Rahul and his family just moved to California where everyone is talking about the drought. But it's July and he's seen more rain in the past month than he ever did in Arizona where they used to live. He finds a chart that gives the annual rainfall for the past 50 years as a decimal amount above or below the average rainfall. How can he organize the data to see if there is a general trend?
In this concept, you will learn to compare positive and negative fractions and decimals.
Comparing Positive and Negative Fractions and Decimals
An integer is any positive whole number or its opposite.
A fraction is any number that is written as a ratio of one number to another. The top number in the fraction is the numerator. The bottom number is the denominator.
A decimal is another type of number that can be expressed as a ratio. 0.5 and 1.327 are two examples of decimals.
Fractions and decimals are examples of rational numbers. A rational number is any number that can be expressed as a ratio. Integers are rational numbers, but so are many of the numbers between them.
In order to compare fractions, you need to give them a common denominator. This allows you to directly compare the numerators. There are many different common denominators for any pair of fractions, but the easiest one to use is usually the Least Common Denominator, that is, the smallest number that both denominators divide into evenly.
Here is an example.
\begin{align*}-\frac{1}{2}>-\frac{3}{4}\end{align*}
The denominators are 2 and 4. The smallest number that both divide into here is 4.
Next, you need to convert one or both fractions so that they have the common denominator.
In this case, the second fraction already has the common denominator, so it can be left alone. But the denominator of the first fraction must be multiplied by 2 in order to make it 4. However, in mathematics you aren't allowed to just multiply the bottom by something because that would change the fraction into a different, non-equivalent, fraction. You are only allowed to multiply the fraction by 1, in fact. Because any number multiplied by 1 is itself. However, you can be creative about what version of 1 you choose. In this example, choose the version of 1=
Then, multiply it out.
\begin{align*}-\frac{1}{2}\times \frac{2}{2}=-\frac{2}{4}\end{align*}
Another way to think of it is that whatever you do to the bottom of the fraction, you also need to do to the top.
After converting, both fractions have the same denominator so their numerators can be compared.
Then, check the signs. In this case, both numbers are negative. When comparing negative numbers, the larger number is further from zero and therefore less.
In this case, that means that -3<-2.
That means\begin{align*}-\frac{1}{2}>-\frac{3}{4}\end{align*}
The answer is >. (The sideways V always opens in the direction of the larger number.)
Rational numbers with decimals are easier to compare because they don't have to be converted. Just look to see which number is bigger. Remember, a positive number is always greater than a negative number, and when comparing negative numbers, the one farther from zero is less.
Here is an example.
-.29 ____ -.56
First, check the signs.
In this case, they are both negative, so the bigger number is less.
Then compare them to see which is bigger.
In this case, .56 is bigger than .29, so it is further from zero.
Therefore, -.29 is greater than -.56: -.29>-.56.
The answer is >.
Examples
Example 1
Earlier, you were given a problem about Rahul and his question about the California drought.
He wants to write the past fifty years in order of rainfall to see if he can tell if California is actually getting drier. Here is his data:
Season (July 1-June 30) (Year given represents end of season) |
Inches Above/Below (+/-) 135 Year Average |
2012 |
-6.29 |
2011 |
+5.22 |
2010 |
+1.38 |
2009 |
-5.90 |
2008 |
-1.45 |
2007 |
-11.77 |
2006 |
-1.79 |
2005 |
+22.98 |
2004 |
-5.73 |
2003 |
+1.44 |
2002 |
-10.56 |
2001 |
+2.96 |
2000 |
-3.41 |
1999 |
-5.89 |
1998 |
+16.03 |
1997 |
-2.58 |
1996 |
-2.54 |
1995 |
+9.37 |
1994 |
-6.87 |
1993 |
+12.38 |
1992 |
+6.02 |
1991 |
-2.99 |
1990 |
-7.63 |
1989 |
-6.90 |
1988 |
-2.50 |
1987 |
-7.32 |
1986 |
+2.88 |
1985 |
-2.16 |
1984 |
-4.55 |
1983 |
+16.3 |
1982 |
-4.27 |
1981 |
-6.02 |
1980 |
+12.00 |
1979 |
+4.69 |
1978 |
+18.46 |
1977 |
-2.68 |
1976 |
-7.77 |
1975 |
-0.63 |
1974 |
-0.06 |
1973 |
+6.28 |
1972 |
-7.81 |
1971 |
-2.66 |
1970 |
-7.24 |
1969 |
+12.49 |
1968 |
+1.60 |
1967 |
+7.02 |
1966 |
+5.46 |
1965 |
-1.30 |
1964 |
-7.05 |
1963 |
-6.60 |
In order to arrange the years from least rainfall to most, first Rahul looks for the year with the least rainfall. He knows this will be a negative value. He looks for the biggest negative whole numbers first, ignoring the decimals.
He finds that 2007 has a -11. That is the biggest negative whole number, so 2007 has the least.
Next, he goes through the rest of the years looking for the next biggest negative whole numbers.
2002 has a -10, so it's next.
