Basic Math and Pre-Algebra For Dummies (68 page)

A little street smarts should tell you that the words
salary increase
or
raise
mean more money, so get ready to do some addition. Here's an example:

• Alison's salary was $40,000 last year, and at the end of the year, she received a 5% raise. What will she earn this year?
  

To solve this problem, first realize that Alison got a raise. So whatever she makes this year, it will be more than she made last year. The key to setting up this type of problem is to think of percent increase as “100% of last year's salary plus 5% of last year's salary.” Here's the word equation:

• this year's salary = 100% of last year's salary + 5% of last year's salary
  

Now you can just add the percentages (see the nearby sidebar for why this works):

• this year's salary = 105% of last year's salary
  

Change the percent to a decimal and the word
of
to a multiplication sign; then fill in the amount of last year's salary:

• this year's salary = 1.05 × $40,000
  

Now you're ready to multiply:

• this year's salary = $42,000
  

So Alison's new salary is $42,000.

The word
interest
means more money. When you receive interest from the bank, you get more money. And when you pay interest on a loan, you pay more money. Sometimes people earn interest on the interest they earned earlier, which makes the dollar amounts grow even faster. Here's an example:

Bethany placed $9,500 in a one-year CD that paid 4% interest. The next year, she rolled this over into a bond that paid 6% per year. How much did Bethany earn on her investment in those two years?

This problem involves interest, so it's another problem in percent increase — only this time, you have to deal with two transactions. Take them one at a time.

The first transaction is a percent increase of 4% on $9,500. The following word equation makes sense:

Now, substitute $9,500 for the initial deposit and calculate:

At this point, you're ready for the second transaction. This is a percent increase of 6% on $9,880:

Then subtract the initial deposit from the final amount:

So Bethany earned $972.80 on her investment.

When you hear the words
discount
or
sale price,
think of subtraction. Here's an example:

Greg has his eye on a television with a listed price of $2,100. The salesman offers him a 30% discount if he buys it today. What will the television cost with the discount?

In this problem, you need to realize that the discount lowers the price of the television, so you have to subtract:

Thus, the television costs $1,470 with the discount.

Part IV

 For find out how to use probability to calculate the odds in dice games, go to
www.dummies.com/extras/basicmathandprealgebra
.

In this part…

• Represent very large and very small numbers with scientific notation
  
• Weigh and measure with both the English and metric systems
  
• Understand basic geometry, including points, lines, and angles, plus basic shapes and solids
  
• Present math info visually, using bar graphs, pie charts, line graphs, and the
  xy
  -graph
  
• Solve word problems involving measurement and geometry
  
• Answer real-world questions with statistics and probability
  
• Get familiar with some basic set theory, including union and intersection
  

Chapter 14

In This Chapter

Knowing how to express powers of ten in exponential form

Appreciating how and why scientific notation works

Understanding order of magnitude

Multiplying numbers in scientific notation

Scientists often work with very small or very large measurements — the distance to the next galaxy, the size of an atom, the mass of the Earth, or the number of bacteria cells growing in last week's leftover Chinese takeout. To save on time and space — and to make calculations easier — people developed a sort of shorthand called
scientific notation.

Scientific notation uses a sequence of numbers known as the powers of ten, which I introduce in Chapter
2
:

• 1 10 100 1,000 10,000 100,000 1,000,000 10,000,000 ...
  

Each number in the sequence is 10 times more than the preceding number.

Powers of ten are easy to work with, especially when you're multiplying and dividing, because you can just add or drop zeros or move the decimal point. They're also easy to represent in exponential form (as I show you in Chapter
4
):

Scientific notation is a handy system for writing very large and very small numbers without writing a bunch of 0s. It uses both decimals and exponents (so if you need a little brushing up on decimals, flip to Chapter
11
). In this
chapter, I introduce you to this powerful method of writing numbers. I also explain the order of magnitude of a number. Finally, I show you how to multiply numbers written in scientific notation.

Scientific notation uses powers of ten expressed as exponents, so you need a little background before you can jump in. In this section, I round out your knowledge of exponents, which I first introduce in Chapter
4
.

Numbers starting with a 1 and followed by only 0s (such 10, 100, 1,000, 10,000, and so forth) are called
powers of ten,
and they're easy to represent as exponents. Powers of ten are the result of multiplying 10 times itself any number of times.

 To represent a number that's a power of 10 as an exponential number, count the zeros and raise 10 to that exponent. For example, 1,000 has three zeros, so 1,000 = 10
³
(10
³
means to take 10 times itself three times, so it equals 10 × 10 × 10). Table 
14-1
shows a list of some powers of ten.

Table 14-1 Powers of Ten Expressed as Exponents

When you know this trick, representing a lot of large numbers as powers of ten is easy — just count the 0s! For example, the number 1 trillion — 1,000,000,000,000 — is a 1 with twelve 0s after it, so

This trick may not seem like a big deal, but the higher the numbers get, the more space you save by using exponents. For example, a really big number is a googol, which is 1 followed by a hundred 0s. You can write this:

• 10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
  

As you can see, a number of this size is practically unmanageable. You can save yourself some trouble and write 10
¹⁰⁰
.

 A 10 raised to a negative number is also a power of ten.

You can also represent decimals using negative exponents. For example,

Although the idea of negative exponents may seem strange, it makes sense when you think about it alongside what you know about positive exponents. For example, to find the value of 10
⁷
, start with 1 and make it larger by moving the decimal point seven spaces to the right: