Applied Math: Rational Numbers

Rational Numbers
The term rational means that the number can be expressed as a ratio or fraction. That is, a number, r, is rational if and only if it can be represented by a fraction  where a and b are integers and b does not equal 0. The set of rational numbers includes integers and decimals.

If there is no finite way to represent a value with a fraction of integers, then the number is irrational.

Common examples of irrational numbers include: .

Fractions
A fraction is a number that is expressed as one integer written above another integer, with a dividing line between them .

It represents the quotient of the two numbers: “x divided by y.” It can also be thought of as x out of y equal parts.
The top number of a fraction is called the numerator, and it represents the number of parts under consideration.

The 1 in  means that 1 part out of the whole is being considered in the calculation.

The bottom number of a fraction is called the denominator, and it represents the total number of equal parts. The 4 in  means that the whole consists of 4 equal parts. A fraction cannot have a denominator of zero; this is referred to as “undefined.”

Fractions can be manipulated, without changing the value of the fraction, by multiplying or dividing (but not adding or subtracting) both the numerator and denominator by the same number. If you divide both numbers by a common factor, you are reducing or simplifying the fraction. Two fractions that have the same value but are expressed differently are known as equivalent fractions. For example,  are all equivalent fractions.

They can also all be reduced or simplified to .

When two fractions are manipulated so that they have the same denominator, this is known as finding a common denominator.

The number chosen to be that common denominator should be the least common multiple of the two original denominators.

Example:  the least common multiple of 4 and 6 is 12. Manipulating to achieve the common denominator: .

Proper Fractions and Mixed Numbers
A fraction whose denominator is greater than its numerator is known as a proper fraction, while a fraction whose numerator is greater than its denominator is known as an improper fraction. Proper fractions have values less than one and improper fractions have values greater than one.

A mixed number is a number that contains both an integer and a fraction. Any improper fraction can be rewritten as a mixed number.

Example: .

Similarly, any mixed number can be rewritten as an improper fraction. Example: . and Improper Fractions and Mixed Numbers

Adding and Subtracting Fractions
If two fractions have a common denominator, they can be added or subtracted simply by adding or subtracting the two numerators and retaining the same denominator. If the two fractions do not already have the same denominator, one or both of them must be manipulated to achieve a common denominator before they can be added or subtracted.

Example: .

Multiplying Fractions
Two fractions can be multiplied by multiplying the two numerators to find the new numerator and the two denominators to find the new denominator.

Example: .

Dividing Fractions
Two fractions can be divided by flipping the numerator and denominator of the second fraction and then proceeding as though it were a multiplication.

Example: . 

Multiplying a Mixed Number by a Whole Number or a Decimal
When multiplying a mixed number by something, it is usually best to convert it to an improper fraction first.

Additionally, if the multiplicand is a decimal, it is most often simplest to convert it to a fraction.

For instance, to multiply  by 3.5, begin by rewriting each quantity as a whole number plus a proper fraction. Remember, a mixed number is a fraction added to a whole number and a decimal is a representation of the sum of fractions, specifically tenths, hundredths, thousandths, and so on:

Next, the quantities being added need to be expressed with the same denominator. This is achieved by multiplying and dividing the whole number by the denominator of the fraction.

Recall that a whole number is equivalent to that number divided by 1:

When multiplying fractions, remember to multiply the numerators and denominators separately:
 

Now that the fractions have the same denominators, they can be added:

Finally, perform the last multiplication and then simplify:

 

Decimals
Decimals are one way to represent parts of a whole. Using the place value system, each digit to the right of a decimal point denotes the number of units of a corresponding negative power of ten.

For example, consider the decimal 0.24. We can use a model to represent the decimal. Since a dime is worth one-tenth of a dollar and a penny is worth one-hundredth of a dollar, one possible model to represent this fraction is to have 2 dimes representing the 2 in the tenths place and 4 pennies representing the 4 in the hundredths place:

 

To write the decimal as a fraction, put the decimal in the numerator with 1 in the denominator. Multiply the numerator and denominator by tens until there are no more decimal places.
Then simplify the fraction to lowest terms. For example, converting 0.24 to a fraction:

Operations with Decimals
Adding and Subtracting Decimals

When adding and subtracting decimals, the decimal points must always be aligned.

Adding decimals is just like adding regular whole numbers. Example: .
If the problem-solver does not properly align the decimal points, an incorrect answer of 4.7 may result.

An easy way to add decimals is to align all of the decimal points in a vertical column visually. This will allow one to see exactly where the decimal should be placed in the final answer. Begin adding from right to left. Add each column in turn, making sure to carry the number to the left if a column adds up to more than 9.

