Discovering irrational numbers
4. Subtract the equation in Step 2 from the equation in Step 3, and then solve for N
, . . . .
. . . .
N N N N
10 000 1235 555555555 1000 123 555555555 9000 1112
9000 1112
9000 1112
1125 139
1125 139
=
- =
=
=
= =
I divided the numerator and denominator by 8 to reduce the fraction to lowest terms.
Understanding the Relationship between Percents and Decimals
Percents are found daily in newspaper ads, financial statements, medical reports, and so on. However, percentages aren’t really the same as the rest of our numbers. You need to change from a percent to a decimal before doing any computations involving percents of other quantities.
A percentis a number that represents how many compared to 100.Percents are usually much quicker and easier to understand or know the worth of than their fractional counterparts. Sure, you have some easier equivalents, such as 1⁄2and 50% or 3⁄4and 75%. But a fraction’s worth may not jump out at you.
For instance, you may not immediately realize that 20
7 is 35%.
In the earlier section, “Changing from Fractions to Decimals,” I show you how to get from a fraction to its decimal equivalent. In this section, you see how to change from decimals to percents and from percents to decimals.
Percents are easy to understand, because you can quickly compare the per- cent amount to 100. If you’re told that 89% of the people have arrived at an event, you know that most of the people have come. After all, 89 out of 100 is a lot of those expected. Similarly, if you’re told that the chance of rain is 1%, you know that the chances are slim-to-none that you’ll have precipitation, because 1 out of 100 isn’t a large amount.
Decimals are fairly easy to deal with, but you may have more trouble wrap- ping your head around how much 0.125 of an estate is. Did you realize that 0.125 is the same as 12.5%? If so, you’re probably in the minority. But good for you for knowing!
Transforming from percents to decimals
A percentageis recognizable by the % symbol, or just by the word, percent.
Percents are sort of pictures of amounts, and so they need to be changed to a decimal or fractional equivalent before combining them with other numbers.
If I told you that 89% of the people had arrived at a meeting, and you’re expecting 500 people, you can’t just multiply 89 by 500 to get the number of people who have arrived. Why? Well, 89 ×500 = 44,500. So you can see that the answer doesn’t make any sense. Instead, you need to change percents to decimals before trying to do computations.
To change a percent to its decimal equivalent, move the decimal point in the percent two places to the left. Moving the decimal point two places to the left gives you the same result as multiplying the percent amount by 0.01 (one hundredth). Why are you multiplying by 0.01? Because percents are compar- isons to 100; you’re changing the percent to how many out of 100.
The decimal equivalent of 12.5% is 0.125, and the decimal equivalent of 6% is 0.06. Notice that with 6%, I had to add a 0 in front of the 6 to be able to move the decimal point the two decimal places. You assume that there’s a decimal point to the right of the 6 in 6%.
If you’re told that 89% of the people have arrived at the company-wide meet- ing, and you’re expecting 500 people, how many people have arrived?
To find out, simply change 89% to its decimal equivalent, 0.89. Then multiply 0.89 ×500 = 445 people.
When checking on the latest shipment of eggs, your dairy manager tells you that 1⁄2% of them are cracked. The shipment contained 3,000 dozen eggs. How many were cracked?
First, change 3,000 dozen to an actual number of eggs by multiplying 3,000 × 12 = 36,000 eggs. Next, change 1⁄2% to a decimal. The fraction 1⁄2is 0.5 as a deci- mal, so 1⁄2% is 0.5%. Now, to change 0.5% to a decimal, move the decimal point two places to the left to get 0.005. Multiply 36,000 by 0.005, and you discover that 180 eggs are cracked. That’s 15 dozen eggs in all.
Moving from decimals to percents
When you change a number from a decimal to a percent, it’s probably because you have a task in mind. Most likely, you’ve started with a fraction and are using the decimal as the transition number— the number between fractions and percents. If the decimal terminates, you can just carry all the digits along in the percent, or you can round off to a predetermined number
25
Chapter 2: Fractions, Decimals, and Percents
of digits. If the decimal repeats, you want to decide how many decimal points you want in the percent value.
To change a decimal to its percent equivalent, move the decimal point in the decimal two places to the right.
