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1 SUBAREA I. NUMBER SENSE AND OPERATIONS Competency 000 Understand the structure of numeration systems and ways of representing numbers. A. Natural numbers--the counting numbers, 23,,,... B. Whole numbers--the counting numbers along with zero, 02,,... C. Integers--the counting numbers, their opposites, and zero,..., 0,,,... D. Rationals--all of the fractions that can be formed from the whole numbers. Zero cannot be the denominator. In decimal form, these numbers will either be terminating or repeating decimals. Simplify square roots to determine if the number can be written as a fraction. E. Irrationals--real numbers that cannot be written as a fraction. The decimal forms of these numbers are neither terminating nor repeating. Examples: π,, e 2, etc. F. Real numbers--the set of numbers obtained by combining the rationals and irrationals. Complex numbers, i.e. numbers that involve i or, are not real numbers. Real Numbers Rational Irrational Integers Positive and Negative Fractions Negative Integers Whole Numbers Natural Numbers Zero MIDDLE SCHOOL MATH.

2 Fraction, Decimals and Percentages: To convert a fraction to a decimal, simply divide the numerator (top) by the denominator (bottom). Use long division if necessary. If a decimal has a fixed number of digits, the decimal is said to be terminating. To write such a decimal as a fraction, first determine what place value the farthest right digit is in, for example: tenths, hundredths, thousandths, ten thousandths, hundred thousands, etc. Then drop the decimal and place the string of digits over the number given by the place value. If a decimal continues forever by repeating a string of digits, the decimal is said to be repeating. To write a repeating decimal as a fraction, follow these steps. a. Let x = the repeating decimal (ex. x = ) b. Multiply x by the multiple of ten that will move the decimal just to the right of the repeating block of digits. (ex.000x = ) c. Subtract the first equation from the second. (ex.000x x = ) d. Simplify and solve this equation. The repeating block of digits will subtract out. (ex.999x = 76 so x = 76 ) 999 e. The solution will be the fraction for the repeating decimal. Percent means parts of one hundred. Numbers may be represented interchangeably as fractions, decimals or percents. 00% = In order to express a fraction as a decimal, convert it into an equivalent fraction whose denominator is a power of 0 (for example, 0, 00, 000). Examples: = 0.0 = 0% = 4 = 0.40 = 40% 0 4 = 25 = 0.25 = 25% 00 Alternatively, a fraction can be converted into a decimal by dividing the numerator by the denominator. MIDDLE SCHOOL MATH. 2

3 Example: = = 37.5% The shaded region below represents 47 out of 00 or 0.47 or 47 or 47% = 0.5 = 50% 0 = 0. = 0% 3 = = 33 3 % 2 3 = = % 4 = 0.25 = 25% 5 6 = = 83 3 % 5 = 0.2 = 20% 3 8 = = 37 2 % 6 = = % 3 8 = = 37 2 % 8 = 0.2 = 2% 7 8 = = 87 2 % =.0 = 00% MIDDLE SCHOOL MATH. 3

4 Unit Rates, Ratios and Proportions: The unit rate for purchasing an item is its price divided by the number of pounds/ounces, etc. in the item. The item with the lower unit rate is the lower price. Example: Find the item with the best unit price: $.79 for 0 ounces $.89 for 2 ounces $5.49 for 32 ounces = per ounce = 0.72 per ounce 32 = per ounce $.89 for 2 ounces is the best price. A second way to find the better buy is to make a proportion with the price over the number of ounces, etc. Cross multiply the proportion, writing the products above the numerator that is used. The better price will have the smaller product. Example: Find the better buy: $8.9 for 40 pounds or $4.89 for 22 pounds Find the unit price = = 8.9 x 4.89 x 40x = x = 4.89 x = x = Since < , $8.9 is less and is a better buy. To find the amount of sales tax on an item, change the percent of sales tax into an equivalent decimal number. Then multiply the decimal number times the price of the object to find the sales tax. The total cost of an item will be the price of the item plus the sales tax. Example: A guitar costs $20 plus 7% sales tax. How much are the sales tax and the total bill? 7% =.07 as a decimal (.07)(20) = $8.40 sales tax MIDDLE SCHOOL MATH. 4

5 $20 + $8.40 = $28.40 total cost An alternative method to find the total cost is to multiply the price times the factor.07 (price + sales tax): $20.07 = $8.40 This gives you the total cost in fewer steps. Example: A suit costs $450 plus 6½% sales tax. How much are the sales tax and the total bill? 6½% =.065 as a decimal (.065)(450) = $29.25 sales tax $450 + $29.25 = $ total cost An alternative method to find the total cost is to multiply the price times the factor.065 (price + sales tax): $ = $ This gives you the total cost in fewer steps. A ratio is a comparison of two numbers. If a class had boys and 4 girls, the ratio of boys to girls could be written one of three ways: :4 or to 4 or The ratio of girls to boys is: 4 4:, 4 to or 4 Ratios can be reduced when possible. A ratio of 2 cats to 8 dogs would reduce to 2:3, 2 to 3 or 23. Note: Read ratio questions carefully. Given a group of 6 adults and 5 children, the ratio of children to the entire group would be 5:. A proportion is an equation in which a fraction is set equal to another. To solve the proportion, multiply each numerator times the other fraction's denominator. Set these two products equal to each other and solve the resulting equation. This is called cross-multiplying the proportion. MIDDLE SCHOOL MATH. 5

6 Example: 4 x = is a proportion To solve this, cross multiply. (4)(60) = (5)( x) 240 = 5x 6 = x Example: x + 3 = 2 3x is a proportion. To solve, cross multiply. 5( x + 3) = 2(3x + 4) 5x + 5 = 6x = x Example: x = 8 x 4 is another proportion. To solve, cross multiply. ( x )( x ) x x = 8( 2) 2x 8 = 6 2x 24 = 0 ( x 6)( x+ 4) = 0 x = 6 or x = 4 Both answers work. MIDDLE SCHOOL MATH. 6

7 Fractions, decimals, and percents can be used interchangeably within problems. To convert a fraction to a decimal, simply divide the numerator (top) by the denominator (bottom). Use long division if necessary. If a decimal has a fixed number of digits, the decimal is said to be terminating. To write such a decimal as a fraction, first determine what place value the farthest right digit is in, for example: tenths, hundredths, thousandths, ten thousandths, hundred thousands, etc. Then drop the decimal and place the string of digits over the number given by the place value. If a decimal continues forever by repeating a string of digits, the decimal is said to be repeating or non-terminating. "Terminating" means "it ends", unlike, say, the decimal for / 3, that goes on to infinity. A non-terminating decimal can t be converted to a fraction, because it is a non-fractional number. Memorize some of the more basic repeating decimals, like = / 3, = 2 / 3, / 6 = 0.666, 5 / 6 = , / 9 = 0... To write a repeating decimal as a fraction, follow these steps.. Let x = the repeating decimal (e.g. x= ) 2. Multiply x by the multiple of ten that will move the decimal just to the right of the repeating block of digits. (e.g. 000x= ) 3. Subtract the first equation from the second. (e.g. 000x-x= ) 4. Simplify and solve this equation. The repeating block of digits will subtract out. (e.g. 999x = 76 so x= ) The solution will be the fraction for the repeating decimal. Another method to convert any repeating decimal into a fraction is by counting the number of decimal places, and putting the decimal's digits over followed by the appropriate number of zeroes. Example: MIDDLE SCHOOL MATH. 7

8 In the case of a repeating decimal, the following procedure is often used. Suppose you have a number like This number is equal to some fraction; call this fraction "x". That is: x = There is one repeating digit in this decimal, so multiply x by "" followed by one zero; that is, multiply by 0: 0x = Now subtract the former from the latter: That is, 9x = 5.2 = 52/0 = 26/5. Solving this, we get x = 26/45. The answer can be verified by dividing 26/45 on your calculator. A decimal can be converted to a percent by multiplying by 00%, or merely moving the decimal point two places to the right. A percent can be converted to a decimal by dividing by 00%, or moving the decimal point two places to the left. Examples: Convert the following decimals into percents = 37.5% 0.7 = 70% 0.04 = 4 % 3.5 = 35 % Examples: Convert the following percents into decimals. 84% = % = % = 0.6 MIDDLE SCHOOL MATH. 8

9 0% =. % = 0.5% = A percent can be converted to a fraction by placing it over 00 and reducing to simplest terms. Examples: Convert the following percents into fractions. 32% = 32 = % = 6 = % = = To find the decimal equivalent of a fraction, use the denominator to divide the numerator. Note decimal comes from deci or part of ten. Example: Find the decimal equivalent of Since 0 cannot divide into 7 evenly, put a decimal point in the answer row on top; put a zero behind 7 to make it 70. Continue the division process. If a remainder occurs, put a zero by the last digit of the remainder and continue the division. Thus = It is a good idea to write a zero before the decimal point so that the decimal point is emphasized. Example: Find the decimal equivalent of MIDDLE SCHOOL MATH. 9

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