Then, he hits a snag because there are 6 years with a -7. 1990=-7.63; 1987=-7.32; 1976=-7.77; 1972=-7.81; 1970=-7.24; 1964=-7.05. In order to order these, he looks at the decimals. The biggest decimal is the most negative. The decimals thus ordered are: .81, .77, .63, .32, .24, .05. Of the -7 years, the least rainfall is 1972 and the most is1964.
So far, his list of rainfall from least to greatest is: 2007, 2002, 1972, 1976, 1990, 1987, 1970, 1964.
Carrying out this process, he makes the following list of years under the average rainfall:
2007, 2002, 1972, 1976, 1990, 1987, 1970, 1964, 1989, 1994, 2012, 1963, 1981, 2009, 1999, 2004, 1984, 1982, 2000, 1999, 1977, 1971, 1997, 1996, 1988, 1985, 2006, 2008, 1965, 1975, 1974.
At this point, he's getting tired of sorting data. He looks at the list so far to see if he can see any trends in it. He sees that 6 out of the last 10 years are under the average rainfall. 13 of the last 20 years are below the average. 16 of the last 25 years are.
He writes those as fractions:
\begin{align*}\frac{6}{10} \frac{13}{20} \frac{16}{25}\end{align*}
Because he's still confused, and he really wants to figure out what everyone is talking about, he converts these fractions to a common denominator to compare them.
First, he decides on a common denominator.
In this case, 100 works.
Then he figures out that the first one must be multiplied by 10/10, the second one by 5/5, and the third one by 4/4 to get them to have the common denominator.
When he does this, he gets these fractions:
\begin{align*}\frac{60}{100} \frac{65}{100} \frac{64}{100}\end{align*}
He concludes that more than half of the driest years in the past 50 years have been in the more recent 25 years.
In the following examples, compare the two fractions.
Example 2
\begin{align*}-\frac{2}{5}\end{align*} _____\begin{align*}-\frac{6}{7}\end{align*}
First, create a common denominator by multiplying the two denominators together.
In this case,
Then, re-write the initial problem.
\begin{align*}-\frac{14}{35}\end{align*}____
Finally, compare the two.
Both are between 0 and -1. But \begin{align*}-\frac{30}{35}\end{align*} is closer to -1 which means that it is more negative. And more negative is smaller. Therefore
The answer is \begin{align*}-\frac{2}{5} > -\frac{6}{7}\end{align*}.
Example 3
-.98 ____ -.88
First, notice the signs of the two decimals.
Because they are both negative, the one that is the bigger number is less than the other.
Next, determine which number is bigger.
.98 is bigger than .88.
Therefore, -.98 < -.88
Example 4
\begin{align*}-\frac{1}{4} \underline{\;\;\;\;\;\;\;\;\;\;\;\;} -\frac{1}{2}\end{align*}
First, determine the common denominator.
In this case, 4 works.
Next, determine which version of "one" each fraction needs to be multiplied by so that it has the common denominator.
The first fraction already has 4 in the denominator, so it can be left alone. 2x2=4, so the second fraction must be multiplied by
.Then, re-write the second fraction with the common denominator.
\begin{align*}-\frac{1}{2}\times \frac{2}{2}=-\frac{2}{4}\end{align*}
Then, re-write the initial problem with the new fraction.
\begin{align*}-\frac{1}{4}\end{align*}____
Then, note the signs.
Since they are both negative, the bigger number is less.
-2 is bigger than -1 so the answer is
Example 5
.67 ____ -.67
First, note the signs.
.67 is positive. -.67 is negative.
Then, remember that a positive number is always greater than a negative.
Therefore, the answer is .67>-.67
Review
Compare each pair of values using <, > or =.
- -.18 ____ -.27
- -.23 ____ -.98
- -9 ____ -11
- -18 ____ -29
- -67 ____ -89
- \begin{align*}-\frac{1}{4} \underline{\;\;\;\;\;\;\;\;\;\;\;\;} -\frac{4}{5}\end{align*}
- \begin{align*}-\frac{3}{4} \underline{\;\;\;\;\;\;\;\;\;\;\;\;} -\frac{1}{3}\end{align*}
- \begin{align*}-\frac{5}{10} \underline{\;\;\;\;\;\;\;\;\;\;\;\;} -\frac{1}{2}\end{align*}
- \begin{align*}-\frac{3}{4} \underline{\;\;\;\;\;\;\;\;\;\;\;\;} -.75\end{align*}
- \begin{align*}-\frac{1}{4} \underline{\;\;\;\;\;\;\;\;\;\;\;\;} -.25\end{align*}
- \begin{align*}-.25 \underline{\;\;\;\;\;\;\;\;\;\;\;\;} -\frac{3}{4}\end{align*}
- \begin{align*}-\frac{18}{20} \underline{\;\;\;\;\;\;\;\;\;\;\;\;} -\frac{1}{2}\end{align*}
Write the following integers in order from least to greatest.
- -4, -12, -19, -8, 0, -2, -1
- 5, 7, 23, 8, -9, -11
- \begin{align*}\frac{-1}{2}, \frac{-1}{4}, \frac{-5}{6}, \frac{-3}{4}\end{align*}
Review (Answers)
To see the Review answers, open this PDF file and look for section 11.4.