The same rules apply to the subtraction of decimals.

Multiplying Decimals

A simple multiplication problem has two components: a multiplicand and a multiplier. When multiplying decimals, work as though the numbers were whole rather than decimals. Once the final product is calculated, count the number of places to the right of the decimal in both the multiplicand and the multiplier. Then, count that number of places from the right of the product and place the decimal in that position.

For example,  has a total of three places to the right of the respective decimals. Multiply  to get 31488.
Now, beginning on the right, count three places to the left and insert the decimal. The final product will be 31.488.

Dividing Decimals

Every division problem has a divisor and a dividend. The dividend is the number that is being divided. In the problem , 14 is the dividend and 7 is the divisor. In a division problem with decimals, the divisor must be converted into a whole number. Begin by moving the decimal in the divisor to the right until a whole number is created. Next, move the decimal in the dividend the same number of spaces to the right.

For example, 4.9 into 24.5 would become 49 into 245. The decimal was moved one space to the right to create a whole number in the divisor, and then the same was done for the dividend. Once the whole numbers are created, the problem is carried out normally:

Returning Correct Change to a Customer
Determining the correct change to return to a customer is an exercise in mental math. As such, it can be helpful to learn shortcuts, but most often, such skills are developed through repeated practice. To be clear, the term ‘correct change’ is usually used to mean two things: the correct value and the fewest number of coins or bills necessary. Obviously, the most important part is giving the right amount in change, but it would be considered poor service, for instance, to give someone 110 pennies rather than a dollar bill and a dime.

However, there are situations where your options may be limited and you must be creative in the way you put the change together.

For example, if you are working at a register and run out of dollar bills, you could substitute 4 quarters for each dollar in change you give out until you get some more dollar bills.

The process for finding the correct change begins with a purchase total and the amount of cash given.

In other words, .

Once you determine the change amount, you can then divide by each monetary unit starting with the largest one that is less than the change amount, then divide the remainder by the next largest, and so on.

For instance, if the purchase total is $7.69 and the customer gives you a ten-dollar bill, the change amount is . To determine the correct change, divide by $1 to get 2 with a remainder of $0.31. This means that you will need two dollar bills. The remainder is what is left after dividing; in this case that is $0.31. Now, divide the remainder by $0.25 to get
1 with a remainder of $0.06. This means that you will need one quarter. Next, divide $0.06 by $0.05 to get 1 with a remainder of $0.01. This means that you will need one nickel. Finally, a remainder of $0.01 means you will need one penny.

Another way to think about this is to break the change amount into parts based on the coin or bill:

 
Check that this selection actually equals the desired amount:


Percentages
Percentages can be thought of as fractions that are based on a whole of 100; that is, one whole is equal to 100%. The word percent means "per hundred."

Percentage problems are often presented in three main ways:
- Find what percentage of some number another number is. o   Example: What percentage of 40 is 8?
- Find what number is some percentage of a given number. o   Example: What number is 20% of 40?
- Find what number another number is a given percentage of. o   Example:

What number is 8 20% of?
There are three components in each of these cases: a whole
(W), a part (P), and a percentage (%). These are related by the equation: .

This can easily be rearranged into other forms that may suit different questions better:  and . Percentage problems are often also word problems.

As such, a large part of solving them is figuring out which quantities are what.

For example, consider the following word problem:
In a school cafeteria, 7 students choose pizza, 9 choose hamburgers, and 4 choose tacos. What percentage of student choose tacos?
To find the whole, you must first add all of the parts: .
The percentage can then be found by dividing the part by the whole ):.

Converting Between Percentages, Fractions, and Decimals
Converting decimals to percentages and percentages to decimals is as simple as moving the decimal point. To convert from a decimal to a percentage, move the decimal point two places to the right.

To convert from a percentage to a decimal, move it two places to the left. It may be helpful to remember that the percentage number will always be larger than the equivalent decimal number. For example:

To convert a fraction to a decimal, simply divide the numerator by the denominator in the fraction.

To convert a decimal to a fraction, put the decimal in the numerator with 1 in the denominator. Multiply the numerator and denominator by tens until there are no more decimal places. Then simplify the fraction to lowest terms.

For example, converting 0.24 to a fraction:

Fractions can be converted to a percentage by finding equivalent fractions with a denominator of 100.

For example,

To convert a percentage to a fraction, divide the percentage number by 100 and reduce the fraction to its simplest possible terms.

For example,