If 13 out of 20 people received a flu shot, what percentage of people have received the shot?
To find the percentage, first write 13 out of 20 as a fraction. Then you can divide 13 by 20 to get the decimal equivalent: 0.65. (You can see how to change fractions to decimals in the earlier section “Changing from Fractions to Decimals.”) Now change 0.65 to a percent by moving the decimal point two places to the right, which gives you 65%. (I don’t show the decimal point to the right of the 5 in 65%, because it’s understood to be there when it’s not otherwise shown.)
The evening crew has completed 67 out of the 333 packets that need to be filled for the big order. What percentage of the packets has been completed?
The decimal equivalent of the fraction 333
67 is 0.201201 . . . . As you can see, this is a repeating decimal. Moving the decimal point two places to the right gives you 20.1201201 . . . . Now you need to decide how many decimal places to keep.
Let’s say that you want just the nearer percent. In that case, you round off all the digits to the right of the decimal point. The number 1,201,201 is less than half of 5,000,000, so you can just drop all the digits to the right of the decimal point and call the percentage 20%. (See “Rounding decimals up or down,” ear- lier in this chapter, for more on how to round numbers and drop digits.)
Dealing with more than 100%
Sometimes you may find it difficult to get your head around the report that there’s a 300% increase in the number of travelers in a company or that you’re getting a 104% increase in your salary. Just what do the percents really mean, and how do you work with them in computations?
First, think about 100% of something. For instance, if you’re told that 100% of a job is complete, you know that the whole thing is wrapped up. But it doesn’t make sense that 200% of the job is complete — you can’t do more than all of the job. So, instead, think of 100% in terms of money. If I pay you 100% of what I owe you, say $800, I pay you all the money — $800. But if I pay you 200% of what I owe you, I pay you all of it twice — or $1,600. (Now, why would I do that? Because I really like you!)
For percents that exceed 100%, apply the same rule for changing percents to decimals. The equivalent of 200% is 2.00, and the equivalent of 104% is 1.04.
Try this example to better understand how to work with percentages that are equal to more than 100. Suppose you’re promised a 104% increase in salary if you agree to stay in your position for another 5 years. If you’re currently earning $87,000, what will your salary be for the next 5 years?
To start, multiply 1.04 by $87,000 to get $90,480. Now the question is: Does the $90,480 actually represent an increase— an amount to be added to your current salary? If so, you’ll be earning $87,000 + $90,480 = $177,480. Or did the person offering the increase actually mean to just increase your salary by 4%, in which case you’d be earning the $90,480? You need to be sure you under- stand the terms exactly. In Chapter 3, you can see how to be sure of your fig- ures when talking about percentages and their ups and downs.
Coming to Grips with Fractions
I’ll bet that fractions are your favorite things. Everyone I know just loves frac- tions. Oh, who am I kidding? I didn’t have you fooled for a minute. So frac- tions are way down on your list of things to talk about at a party, but they can’t be ignored. Fractions are a way of life. You cut a pie into eighths, sixths, or fourths (or, heaven forbid, sevenths). You whip out your 7⁄8-inch wrench.
You measure a board to be 5 feet and 9⁄16inches long.
See what happened when the United States decided not to go metric? We got stuck using all these fractions when computing. So, until that happy day when the country changes its mind regarding the metric system, you candeal with fractions and the operations that go with them.
Fractions are still useful for describing amounts. They’re exact numbers, so a sale proclaiming 1⁄3off the original price tells you to divide by 3 and take one of them away from the original price. Do you get the same thing changing the fraction to a decimal? No, not really. The decimal for 1⁄3is 0.3333 . . ., which approximates 1⁄3but isn’t exactly the same. You don’t really notice the differ- ence unless you’re dealing with large amounts of money or assets.
Adding and subtracting fractions
Fractions can be added and subtracted only if they have the same denomina- tor. So before adding or subtracting them, you have to find equivalent values for the fractions involved so that they have the same denominator. One frac- tion is equivalent to another if the numerator and denominator of the one
27
Chapter 2: Fractions, Decimals, and Percents
fraction are the same multiple as the numerator and denominator of the other fraction. For example, the following three fractions are equivalent to the fraction
18 6 :
, ,
18 6
5 5
90 30
18 6
8 8
144 48
18 6
6 1 6 1
3
= = =1
$ $ $
You could find infinite numbers of fractions that are equivalent to the given fraction. The three I chose are found by multiplying the numerator and denominator by 5, 8, and 1⁄6.
