Understanding Fractions: Numerators, Types, Equivalents

• A fraction represents a part of a whole.
• It is written as one number over another, separated by a line.

Numerator and Denominator

• The numerator is the top number, indicating how many parts of the whole are taken.
• The denominator is the bottom number, indicating the total number of equal parts the whole is divided into.
• In the fraction 3/4, 3 is the numerator and 4 is the denominator.

Types of Fractions

• Proper fractions have a numerator less than the denominator and a value less than 1, for example, 2/5.
• Improper fractions have a numerator greater than or equal to the denominator and a value greater than or equal to 1, for example, 7/3.
• Mixed numbers consist of a whole number and a proper fraction, for example, 1 2/3.

Equivalent Fractions

• Equivalent fractions represent the same value, even though they have different numerators and denominators, for example, 1/2 and 2/4.
• Equivalent fractions are found by multiplying or dividing both the numerator and denominator by the same non-zero number.

Adding Fractions with Like Denominators

• When adding fractions with the same denominator, add the numerators and keep the denominator the same.
• For example, 1/5 + 2/5 = (1+2)/5 = 3/5.

Adding Fractions with Unlike Denominators

• Fractions with different denominators require finding a common denominator before adding.
• The least common denominator (LCD) represents the smallest common multiple of the denominators.
• Convert each fraction to an equivalent fraction using the LCD as the denominator.
• Add the numerators while keeping the common denominator.
• Simplify the resulting fraction, if possible.
• For example, to add 1/3 + 1/4, the LCD is 12. Convert 1/3 to 4/12 and 1/4 to 3/12, resulting in 4/12 + 3/12 = 7/12.

Adding Mixed Numbers

• Mixed numbers should be converted to improper fractions before adding.
• Find a common denominator if necessary and add the fractions.
• Add the whole numbers.
• Convert the resulting fraction back to a mixed number if it is improper.
• Simplify the fraction if possible.

Multiplying Fractions

• To multiply fractions, multiply the numerators together to get the new numerator.
• Multiply the denominators together to get the new denominator.
• Simplify the resulting fraction, if possible.
• For example, 2/3 * 3/4 = (23)/(34) = 6/12, which simplifies to 1/2.

Multiplying Fractions and Whole Numbers

• Whole numbers are converted to a fraction by placing them over 1.
• Proceed to multiply the fractions as described above.
• For example, 5 * 1/2 = 5/1 * 1/2 = 5/2.

Multiplying Mixed Numbers

• Convert mixed numbers to improper fractions before multiplying.
• Multiply the improper fractions as described above.
• Convert the resulting improper fraction back to a mixed number if needed.
• Simplify the result if possible

Converting Fractions to Decimals

• Divide the numerator by the denominator to convert a fraction to a decimal.
• The result of the division is the decimal equivalent of the fraction.
• To convert 1/4 to a decimal, divide 1 by 4, which equals 0.25.

Terminating Decimals

• A terminating decimal is a decimal that ends.
• Fractions with denominators that can be expressed as a product of only 2s and 5s can be converted into terminating decimals.
• For example: 3/8 = 0.375 (8 = 222)

Repeating Decimals

• A repeating decimal contains a digit or a group of digits that repeat infinitely.
• A bar is placed over the repeating digits to indicate the repeating pattern.
• For example: 1/3 = 0.3333... = 0.3 (with a bar over the 3)

Subtracting Fractions with Like Denominators

• Subtract the numerators while keeping the denominator the same.
• For example, 3/5 - 1/5 = (3-1)/5 = 2/5.

Subtracting Fractions with Unlike Denominators

• Find a common denominator for both fractions before subtracting.
• Convert each fraction to an equivalent fraction with the common denominator.
• Subtract the numerators and keep the common denominator.
• Simplify the resulting fraction if possible.
• For example, to subtract 1/3 - 1/4, the LCD is 12. Convert 1/3 to 4/12 and 1/4 to 3/12. Then, 4/12 - 3/12 = 1/12.

Subtracting Mixed Numbers

• Before subtracting, convert mixed numbers to improper fractions.
• Find a common denominator if necessary and subtract the fractions.
• Subtract the whole numbers.
• If the resulting fraction is improper, convert it back to a mixed number.
• Simplify the fraction if possible.
• Borrowing may be required if the fraction part of the first mixed number is less than the fraction part of the second mixed number; borrow 1 from the whole number part and add it to the fraction part as an equivalent fraction with the common denominator.