Fractions, Decimals, and Percents
Chapter 2: Fractions, Decimals, and Percents
Suppose, for example, that you’re planning to cut a piece of wood into four equal pieces. So you measure the wood, divide by four, and cut the pieces to size. Consider a piece of wood measuring 6 feet, 33⁄8inches. One-fourth of the length of the board is 1 foot, 6.84375 inches. (Chapter 7 shows you how to deal with feet and inches, and later in this chapter I show how to divide frac- tions.) With that measurement in hand, you’re all set to create four pieces of wood that measure 1 foot, 6.84375 inches each, right? Of course not. You know that the sawdust accounts for at least the last three decimal digits of the number. So, to make things a bit easier, you round off the measure to about 1 foot, 6.8 inches, which is equal to about 1 foot, 67⁄8inches.
When rounding decimals, you decide how many decimal places you want in your number, and then you round to the nearer of that place and lop off the rest of the digits. You can also use the Rule of 5 when rounding. I explain the different ways to round in the following sections.
Rounding to the nearer digit
The rule for rounding is that you choose how many decimal digits you want in your answer, and then you get rid of the excess digits after rounding up or down. Before discarding the excess digits, use the following steps:
1. Count the number of digits to be discarded, and think of the power of 10 that has as many zeros as digits to discard.
2. Use, as a comparison, half of that power of 10.
For example, half of 10 is 5, half of 100 is 50, and so on.
3. Now consider the discarded digits.
If the amount being discarded is smaller than 5, 50, or 500 (and so on), you just drop the extra digits. If the amount being discarded is larger than 5, 50, or 500 (and so on), you increase the last digit of the number you’re keeping by 1 and discard the rest. If you have exactly 5, 50, or 500 (and so on), you use a special rule, the Rule of 5,which I cover later in this chapter.
In the following examples, I talk about rounding a number to the “nearer tenth” or “nearer hundredth.” This naming has to do with the number of placeholders present. To read more about this naming convention, check out the later section, “Ending up with terminating decimals.”
Round the number 45.63125 to the nearer hundredth (leave two decimal places).
You want to keep the whole number, 45, and the first three digits to the right of the decimal point — the 6 and, possibly, the 3. The digits left over form the number 125. Because 125 is less than 500, just drop off those three digits. So your answer is that 45.63125 rounds to 45.63 when it’s rounded to the nearer hundredth.
Round the number 645.645645 . . . to the nearer thousandth (leave three deci- mal places).
You want to keep the whole number, 645, to the left of the decimal point and at least the first two digits to the right of the decimal point. You keep the 6 and 4 and make a decision about the 5. Because the numbers to be dropped off represent 645 (or 6,456, 64,564, and so on), you see that the numbers to be dropped off represent a number bigger than 500 (or 5,000, 50,000, and so on).
So you add 1 to the 5, the last digit, forming the rounded number 645.646 to replace the repeating decimal part of the original number.
Round the number 1.97342 to the nearer tenth (leave one decimal place).
When you round, you’ll be dropping off the digits 7342, which represent a number bigger than 5,000. So you add 1 to the 9. But 1 + 9 = 10, so you have to carry the 1 from the 10 to the digit to the left of the decimal point. Your resulting number is 2.0 after dropping the four digits off. You leave the 0 after the decimal point to show that the number is correct (has been rounded) to the nearer tenth.
Rule of 5
When you’re rounding numbers and the amount to drop off is exactly 5, 50, or 500 (and so on), you round either up or down, depending on which choice cre- ates an even number (a number ending in 0, 2, 4, 6, or 8). The reason for this rule is that half the time you round up and half the time you round down — making the adjustments due to rounding more fair and creating less of an error if you have to repeat the process many times.
Round the number 655.555 to the nearer hundredth (leave two decimal places).
Because there are only three digits, only the last digit needs to be dropped off. And because you have exactly 5, you round the hundredth place to a 6, giving you 655.56. Why? Rounding up gives you an ending digit that’s an even number.
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Chapter 2: Fractions, Decimals, and Percents
Converting Decimals to Fractions
Decimals have a much better reputation than fractions. Why, for the life of me, I can’t understand. But many people do prefer the decimal point followed by a neat row of digits. For most applications, the decimal form of a number is just fine. For many precision computations, though, the exact form,a number’s fractional value, is necessary. Those working with tools know that fractions describe the different sizes (unless you’re working with the metric system).
Changing a decimal into a fraction takes a decision on your part. First, you decide whether the decimal is terminating or repeating. Then, in the case of the repeating decimal, you decide whether all the digits repeat or just some of them do. What if the decimal doesn’t terminate or repeat and just goes on forever and ever? Then you’re done. Only decimals that terminate or repeat have fractional equivalents. The decimals that neither terminate nor repeat are called irrational(how appropriate).
In the following sections, I explain how to change terminating and repeating decimals into fractions. (For more information on irrational numbers, check out the nearby sidebar “Discovering irrational numbers.”)
Ending up with terminating decimals
Terminating decimalsoccur when a fraction has the following in its denominator:
Only factors of 2 or 5 Powers of 2 or 5
Products of powers of 2 or 5
A terminating decimal has a countable number of decimal places, and the number formed has a name based on the number of decimal places being used. The names of the first eight decimal places with a 1 place holder are:
0.1 one tenth
0.01 one hundredth
0.001 one thousandth
0.0001 one ten-thousandth
0.00001 one hundred-thousandth
0.000001 one millionth
0.0000001 one ten-millionth
0.00000001 one hundred-millionth
So, for example, you read the number 0.003456 as three thousand four hundred fifty-six millionths.The millionths comes from the position of the last digit in the number. And, as a bonus, the names of the decimal places actually tell you what to put in the denominator when you change a terminating decimal to a fraction. How?
When changing a terminating decimal to its fractional equivalent, you place the decimal digits over a power of 10 that has the same number of 0s as the number of decimal digits. Then you reduce the fraction to lowest terms.
Check out the following example to see what I mean.
Rewrite the decimal 0.0875 as a fraction in its lowest terms.
The number 0.0875 is eight hundred seventy-five ten-thousandths. To rewrite this number as a fraction, you put the digits 875 (you don’t need the lead zero because 875 and 0875 are the same number) in the numerator of a frac- tion and 10,000 in the denominator (you use 10,000 because there are 4 digits to the right of the decimal point and 4 zeros in 10,000 ). Now reduce the frac- tion. To do so, first divide the numerator and denominator by 25, and then reduce the numerator and denominator by 5. The math should look like this:
, ,
10 000 875
10 000 875
400 35
80 7
400 35
80 7
= = =
Yes, I know that I could have reduced the fraction in one step by dividing both the numerator and denominator by 125, but I don’t really know my mul- tiples of 125 that well, so I chose to reduce the fractions in stages.