Chapter+Four


 * Chapter Four

Sec 4.1 – 4.3 ** **Sec 4.5**
 * Fractions are part of a whole. If the whole was cut into the number of pieces listed on the bottom of the fraction, the top number represents the number of those pieces.
 * “Numerator” is the number on the top
 * “Denominator” is the number on the bottom
 * Proper fractions are fractions that are between 1 and -1, meaning the number on the top is less than the number on the bottom.
 * Improper fractions are when the number on the top is more than the number on the bottom, so improper fractions are either more than 1 or less than -1.
 * Graphing fractions means that you circle the spot on the numberline where that fraction would be. Most fractions are in between the integers, so you have to guess (estimate) how far between the two integers the mark actually is.
 * Equivalent fractions are fractions that are mathematically equal. They have the same value, but do not look the same because they have different numerators and denominators.
 * You can write equivalent fractions, or find out if fractions are equivalent, by setting them up side by side, figuring out what you have to multiply or divide the denominator of the first one by to get the second denominator, then multiply or divide the numerator by the same thing.
 * Simplifying fractions, or put into lowest terms (simplifying is the process, lowest terms is the result)
 * There are a few ways to do this.
 * The most common way is to divide the numerator and the denominator by the same number (called a factor). You have to find a factor that will evenly go into both the top and the bottom. It doesn’t have to be the biggest number, and usually it is easiest to pick 2 if both numbers are even. You may have to do this several times, but you keep dividing both until you can’t find any more factors that go into both numerator and denominator.
 * The other way to do this works well with very large numbers, or with terms that include variables. It is using prime factorization (explained below and again in section 4.4) on both the numerator and denominator individually, then cross out any factors that are the same in both top and bottom. If the top has two factors of 3 and the bottom has 3 factors of three, you can only cross out two of them, leaving a three on the bottom. After you have crossed out all common factors, multiply any remaining factors in the numerator and denominator and that is the fraction in lowest terms.
 * Unless the book or test asks for something else, always reduce fractions to lowest terms to get the correct answer.
 * Prime numbers are numbers that have no factors except 1 and itself. An example would be 5 because no other factors can go evenly into 5 except 1 and 5. 6 is not a prime number because 2 and 3 are also factors.
 * Composite numbers are numbers that have more factors than 1 and itself (the opposite of prime numbers).
 * Zero and 1 are considered neither prime nor composite.
 * Prime factorization means that you take a composite number and break it down into each of its factors and those factors are broken down until all of the factors are prime. This usually looks like a tree with the number at the top being split into two of its factors, then each composite factor split into two more, etc., until the only numbers remaining are prime. The prime factors that remain after the tree is done are written in order from lowest to greatest, with exponents. So, for example, 28 is written as 22 • 7
 * Multiplying fractions
 * Multiplying fractions with integers by putting the two fractions side by side and multiplying both numerators straight across and both denominators straight across the bottom. Then reduce to lowest terms.
 * Multiplying fractions with variables is done the same way. Remember that you can reduce fractions that have the same variable in the numerator and denominator.
 * A reciprocal of a fraction means that you switch the numerator and denominator.
 * Dividing fractions
 * Write the first fraction, then write the reciprocal of the second fraction and then multiply them straight across top and bottom and reduce.
 * Sec 4.4 **
 * Finding a common denominator (both fractions will be equivalent to their original values, and the denominators will be the same, but the numerators probably won’t be.)
 * Inspection (just looking or trying multiples of one of the denominators). Remember that you have to multiply the numerator and denominator by the same thing!
 * Multiply both denominators by each other and reducing later. You still have to multiply numerator and denominator by the same thing.
 * Use prime factorization on each denominator. Cross out one of each shared factor. Then multiply all remaining factors. That will be the common denominator, but you still have to multiply the denominator by whatever is necessary to get to that denominator, and then multiply the numerator by the same number.
 * Adding and subtracting fractions
 * You must have common denominators in order to add or subtract fractions!
 * Put the fractions side by side. Add or subtract the numerators, but the denominator stays the same (whatever the common denominator was.)
 * Adding and subtracting fractions with variables
 * You still have to have a common denominator.
 * Add or subtract the numerators like normal, but you might end up with something like X + 4 on the top. That is fine. Leave the denominator the same.
 * Add or subtract the numerators like normal, but you might end up with something like X + 4 on the top. That is fine. Leave the denominator the same.
 * Mixed numbers are numbers that have an integer part and a fraction part. They could be positive or negative.
 * Converting a mixed number to an improper fraction: Multiply the denominator by the integer, then add the numerator. This becomes the new numerator and the denominator stays the same.
 * · Converting an improper fraction to a mixed number: Divide the numerator by the denominator. The whole part of the answer is the integer, the remainder is the new numerator. The denominator stays the same.
 * Rounding mixed numbers
 * The answer will either be the inter part, or one more than the integer part, depending on whether the fraction part is ½ or more.
 * There are different ways to figure it out, but you are trying to decide if the fraction is more or less than one half.
 * One way is to divide the denominator by two. If the numerator is more than this number, the fraction is more than ½ and the mixed number rounds up. If less, it rounds down.
 * Another way is to multiply the numerator by two. If this number is more than the denominator, the fraction is more than ½ and it rounds up. If less, it rounds down.
 * You could use a fraction of ½ and get a common denominator with the fraction being evaluated. Once you have common denominators, whichever numerator is larger is the largest fraction.
 * Sec 4.6**
 * Fractions with exponents are written as a fraction inside of parentheses with an exponent after the closing parenthesis.
 * You can just put the same exponent on the numerator and on the denominator and figure each one individually. Then reduce the fraction.
 * Order of operations with fractions: The same rules (PEMDAS) apply with fractions.
 * Simplifying complex fractions
 * Complex fractions are a fraction divided by another fraction, but they are written on top of each other with a large division bar between them.
 * You could just rewrite them as side by side fractions and divide as normal.
 * The normal way, however, is to multiply the top fraction and the bottom by the reciprocal of the bottom one. This will give you a denominator of 1 on the complex fraction, which can just be ignored.
 * Sec 4.7**
 * Solving equations with fractions: All of the same rules and procedures apply with fractions as they do with integers, it just takes longer.
 * Clearing Fractions - Multiply each term by the lowest common denominator of all terms, then solve for x.
 * Sec 4.8**
 * Area of a triangle = ½ (Base) ( Height)
 * The height might be the same as one of the sides, or it might not.
 * Volume of rectangular solids
 * Volume is a measure of capacity, like how much water would a glass hold, or how much sand can a box hold?
 * Volume of a rectangular solid = (Length) (Width) (Height)
 * All three measurements must be in the same units (all inches, all centimeters, etc.) If they aren’t you must convert first.
 * The units are always cubed in the answer.
 * Volume of pyramid = 1/3 (base) (height) The base is measured as length times width, then the answer is multiplied by the height and then multiplied by 1/3.