Pre-Algebra Notes – Unit Five: Rational Numbers and Equations
Rational Numbers
Rational numbers are numbers that can be written as a quotient of two integers.
Since decimals are special fractions, all the rules we learn forfractions will work for decimals. The only difference is the denominators for decimalsare powers of 10; i.e., 101,102,103,104, etc.... Students normally think of powers of 10 in standard form: 10, 100, 1000, 10,000, etc.
In a decimal, the numerator is the number to the right of the decimal point. Thedenominator is not written, but is implied by the number of digits to the right of thedecimal point. The number of digits to the right of the decimal point is the same as thenumber of zeros in the power of 10: 10, 100, 1000, 10,000…
Therefore, one place is tenths, two places are hundredths, and three places are thousandths.
Examples:1) 0.562 places →
2) 0.5323 places →
3) 3.21 place →
The correct way to say a decimal numeral is to:
1)Forget the decimal point (if it is less than one).
2)Say the number.
3)Then say its denominator and add the suffix “ths”.
Examples:1)0.53Fifty-three hundredths
2)0.702Seven hundred two thousandths
3)0.2Two tenths
4)5.63Five and sixty-three hundredths
When there are numbers on both sides of the decimal point, the decimal point is read as“and”. You say the number on the left side of the decimal point, and then the decimal point is read as “and”. You then saythe number on the right side with its denominator.
Examples:1) Write 15.203 in word form.
Fifteen and two hundred three thousandths
2) Write 7.0483 in word form.
Seven and four hundred eighty-three ten-thousandths
3) Write 247.45 in word form.
Two hundred forty-seven and forty-five hundredths
Converting Fractions to Decimals:
Terminating and Repeating Decimals
Syllabus Objective: (2.22) The student will write rational numbers in equivalent forms. (2.3) the student will translate among different forms of rational numbers.
CCSS 8.NS.1-2: Understand informally that every number has a decimal expansion; show that the decimal expansion of a rational number repeats eventually or terminates.
A rational number, a number that can be written in the form of (quotient of two integers), will either be a terminating or repeating decimal. A terminating decimal has a finite number of decimal places; you will obtain a remainder of zero. A repeating decimal has a digit or a block of digits that repeat without end.
One way to convert fractions to decimals is by making equivalent fractions.
Example:Convert to a decimal.
Since a decimal is a fraction whose denominator is a power of 10, look for a power of 10 that 2 will divide into evenly.
Since the denominator is 10, we need only one digit to the right of the decimal point, and the answer is 0.5.
Example:Convert to a decimal.
Again, since a decimal is a fraction whose denominator is a power of 10, we look for powers of 10 that the denominator will divide into evenly. 4 won’t go into 10, but 4 will go into 100 evenly.
Since the denominator is 100, we need two digits to the right of the decimal point, and the answer is 0.75.
There are denominators that will never divide into any power of 10 evenly. Since thathappens, we look for an alternative way of converting fractions to decimals. Could yourecognize numbers that are not factors of powers of ten? Using your Rules ofDivisibility, factors of powers of ten can only have prime factors of 2 or 5. That wouldmean 12, whose prime factors are 2 and 3, would not be a factor of a power of ten. Thatmeans that 12 will never divide into a power of 10 evenly. For example, a fraction such as will not terminate – it will be a repeating decimal.
Not all fractions can be written with a power of 10 as the denominator. We need to look at another way to convert a fraction to a decimal: divide the numerator by the denominator.
Example:Convert to a decimal.
This could be done by equivalent fractions since the only prime factor of 8 is 2.
However, it could also be done by division.
Doing this division problem, we get 0.375 as the equivalent decimal.
Example:Convert to a decimal.
This could not be done by equivalent fractions since one of the factors of 12 is 3. We can still convert it to a decimal by division.
Six is repeating, so we can write it as .
The vinculum is written over the digit or digits that repeat.
Example:Convert to a decimal.
This would be done by division.
Converting Decimals to Fractions
Syllabus Objective: (2.22) The student will write rational numbers in equivalent forms. (2.3) The student will translate among different forms of rational numbers.
CCSS 8.NS.1-3: Convert a decimal expansion which repeats eventually into a rational number.
To convert a decimal to a fraction:
1)Determine the denominator by counting the number of digits to the right of the decimal point.
2)The numerator is the number to the right of the decimal point.
3)Simplify, if possible.
Examples:1) Convert 0.52 to a fraction.
2) Convert 0.613 to a fraction.
3) Convert 8.32 to a mixed number and improper fraction.
But what if we have a repeating decimal?
While the decimals 0.3 and look alike at first glance, they are different. They do not have the same value. We know 0.3 is three tenths, . How can we say or write as a fraction?
As we often do in math, we take something we don’t recognize and make it look like a problem we have done before. To do this, we eliminate the repeating part – the vinculum (line over the 3).
Example:Convert to a fraction.
Let’s let x = 0.333333...
Notice, and this is important, that only one number is repeating. If I multiply both sides of the equation above by 10 (one zero), then subtract the two equations, the repeating part disappears.
is the equivalent fraction for
Example:Convert to a fraction.
The difficulty with this problem is the decimal is repeating. So we eliminate the repeating part by letting.
Note, three digits are repeating. By multiplying both sides of the equation by 1000 (three zeros), the repeating parts line up. When we subtract, the repeating part disappears.
Example:Convert to a fraction.
Note, one digit is repeating, but one is not. By multiplying both sides of the equation by 10, the repeating parts line up. When we subtract, the repeating part disappears.
Ready for a “short cut”? Let’s look at some patterns for repeating decimals.
It is easy to generate the missing decimals when you see the pattern!
Let’s continue to look at a few more repeating decimals, converting back into fractional form.
Because we are concentrating on the pattern, we will choose NOT to simplify fractions whereapplicable. This would be a step to add later.
The numerator of the fraction is the same numeral as the numeral under the vinculum. We can also quickly determine the denominator: it is 9thsfor one place under the vinculum, 99thsfor two places under the vinculum, 999thsfor three places under the vinculum, and so on.
But what if the decimal is of a form where not all the numerals are under the vinculum? Let’s look at a few.
Note that again we chose not to simplify fractions where applicable as we want to concentrate on the pattern.
Does????
Do you believe it? Let's look at some reasons why it's true. Using the method we just looked at:
Surely if 9x = 9, then x = 1. But since x also equals .9999999... we get that .9999999... = 1.
But this is unconvincing to many people. So here's another argument. Most people who have trouble with this fact oddly don't have trouble with the fact that 1/3 = .3333333... . Well, consider the following addition of equations then:
This seems simplistic, but it's very, very convincing, isn't it? Or try it with some other denominator:
Which works out very nicely. Or even:
It will work for any two fractions that have a repeating decimal representation and that add up to 1. The problem, though, is BELIEVING it is true.
So, you might think of 0.9999.... as another name for 1, just as 0.333... is another name for 1/3.
Comparing and Ordering Rational Numbers
Syllabus Objectives: (2.24) The student will explain the relationship among equivalent representations of rational numbers.
We will now have fractions, mixed numbers and decimals in ordering problems. Sometimes you can simply think of (or draw) a number line and place the numbers on the line. Numbers increase as you go from left to right on the number line, so this is particularly helpful when you are asked to go from least to greatest.
If placement is not obvious (for instance, when values are very close together), it may be advantageous to write all the number in the same form (decimal or fractional equivalents), and then compare.
Example: Order the numbers from least to greatest.
Let’s first rewrite all improper fractions as mixed numbers.
Now let’s place the values on the number line.
From least to greatest, the order would be .
Sometimes writing the numbers in the same form will assist you in ordering.
Example: Order from least to greatest.
(1)Find the decimal equivalents,
then compare.
Adding and Subtracting Fractions
with Like Denominators
Syllabus Objective: (2.4) The student will add fractions and mixed numbers. (2.5) The student will subtract fractions and mixed numbers.
Let’s add . Will it be ? Why not? If we did, the fraction would indicate that we have two equal size pieces and that 8 of these equal size pieces made one whole unit. That’s not true.
Let’s draw a picture to represent this:
Notice the pieces are the same size. That will allow us to add the pieces together. Each rectangle has 4 equally sized pieces. Mathematically, we say that 4 is the common denominator. Now let’s count the number of shaded pieces.
Adding the numerators, a total of 2equally sized pieces are shaded and 4 pieces make one unit. We can now show:
Example: Find the sum of .
Since the fractions have the same denominator, we write the sum over 9.
Example: Find the difference of .
Since the fractions have the same denominator, we write the difference over 5.
Writing these problems with variables does not change the strategy.
Example: Simplify the variable expression.
Adding and Subtracting Fractions
with Unlike Denominators
Syllabus Objective: (2.4) The student will add fractions and mixed numbers. (2.5) The student will subtract fractions and mixed numbers.
Let’s first review the ways to find a common denominator. We find the least common denominator by determining the least common multiple.
Strategy 1: Multiply the numbers. This is a quick, easy method to use when the numbers are relatively prime (have no factors in common).
Example: Find the LCM of 4 and 5.
Since 4 and 5 are relatively prime, LCM would be .
Strategy 2: List the multiples. Write multiples of each number until there is a common multiple.
Example: Find the LCM of 12 and 16.
12, 24, 36, 48, 60, …
16, 32, 48, 64, …
48 is the smallest multiple of both numbers; therefore, 48 is the LCM.
Strategy 3: Prime factorization. Write the prime factorization of both numbers. The LCM must contain all the factors of both numbers. Write all prime factors, using the highest exponent.
Example: Find the LCM of 60 and 72.
and
The LCM is
This strategy can also be shown by using a Venn diagram.
Example: Find the LCM of 36 and 45.
Draw a Venn diagram, placing common factors in the intersection. The LCM is the product of all the factors in the diagram.