As I note earlier, you can only add or subtract fractions if they have the same denominator. So if two fractions that need to be added together don’t have the same denominator, you have to change the fractions until they do. You change either one or both of the fractions until they have the same (common) denominator. For example, if you want to add 3⁄5and 7⁄10, you change the 3⁄5to an equivalent fractions with 10 in the denominator. The fraction equivalent to
3⁄5is 6⁄10, which has the same (common) denominator as the fraction 7⁄10. Find the sum of 3⁄4and 3⁄8.
Change 3⁄4to a fraction with 8 in the denominator. Then add the two numera- tors together. Your math should look like this:
4 3
2 2
8 6 8 6
8 3
8 9
= + =
$
The sum of the two fractions, 9⁄8, is an improper fraction— it’s bigger than 1.
So, you need to rewrite it as 11⁄8.
To write an improper fraction as a mixed number, you divide the denomina- tor into the numerator. The number of times that the denominator divides is the whole number, in front. The remainder goes in the numerator of the frac- tion that’s left. If you need to borrow from the whole number in order to sub- tract, you add the equivalent of 1 to the fraction.
Find the difference between 91⁄3and 47⁄8.
The common denominator of the two fractions is 24. So write the two as equivalent fractions and then subtract. Here’s what the math looks like:
93 1
8
8 9
24 8 48
7 3
3 4
24 21
=
- = -
$
$
You borrow 1 from the 9, which you write as 24
24 before adding it to the top fraction, like so:
924 8
24
24 8
24 32 424
21 424
21 424 11
8 + =
- = -
Whenever you have to borrow from a whole number in order to subtract frac- tions, you change the 1 that’s borrowed from the whole number to a fraction with the common denominator in both the top and bottom of the fraction that’s added.
Multiplying and dividing fractions
Adding and subtracting fractions is generally more complicated than multi- plying and dividing them. The nice thing about multiplication and division of fractions is that you don’t need a common denominator. And you can reduce the fractions before ever multiplying to make the numbers smaller and more manageable. The challenge, however, is that you have to change all mixed numbers to improper fractions and change whole numbers to fractions by putting them in the numerator with a 1 in the denominator.
Multiply 51⁄3by 41⁄5.
To find the product of this problem, first change the mixed numbers to improper fractions. To do so, multiply the whole number by the denominator, add the numerator, and then write the sum over the denominator, like this:
53 1
3 5 3 1
3 16 45
1 5 4 5 1
5 21
= +
=
= + =
$
$
You’re almost ready to multiply the two improper fractions together. But first, remember that it helps to reduce the multiplication problem. Do so by divid- ing the 3 in the denominator of the first fraction and the 21 in the numerator of the second fraction by 3:
3 16
5 21
3 16
5 21
1 7
$
=$
29
Chapter 2: Fractions, Decimals, and Percents
After you do that, you’re ready to multiply the two numerators and two denominators together. Your math will look like this:
1 16
5 7
5 112 22
5
= = 2
$
The final answer, 222⁄5, is obtained by dividing the numerator by the denomi- nator and writing the quotient and remainder as a mixed fraction.
To divide one fraction by another, you change the problem to multiplication by flipping the second fraction. In other words, division is actually multiplica- tion by the reciprocalof a number. The reciprocal of 2, for example, is 1⁄2, and the reciprocal of 4⁄7is 7⁄4.
Divide 83⁄4by 31⁄8.
To begin, change the mixed numbers to improper fractions. Then change the division to multiplication by multiplying the first fraction by the reciprocal of the second fraction. Your math will look like this:
84
3 3
8 1
4 35
8 25
4 35
25 ' = ' =
$
8Now reduce the multiplication problem by dividing by 5 in the upper left and lower right and dividing by 4 in the other two numbers. Then multiply across the top and bottom and write the final answer as a mixed number. Here’s what the math will look like:
4 35
25 8
4 35
25 8
5 14 2
5 4
1 7
5 2
= = =