this is often referred to as an LCM.) So what’s the LCM of 4, 3 and 2? Turns out to be 12.

So the common denominator is 12, but do you just replace all of the denominators with a 12, adding 1/12, 1/12 and 1/12? No way, Jose. That won’t get you the right answer. What you need to do is change the numerator so that the denominator is 12. And to do that, you need to multiply by 1.

Remember 1 is the same as any fraction that has the same number in the numerator and denominator. So to change 1/4 to a fraction with 12 in the denominator, you’ll need to multiply by 3/3.

So, think ahead: what do you need to do to turn the other fractions into ones with 12 in the denominators? Multiply by 1. But which 1? You need to think about what number multiplied by the denominator will give you 12.

There’s another way to think about this, for sure. Think about the denominator you want: 12. What is one-fourth of 12? 3, right, so 1/4 is the same thing as 3/12. For some folks, that way of thinking is going to work much, much easier. But you can choose what works for you. Now we can solve the problem:

So in this case, you need a little more than one pizza. You can either ask your siblings to eat a little less (and get by on one pizza) or you can order two pizzas and put the rest in the freezer. (Personally, I’d choose the second option.)

Subtraction works the exact same way! Just find the common denominator and change the fractions. Then subtract, and finally, simplify your answer (if necessary).

Got it? If not, ask your questions in the comments section. And make sure you try out these practice problems to see how well you can really do! (Remember, no one’s grading anything, so what have you got to lose?)

If you have questions, don’t forget to ask them in the comments section. I also love to hear about different ways to approach these ideas. Don’t be afraid to tell us how you do things differently.

Here are the answers to Wednesday’s practice problems: 15/4, 7/16, 28/15, 30, 1/3.

New here? You’ve stumbled upon January’s Review of Math Basics here at Math for Grownups. This week, we’re doing a quick refresher of fractions.  Monday was multiplication, so if you missed that, you might want to take a quick look before reading further. Math concepts build, you know. 

Psst! Wanna know a secret? Sure you do. So here you go: There’s a debate among math educators about whether dividing with fractions is useful at all. There. I said it. But don’t tell your kids or they might rebel.

But yes, I’m being somewhat serious here. Among math teachers who really, really think about these things — perhaps too much and I’m often in that camp — dividing with fractions is pretty much unnecessary. Okay, so you might need to divide with fractions (like when you’re halving a recipe). But while the process is stupidly simple (trust me), there are other ways to think about it that may make more sense.

Let’s take a look at that rule:

Dividing by a fraction is the same thing as multiplying by its reciprocal.

If you know what all of those words mean, you can recognize that this is pretty darned easy. But if your days in elementary school are long past, you might have forgotten what the reciprocal is. Luckily, this is no big deal. The reciprocal of a fraction is formed when you switch the numerator and denominator. In layman’s terms, you turn the fraction upside down. Like this:

It couldn’t be easier, right? So let’s put it all in context with an example.

See what we did there? We turned the second fraction over and multiplied instead of divided. This is called the “invert and multiply” process. Now, all we need to do is simplify the answer.

Notice how the 4 and 6 are both divisible by 2? Well, that means the fraction can be simplified. On a 4th-grade math test, this means your teacher wants you to do more work. In the real world, it just means that the fraction will be easier to work with or even understand. (When you see the result, you’ll know what I mean.)

Doesn’t 2/3 seem a lot easier to understand than 4/6? Think of recipes. Do you have a 1/6-cup measure in your cabinet? (I don’t.)

So let’s consider how this works (or why, if you’d rather) by considering a really basic division problem: 1 ÷ 1/2.

How many ½s fit into 1? That’s the question that division asks, right? Think about those measuring cups. If you had two ½ cup measuring cups, you would have the equivalent of 1 cup. In other words:

Make sense? Now here’s another way to look at it:

Let me summarize: 2 ½s fits into 1. In other words, 1÷ ½ is 2. And that turns out to be the same thing as multiplying by the reciprocal of ½, which is 2.

That’s a lot to take in, and you don’t have to know it by heart – or even fully understand. It just explains why this crazy rule works. And here’s another secret – there are lots of other ways to divide fractions. You can do it in your head. (It’s pretty easy to solve this problem without any arithmetic: ½ ÷ ¼. Right?) Or you could even find a common denominator (more on that Friday) and then just divide the numerators. (I’ll leave that process for you to figure out if you’re so inclined.)

The thing is, there aren’t many times in the real world that dividing by fractions is really necessary. Here’s an example to explain what I mean. Let’s say I’m cutting a recipe in half. The recipe calls for ¾ cup of sugar. How much will I actually need? Well, I can look at the question in a couple of different ways. (See which one jumps out at you.)

I would bet – and I can’t prove it – that most of you thought about the second option. That’s because you’re cutting the recipe in half, not dividing the recipe by 2.

In short, dividing by fractions is pretty darned simple, compared to other things you have been required to do in math. Too bad it doesn’t show up much in the real world, right?

Just for fun, try these problems on for size – using whatever method works for you. (No need to show your work!) Bonus points if you can simplify your answer, when necessary. (And no, there are no bonus points, because there are no points.)