As the numbers in the denominator become larger, this strategy can become cumbersome. That is when the value of the following strategy becomes evident.
Strategy 4: Simplifying/Reducing Method. Write the two numbers as a single fraction; then reduce and find the cross products. The product is the LCM.
Example: Find the LCM of 18 and 24.
When adding or subtracting fractions, LCM is referred to as the Least Common Denominator (LCD). We have several waysto find a common denominator.
Methods of Finding a Common Denominator- Multiply the denominators.
- List multiples of each denominator, use a common multiple.
- Find the prime factorization of the denominators, and find the Least Common Multiple.
- Use the Simplifying/Reducing Method.
- the denominators are prime numbers or relatively prime.
- the denominators are small numbers.
- the denominators are small numbers; some will advise to never or seldom use this method.
- the denominators are composite numbers/ large numbers.
Let’s add . Will it be ? Why not? If we did, the fraction would indicate that we have two equal size pieces and that 7 of these equal size pieces made one whole unit. That’s just not true.
Let’s draw a picture to represent this:
Notice the pieces are not the same size.
Making the same cuts in each rectangle will result in equally sized pieces. That will allow us to add the pieces together. Each rectangle now has 12 equally sized pieces. Mathematically, we say that 12 is the common denominator. Now let’s count the number of shaded pieces.
From the drawing we can see that is the same as and has the same value as .
Adding the numerators, a total of 7 equally sized pieces are shaded and 12 pieces make one unit.
If we did a number of these problems, we would be able to find a way of adding and subtracting fractions without drawing the picture.
Using the algorithm, let’s try one.
Example:
Multiply the denominators to find the least common denominator,. Now make equivalent fractions and add the numerators.
These problems can also be written horizontally.
Let’s try a few. Using the algorithm, first find the common denominator, and then make equal fractions. Once you complete that, you add the numerators and place that result over the common denominator and simplify, if possible.
Remember, the reason you are finding a common denominator is so you have equally sized pieces. To find a common denominator, use one of the strategiesshown. Since the denominators are relatively prime, use the “multiply the denominators” method.
Example:
Example:
Writing these problems with variables does not change the strategy.
Example: Simplify the expression.
The LCD is 15. Making equivalent fractions, we have:
If the denominators are larger composite numbers, using the reducing method to find the common denominator may make the work easier.
Example: Simplify the expression.
Using the Simplifying/Reducing method:, , so the LCD is 72.
Another nice feature of using the Simplifying/Reducing Method is that you do not need to compute what or because we can see the number in the cross products. That is, we can identify 18 times 4is 72, so we multiply −5c by 4 to obtain the new numerator (). Likewise, since 24 times 3 is 72, we determine the other numerator as .
Example:Evaluate the expression.
Regrouping To Subtract Mixed Numbers
Syllabus Objective: (2.5) The student will subtract fractions and mixed numbers.
The concept of borrowing when subtracting with fractions has been typically a difficult area for kids to master. For example, when subtracting , students usually answer if they subtract this problem incorrectly. In order to ease the borrowing concept for fraction, it would be a good idea to go back and review borrowing concepts that kids are familiar with.
Example: Take away 3 hours 47 minutes from 5 hours 16 minutes.
5 hrs 16 min
3 hrs 47 min
?????????
Subtracting the hours is not a problem but students will see that 47 minutes cannot be subtracted from 16 minutes. In this case, students will see that 1 hour must be borrowed from 5 hrs and added to 16 minutes:
5 hrs 16 min
3 hrs 47 min
?????????
Now the subtraction problem can be rewritten as:
4 hrs 76 min4 hrs 76 min
3 hrs 47 min 3 hrs 47 min
??????????? 1 hr 29 min
If students can understand the borrowing concept from the previous example, the same concept can be linked to borrowing with mixed numbers. Lets go back to the first example: .It may be easier to link the borrowing concept if the problem
is rewritten vertically:
???????
Example: Subtract Step 1. Find a common denominator:
The common denominator is 10.
Step 2. Make Equivalent fractions using 10 as
the denominator.
Step 3. It is not possible to subtract the numerators. You cannot take 5 from 4!! Use the concept of borrowing as described in the above examples to re-write this problem. Borrow from 1 from 13 and add 1 () to .
Example: Catherine has a canister filled with cups of flour. She used cups of flour to bake a cake. How much flour is left in the canister?
Subtract .
Step 1. Find a common denominator: The common denominator between 2 and 4 is 4.
Step 2. Make equivalent fractions using 4 as the common denominator.
Step 3. When subtracting the numerators, it is not possible to take 3 from 2, therefore
borrow. It may be easier to follow the borrowing if the problem is rewritten
vertically .
There are cups of flour left in the canister.
Multiplying Fractions and Mixed Numbers
Syllabus Objective: (2.6) The student will multiply fractions and mixed numbers.
Multiplying fractions is pretty straight-forward. So, we’ll just write the algorithm for it, give an example and move on.
Example:
Since is not a fraction, we convert it to