Throughout your stuy of science, you will often nee to solve math problems. This appenix is esigne to help you quickly review the basic math skills you will use most often. Fractions Aing an Subtracting Fractions To a or subtract fractions that have the same enominator, a or subtract the numerators, an then write the sum or ifference over the enominator. Express the answer in lowest terms. s 0 0 0 0 To a or subtract fractions with ifferent enominators, fin the least common enominator. Write an equivalent fraction for each fraction using the least common enominator. Then a or subtract the numerators. Write the sum or ifference over the least common enominator an express the answer in lowest terms. s Multiplying Fractions When multiplying two fractions, multiply the numerators to fin the prouct s numerator. Then multiply the enominators to fin the prouct s enominator. It helps to ivie any numerator or enominator by the greatest common factor before multiplying. Express the answer in lowest terms. s 6 6 Diviing Fractions To ivie one fraction by another, invert an multiply. Express the answer in lowest terms. s 6 6 Ratios an Proportions A ratio compares two numbers or quantities. A ratio is often written as a fraction expresse in lowest terms. A ratio also may be written with a colon. s The ratio of to is written as to,, or :. The ratio of 0 to is written as 0, or :. A proportion is a mathematical sentence that states that two ratios are equivalent. To write a proportion, place an equal sign between the two equivalent ratios. s The ratio of 6 to is the same as the ratio of to. 6 The ratio of to is the same as the ratio of to. You can set up a proportion to etermine an unknown quantity. Use x to represent the unknown. To fin the value of x, cross multiply an then ivie both sies of the equation by the number that comes before x. Two out of five stuents have blue notebooks. If this same ratio exists in a class of twenty stuents, how many stuents in the class have blue notebooks? x 0 0 x x 0 z Cross multiply. z Divie. 6
Percents an Decimals To convert a percent to a ecimal value, write the number without the percent sign an move the ecimal point two places to the left. A a zero before the ecimal point. s % 0..% 0. You can convert a ecimal value to a percent value by moving the ecimal point two places to the right an aing the percent sign. s 0.6 6% 0..% Exponents A base is a number that is use as a factor. An exponent is a number that tells how many times the base is to be use as a factor. A power is any number that can be expresse as a prouct in which all of the factors are the same. Any number raise to the zero power is. Any number raise to the first power is that number. The only exception is the number 0, which is zero regarless of the power it is raise to. Powers of = = 0 = = = Exponents Powers of 0 0 = 00 0 = 0 0 0 = 0 = 0 = 0 00 Multiplying Exponents To multiply exponential expressions with the same base, a the exponents. The general expression for exponents with the same base is x a x b x a + b. ( ) ( ) 6 To raise a power to a power, keep the base an multiply the exponents. The general expression is (x a ) b x ab. ( ) ( ) ( ) ( ) 6 To raise a prouct to a power, raise each factor to the power. The general expression is (xy) n x n y n. ( ) Diviing Exponents To ivie exponential expressions with the same base, keep the base an subtract the exponents. The general expression is: x a x x b a b 6 6 When the exponent of the enominator is greater than the exponent of the numerator, the exponent of the result is negative. A negative exponent follows the general expression: x n x n Math Skills
Scientific Notation Scientific notation is use to express very large numbers or very small numbers. To convert a large number to scientific notation, move the ecimal point to the left until it is locate to the right of the first nonzero number. The number of places that you move the ecimal point becomes the positive exponent of 0.,0,000. 0 To write a number less than in scientific notation, move the ecimal point to the right of the first nonzero number. Use the number of places you move the ecimal point as the negative exponent of 0.. 0.00 0 0 0. 0 - Aing an Subtracting To a or subtract numbers in scientific notation, the exponents must be the same. If they are ifferent, rewrite one of the numbers to make the exponents the same. Then write the answer so that only one number is to the left of the ecimal point..0 0. 0.0 0. 0. 0 =. 0 Multiplying an Diviing To multiply or ivie numbers in scientific notation, the exponents are ae or subtracte. s (. 0 ) (. 0 ) (. 0 ). 0 (.0 0 ) (. 0 6 ) (.0 0 6 ).0 0 Significant Figures When measurements are combine in calculations, the uncertainty of each measurement must be correctly reflecte in the final result. The igits that are accurate in the answer are calle significant figures. When the result of a calculation has more significant figures than neee, the result must be roune off. If the first igit after the last significant igit is less than, roun own. If the first igit after the last significant igit is or more, roun up. s roune to three significant figures is 0. roune to three significant figures is 0..6 roune to three significant figures is.6..6 roune to four significant figures is.. Aing an Subtracting In aition an subtraction, the number of significant figures in the answer epens on the number with the largest uncertainty.. g g.00 g g The measurement with the largest uncertainty is g an it is measure to the nearest gram. Therefore, the answer is given to the nearest gram. Multiplying an Diviing In multiplication an ivision, the measurement with the smallest number of significant figures etermines the number of significant figures in the answer. Mass Density Volume 0. g. ml. g/ml Because. ml has only two significant figures, the answer must be roune to two significant figures.
Formulas an Equations An equation is a mathematical sentence that contains one or more variables an one or more mathematical operators (such as,,,, an ). An equation expresses a relationship between two or more quantities. A formula is a special kin of equation. A formula such as V l w h states the relationship between unknown quantities represente by the variables V, l, w, an h. The formula means that volume (of a rectangular soli) equals length times with times height. Some formulas have numbers that o not vary, such as the formula for the perimeter of a square: P s. In this formula, the number is a constant. To solve for a quantity in an equation or formula, substitute known values for the variables. Be sure to inclue units. An airplane travels in a straight line at a spee of 600 km/h. How far oes it fly in. hours? Write the formula that relates spee, istance, an time. