Symbols of addition 6 pot

Then add the 1 pound to the 4 pounds: 4 pounds 25 ounces = 4 pounds + 1 pound 9 ounces = 5 pounds 9 ounces SUBTRACTION WITH MEASUREMENTS 1. Subtract like units if possible. 2. If not, regroup units to allow for subtraction. 3. Write the answer in simplest form. For example, 6 pounds 2 ounces subtracted from 9 pounds 10 ounces. 9 lb 10 oz Subtract ounces from ounces. – 6 lb 2 o z Then subtract pounds from pounds. 3 lb 8 oz Sometimes, it is necessary to regroup units when subtracting. Example Subtract 3 yards 2 feet from 5 yards 1 foot. Because 2 feet cannot be taken from 1 foot, regroup 1 yard from the 5 yards and convert the 1 yard to 3 feet. Add 3 feet to 1 foot. Then subtract feet from feet and yards from yards: 5 4 ΋ yd 1 4 ΋ ft – 3 y d 2 ft 1 yd 2ft 5 yards 1 foot – 3 yards 2 feet = 1 yard 2 feet MULTIPLICATION WITH MEASUREMENTS 1. Multiply like units if units are involved. 2. Simplify the answer. Example Multiply 5 feet 7 inches by 3. 5 ft 7 in Multiply 7 inches by 3, then multiply 5 feet by 3. Keep the units separate. ϫ 3 15 ft 21 in Since 12 inches = 1 foot, simplify 21 inches. 15 ft 21 in = 15 ft + 1 ft 9 in = 16 ft 9 in – THEA MATH REVIEW– 115 Example Multiply 9 feet by 4 yards. First, decide on a common unit: either change the 9 feet to yards, or change the 4 yards to feet. Both options are explained below: Option 1: To change yards to feet, multiply the number of feet in a yard (3) by the number of yards in this problem (4). 3 feet in a yard ϫ 4 yards = 12 feet Then multiply: 9 feet ϫ 12 feet = 108 square feet. (Note: feet ϫ feet = square feet = ft 2 ) Option 2: To change feet to yards, divide the number of feet given (9), by the number of feet in a yard (3). 9 feet ÷ 3 feet in a yard = 3 yards Then multiply 3 yards by 4 yards = 12 square yards. (Note: yards • yards = square yards = yd 2 ) DIVISION WITH MEASUREMENTS 1. Divide into the larger units first. 2. Convert the remainder to the smaller unit. 3. Add the converted remainder to the existing smaller unit if any. 4. Divide into smaller units. 5. Write the answer in simplest form. Example Divide 5 quarts 4 ounces by 4. 1. Divide into the larger unit: 1 qt r 1 qt 4ͤ5 ෆ q ෆ t ෆ – 4 qt 1 qt 2. Convert the remainder: 1 qt = 32 oz 3. Add remainder to original smaller unit: 32 oz + 4 oz = 36 oz – THEA MATH REVIEW– 116 4. Divide into smaller units: 36 oz ÷ 4 = 9 oz 5. Write the answer in simplest form: 1 qt 9 oz Metric Measurements The metric system is an international system of measurement also called the decimal system. Converting units in the metric system is much easier than converting units in the customary system of measurement. However, mak- ing conversions between the two systems is much more difficult. The basic units of the metric system are the meter, gram, and liter. Here is a general idea of how the two systems compare: Metric System Customary System 1 meter A meter is a little more than a yard; it is equal to about 39 inches 1 gram A gram is a very small unit of weight; there are about 30 grams in one ounce. 1 liter A liter is a little more than a quart. Prefixes are attached to the basic metric units listed above to indicate the amount of each unit. For exam- ple, the prefix deci means one-tenth ( ᎏ 1 1 0 ᎏ ); therefore, one decigram is one-tenth of a gram, and one decimeter is one-tenth of a meter. The following six prefixes can be used with every metric unit: Kilo Hecto Deka Deci Centi Milli (k) (h) (dk) (d) (c) (m) 1,000 100 10 ᎏ 1 1 0 ᎏ ᎏ 1 1 00 ᎏ ᎏ 1,0 1 00 ᎏ Examples ■ 1 hectometer = 1 hm = 100 meters ■ 1 millimeter = 1 mm = ᎏ 1,0 1 00 ᎏ meter = .001 meter ■ 1 dekagram = 1 dkg = 10 grams ■ 1 centiliter = 1 cL* = ᎏ 1 1 00 ᎏ liter = .01 liter ■ 1 kilogram = 1 kg = 1,000 grams ■ 1 deciliter = 1 dL* = ᎏ 1 1 0 ᎏ liter = .1 liter *Notice that liter is abbreviated with a capital letter—L. – THEA MATH REVIEW– 117 The chart below illustrates some common relationships used in the metric system: Length Weight Volume 1 km = 1,000 m 1 kg = 1,000 g 1 kL = 1,000 L 1 m = .001 km 1 g = .001 kg 1 L = .001 kL 1 m = 100 cm 1 g = 100 cg 1 L = 100 cL 1 cm = .01 m 1 cg = .01 g 1 cL = .01 L 1 m = 1,000 mm 1 g = 1,000 mg 1 L = 1,000 mL 1 mm = .001 m 1 mg = .001 g 1 mL = .001 L Conversions within the Metric System An easy way to do conversions with the metric system is to move the decimal point either to the right or left because the conversion factor is always ten or a power of ten. Remember, when changing from a large unit to a smaller unit, multiply. When changing from a small unit to a larger unit, divide. Making Easy Conversions within the Metric System When multiplying by a power of ten, move the decimal point to the right, since the number becomes larger. When dividing by a power of ten, move the decimal point to the left, since the number becomes smaller. (See below.) To change from a larger unit to a smaller unit, move the decimal point to the right. → kilo hecto deka UNIT deci centi milli ← To change from a smaller unit to a larger unit, move the decimal point to the left. Example Change 520 grams to kilograms. 1. Be aware that changing meters to kilometers is going from small units to larger units and, thus, requires that the decimal point move to the left. 118 Metric Prefixes An easy way to remember the metric prefixes is to remember the mnemonic: “King Henry Died of Drinking Chocolate Milk”. The first letter of each word represents a corresponding metric heading from Kilo down to Milli: ‘King’—Kilo, ‘Henry’—Hecto, ‘Died’—Deka, ‘of’—original unit, ‘Drinking’—Deci, ‘Chocolate’—Centi, and ‘Milk’—Milli. 2. Beginning at the UNIT (for grams), note that the kilo heading is three places away. Therefore, the decimal point will move three places to the left. k h dk UNIT d c m 3. Move the decimal point from the end of 520 to the left three places. 520 ← .520 Place the decimal point before the 5: .520 The answer is 520 grams = .520 kilograms. Example Ron’s supply truck can hold a total of 20,000 kilograms. If he exceeds that limit, he must buy stabiliz- ers for the truck that cost $12.80 each. Each stabilizer can hold 100 additional kilograms. If he wants to pack 22,300,000 grams of supplies, how much money will he have to spend on the stabilizers? 1. First, change 2,300,000 grams to kilograms. kg hg dkg g dg cg mg 2. Move the decimal point 3 places to the left: 22,300,000 g = 22,300.000 kg = 22,300 kg. 3. Subtract to find the amount over the limit: 22,300 kg – 20,000 kg = 2,300 kg. 4. Because each stabilizer holds 100 kilograms and the supplies exceed the weight limit of the truck by 2,300 kilograms, Ron must purchase 23 stabilizers: 2,300 kg ÷ 100 kg per stabilizer = 23 stabilizers. 5. Each stabilizer costs $12.80, so multiply $12.80 by 23: $12.80 ϫ 23 = $294.40.  Algebra This section will help in mastering algebraic equations by reviewing variables, cross multiplication, algebraic frac- tions, reciprocal rules, and exponents. Algebra is arithmetic using letters, called variables, in place of numbers. By using variables, the general relationships among numbers can be easier to see and understand. Algebra Terminology A term of a polynomial is an expression that is composed of variables and their exponents, and coefficients. A vari- able is a letter that represents an unknown number. Variables are frequently used in equations, formulas, and in mathematical rules to help illustrate numerical relationships. When a number is placed next to a variable, indi- cating multiplication, the number is said to be the coefficient of the variable. – THEA MATH REVIEW– 119 یی ی یی ی 120 Examples 8c 8 is the coefficient to the variable c. 6ab 6 is the coefficient to both variables, a and b. THREE KINDS OF POLYNOMIALS ■ Monomials are single terms that are composed of variables and their exponents and a positive or nega- tive coefficient. The following are examples of monomials: x,5x,–6y 3 ,10x 2 y,7,0. ■ Binomials are two non-like monomial terms separated by + or – signs. The following are examples of binomials: x + 2, 3x 2 – 5x,–3xy 2 + 2xy. ■ Trinomials are three non-like monomial terms separated by + or – signs. The following are examples of trinomials: x 2 + 2x – 1, 3x 2 – 5x + 4, –3xy 2 + 2xy – 6x. ■ Monomials, binomials, and trinomials are all examples of polynomials, but we usually reserve the word polynomial for expressions formed by more three terms. ■ The degree of a polynomial is the largest sum of the terms’ exponents. Examples ■ The degree of the trinomial x 2 + 2x – 1 is 2, because the x 2 term has the highest exponent of 2. ■ The degree of the binomial x + 2 is 1, because the x term has the highest exponent of 1. ■ The degree of the binomial –3x 4 y 2 + 2xy is 6, because the x 4 y 2 term has the highest exponent sum of 6. LIKE TERMS If two or more terms have exactly the same variable(s), and these variables are raised to exactly the same expo- nents, they are said to be like terms. Like terms can be simplified when added and subtracted. Examples 7x + 3x = 10x 6y 2 – 4y 2 = 2y 2 3cd 2 + 5c 2 d cannot be simplified. Since the exponent of 2 is on d in 3cd 2 and on c in 5c 2 d, they are not like terms. The process of adding and subtracting like terms is called combining like terms. It is important to combine like terms carefully, making sure that the variables are exactly the same. Algebraic Expressions An algebraic expression is a combination of monomials and operations. The difference between algebraic expres- sions and algebraic equations is that algebraic expressions are evaluated at different given values for variables, while algebraic equations are solved to determine the value of the variable that makes the equation a true statement. There is very little difference between expressions and equations, because equations are nothing more than two expressions set equal to each other. Their usage is subtly different. – THEA MATH REVIEW– . because the x term has the highest exponent of 1. ■ The degree of the binomial –3x 4 y 2 + 2xy is 6, because the x 4 y 2 term has the highest exponent sum of 6. LIKE TERMS If two or more terms have. coefficient of the variable. – THEA MATH REVIEW– 119 یی ی یی ی 120 Examples 8c 8 is the coefficient to the variable c. 6ab 6 is the coefficient to both variables, a and b. THREE KINDS OF POLYNOMIALS ■ Monomials. polynomial is the largest sum of the terms’ exponents. Examples ■ The degree of the trinomial x 2 + 2x – 1 is 2, because the x 2 term has the highest exponent of 2. ■ The degree of the binomial x + 2