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Study Guide

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Here's a video that showcases math used in the real world.

*All right, Shmoop*, you're probably thinking. *I've humored you for long enough. I've looked at all your silly diagrams, read your silly guide and done your silly exercises. So what's the payoff? Am I ever actually going to use any of this stuff?*

Yes, indeedy! One pretty major topic in which much of this material will come in handy is money. Moolah. Cash. Cheddar. Dollar, dollar bills. You'd like to have money some day, yes? That's what we figured. Well, the better you understand certain mathematical concepts, the better you'll be with money. And if you're really, *really* good, who knows? Maybe one day you'll have an entire room filled with money, and you can jump in and do laps all day long. And you'll owe it all to algebra.

### Decimals in Use

### Paychecks and Expenses

### Sample Problem

Luis works 2.5 hours a day, 4 days a week. How many hours a week does Luis work?

2.5 hours/day × 4 days/week = 10 hours/week

Only 10 hours a week? Guess who will

*not*be jumping into a roomful of money? Luis.### Prices and Counting Change

### Sample Problem

Nathan buys a sandwich for $6.50. It's one of those "gourmet" sandwiches, which just means that it's a regular turkey sandwich but comes with a packet of dijon mustard. Anyway, he pays with a $10 bill. Assuming the cashier doesn't screw things up, how much change does Nathan get?

$10 – $6.50 = $3.50

So Nathan gets $3.50 in change.

### Tipping

### Sample Problem

Kate and Jim ate at a restaurant. The total bill was $56.80. They want to leave only a 10% tip because their server spilled two drinks and one plate of nachos into their laps. How much of a tip should they leave?

By the way, there's a really quick way to figure out 10% of a bill: just move the decimal point one spot to the left. Kate and Jim's bill was $56.80, so they can slide that decimal point over between the 5 and 6 to get $5.68. That's only if they're leaving a measly 10% tip though, which we

*really*don't recommend doing. Restaurant servers have bills to pay too, y'know?### Sharing Work

### Sample Problem

A librarian has 50 books that need to go back on the shelves. She has her five student helpers divide up the books evenly. How many books will each student put back on the shelves?

books each

Hopefully they're not all enormous bricks like

*War and Peace*.### How to Solve a Math Problem

### How to Solve a Math Problem

Shouldn't we have started out with this? Oh, it's no big deal. You already know how to do all this business. It's just a little review of the general process, so this should all feel like old hat to you.

There are three steps to solving a math problem.

1.

**Figure out what the problem is asking.**

2. Solve the problem.

3. Check the answer.Simple enough.

### Sample Problem

Doug went to the grocery store. He bought a bunch of bananas for $1.37 and a jar of peanut butter for $2.99. There was no tax (because he lives in the state of Simplified Math Problems—it's near North Dakota), and he paid with a ten-dollar bill. How much change will he get?

Let's work through the steps.

1.

**Figure out what the problem is asking.**We want to know how much change Doug will get. In other words, if he starts with $10, spends $1.37, and then spends $2.99, how much will he have left? He wants to know so that he's clear about how much he'll have to blow on scratchers.

2.

**Solve the problem.**There are two ways to write this word problem as an arithmetic problem. We can write:

$10 – ($1.37 + $2.99)

or:

$10 – $1.37 – $2.99

Either way, Doug gets $5.64 in change. That'll get him five $1 scratchers, and he'll still have a bit left for laundry money. You go, Doug.

3.

**Check the answer.**We can check to see if the answer is reasonable or in the right ballpark by estimating. $1.37 is about $1.40, and $2.99 is about $3, so our answer should be about $10 – $4.40 = $5.60. Yep, $5.64 is indeed close to $5.60, which means our answer is probably right on the money.

We can also check to see that our answer is absolutely, perfectly correct by working backwards. In this case, the cost of the bananas plus the cost of the peanut butter plus the change should add up to $10.00. So $1.37 + $2.99 + $5.64 should equal $10.00, which it does. Weird. We just totally got a taste for banana peanut butter sandwiches. Wonder where

*that*came from.### I Like Abstract Things: Summary

Okay, so you're a dreamer. There's plenty in algebra for you, too. Just think: we've touched on the concept of infinity, which is pretty far out there. Even if you were immortal (you're not), you could never finish reading the decimal representation of a single irrational number. Plus, there are so many irrational numbers that you can't possibly list them all. However, we can technically list the

*rational*numbers, which means that, although there are infinitely many rational numbers, there are still more irrational numbers than rational numbers.Mind. Blown.

These ideas are part of set theory, which addresses what it means to "count" collections of infinitely many objects. You know, like Jay Leno's car collection.

We've also started proving theorems, such as the fact that the square root of 2 is irrational. Most of mathematics is more about proofs than about arithmetic, but arithmetic is a good place to find some interesting proofs. So is a photography studio, come to think of it.

Let's continue on to Algebraic Expressions!

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