The answers to Monday’s problems: ⅓, 4/35, 15/8 or 1⅞, 5¼, 9⅔. How did you do? ETA: Me? Not so good. I made a careless error with the last problem. The correct answer is 3 ⅔, which is explained by the comments below. 

Welcome to Week 2 of January’s Back to Math Basics — a quick review of the basic math that you need to do everyday math. Answers to last Friday’s integers questions are at the end of this post.

When kids are first learning about fractions, teachers often turn to something that all but the lactose- or gluten-intolerant can appreciate — pizza! (And I can empathize with the allergy inclined. For you, imagine a dairy-free, vegetable pie with polenta crust — yum!)

This is for very good reason: Fractions are simply parts of the whole. When you cut a pizza into 12 equal parts you are creating twelfths. To count them, you’d start at one piece and count around the pizza (or in random order, makes no diff): one-twelfth (1/12), two-twelfths (2/12), three-twelfths (3/12)… all the way to 12-twelfths (12/12) or the whole pizza (1). Half of the pizza is six-twelfths (6/12) or one-half (1/2). A fourth of the pizza is three-twelfths (3/12) or one-fourth (1/4). Get it?

(Okay, so it’s really, really hard to write a blog post about fractions. In Word, I can depend on something call MathType to write fractions, which I’ll create for examples below. But in paragraphs, this doesn’t work so well.  So please bear with me!)

It might make sense to start with addition and subtraction, but in this case, multiplication and division is the better start. (Spoiler alert: You’ll use multiplication to add and subtract. Really.) But just like with integers, multiplying and dividing fractions are really, really easy.

So let’s go back to those pizzas. Let’s say your son is having a birthday party, and he wants to serve pizza. If each kid can eat 1/4 of a pizza and there are 12 kids at the party, how many pizzas do you need to buy? (Seriously, this is not as dorky a question as it might sound. I have had to figure this out IRL.)

Are you actually multiplying two fractions here? Why, yes. Yes you are! In fact, any whole number can be written as a fraction — just use the number itself as the numerator (the top number in a fraction) and 1 as the denominator (the bottom number in a fraction). So…

Now, here’s the multiplication rule. Just multiply the numerators together and then the denominators together.

How easy is that? But what does 15/4 really mean? This is called an improper fraction — which just means that it’s got a numerator that’s bigger than the denominator. But it has a much, much bigger meaning — improper fractions are bigger than one.

How many pizzas is 15/4? Well this is easy too.

Fractions mean division. So to turn an improper fraction into divide the denominator into the numerator. But 4 doesn’t divide evenly into 15. In fact, 4 goes into 15 three times, with 3 left over. (Or as your third-grade self said: 3, with a remainder of 3.)

The whole number is the number of times 4 divides into 15. The remainder becomes the numerator of a fraction, and 4 stays in the denominator. Like this:

Whew! What this is means is that you need 3 and 3/4 pizzas. I don’t know of any pizzeria that delivers in this way, so round up to 4 pizzas, and you should be good to go.

That’s a lot of information. So here’s a quick summary:

1. Any whole number can be written as a fraction. Just use the number as the numerator and put a 1 in the denominator.

2. To multiply fractions, multiply the numerators together and then multiply the denominators together.

3. To change an improper fraction to a mixed number, divide the denominator into the numerator. The whole number answer is the whole number in the mixed number. The remainder is the numerator, and the denominator stays the same.

Show me (or better yet, yourself) what you’ve got with these examples. I’ll have the answers in Wednesday’s post. Questions? Ask them in the comments section.

Answers to Friday’s challenge questions: -30, -2, 5, 32, -14. How did you do?

I haven’t started my holiday baking yet, but that time is just around the corner. Today, I bring you a post from last year, Cookie Exchange Math, in which I look at the fractions involved in tripling my cow cookie — yes, I said cow cookie — recipe. If you need to feed the masses, check out an easy way to manage those pesky and sometimes strange fractions that come from increasing a recipe.

Ah, the cookie exchange!  What better way to multiply the variety of your holiday goodies.  (You can always give the date bars to your great aunt Marge.)

The problem with this annual event is the math required to make five or six dozen cookies from a recipe that yields three dozen.  That’s what I call “cookie exchange math.”

Never fear! You can handle this task without tossing your rolling pin through the kitchen window. Take a few deep breaths and think things through.

To double or triple a recipe is pretty simple — just multiply each ingredient measurement by the amount you want to increase the recipe by.  But it’s also pretty darned easy to get confused, especially if there are fractions involved.  (And there are always fractions involved.)

The trick is to look at each ingredient one at a time.  Don’t be a hero!  Use a pencil and paper if you need to.  (Better yet, if you alter a recipe often enough, jot down the changes in the margin of your cookbook.)  It’s also a good idea to collect all of your ingredients before you get started.  That’ll save you from having to borrow an egg from your neighbor after your oven is preheated.

Read the rest here — and you’ll avoid fractions-related, messy kitchen mistakes.

While you’re at it, check out this interview I did with fantastic candy-maker, Nicole Varrenti, owner of Nicole’s Treats. (I love her chocolate mustaches, personally.) It shouldn’t be any surprise that she uses math daily.

Finally, if you have some holiday-related math questions, would you mind sharing them with me? What trips you up — mathematically — at this time of year? Comment below!