3: Fractions, Decimals, and Percentages

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3.1: Visualize Fractions (Part 1)
A fraction is a way to represent parts of a whole. The denominator b represents the number of equal parts the whole has been divided into, and the numerator a represents how many parts are included. The denominator, b, cannot equal zero because division by zero is undefined. A mixed number consists of a whole number and a fraction. When a fraction has a numerator that is smaller than the denominator, it is called a proper fraction, and its value is less than one.
3.2: Visualize Fractions (Part 2)
Equivalent fractions are fractions that have the same value. When working with fractions, it is often necessary to express the same fraction in different forms. To find equivalent forms of a fraction, we can use the Equivalent Fractions Property. We can use the inequality symbols to order fractions. Remember that a > b means that a is to the right of b on the number line. As we move from left to right on a number line, the values increase.
3.3: Fraction Multiplication
A fraction is considered simplified if there are no common factors, other than 1, in the numerator and denominator. If a fraction does have common factors in the numerator and denominator, we can reduce the fraction to its simplified form by removing the common factors. To multiply fractions, we multiply the numerators and multiply the denominators. Then we write the fraction in simplified form.
3.4: Fraction Division
The reciprocal of the fraction a/b is b/a, where a ≠ 0 and b ≠ 0. A number and its reciprocal have a product of 1. To find the reciprocal of a fraction, we invert the fraction. This means that we place the numerator in the denominator and the denominator in the numerator. To divide fractions, multiply the first fraction by the reciprocal of the second.
3.5: Add and Subtract Fractions with Matching (Common) Denominators
To add fractions, add the numerators and place the sum over the common denominator. To subtract fractions, subtract the numerators and place the difference over the common denominator.
3.6: Add and Subtract Fractions with Different Denominators
In fraction multiplication, you multiply the numerators and denominators together, respectively. To divide fractions, you multiply the first fraction by the reciprocal of the second. For fraction addition, add the numerators together and place the sum over the common denominator. If the fractions have different denominators, first convert them to equivalent forms with the LCD. Likewise, for fraction subtraction, subtract the numerators and place the difference over the common denominator.
3.7: Percentages
A percent is a ratio whose denominator is 100. Since percents are ratios, they can easily be expressed as fractions. Remember that percent means per 100, so the denominator of the fraction is 100. To convert a percent to a decimal, we first convert it to a fraction and then change the fraction to a decimal. To convert a decimal to a percent, remember that percent means per hundred. If we change the decimal to a fraction whose denominator is 100, it is easy to change that fraction to a percent.
3.8: Solve General Applications of Percent
We will solve percent equations by using the methods we used to solve equations with fractions or decimals. Many applications of percent occur in our daily lives, such as tips, sales tax, discount, and interest. To solve these applications we'll translate to a basic percent equation, just like those we solved in the previous examples in this section. Once you translate the sentence into a percent equation, you know how to solve it.
3.9: Decimals (Part 1)
Decimals are another way of writing fractions whose denominators are powers of ten. To convert a decimal number to a fraction or mixed number, look at the number to the left of the decimal. If it is zero, the decimal converts to a proper fraction. If not, the decimal converts to a mixed number. The numbers to right of the decimal point become the numerator while the place value corresponding to the final digit represent to the denominator. Finally, simplify the fraction if possible.
3.10: Decimals (Part 2)
Since decimals are forms of fractions, locating decimals on the number line is similar to locating fractions on the number line. To round a decimal, locate the given place value and mark it with an arrow. Underline the digit to the right of the place value and determine if it is greater than or equal to 5. If it is, add one to the digit in the given place value. If not, don't change the digit. Finally, rewrite the number, removing all digits to the right of the given place value.
3.11: Decimal Operations (Part 1)
To add or subtract decimals, write the numbers vertically so the decimal points line up. Use zeros for place holders, as needed. Then, add or subtract the numbers as if they were whole numbers. Lastly, place the decimal in the answer under the decimal points in the given numbers. Multiplying decimals is like multiplying whole numbers—we just have to determine where to place the decimal point. The number of decimal places in the product is the sum of the number of decimal places in the factors.
3.12: Decimal Operations (Part 2)
Just as with multiplication, division of decimals is very much like dividing whole numbers. To divide a decimal by a whole number, we place the decimal point in the quotient above the decimal point in the dividend and then divide as usual with long division. Sometimes we need to use extra zeros at the end of the dividend to keep dividing until there is no remainder. To divide decimals, we multiply both the numerator and denominator by the same power of 10 to make the denominator a whole number.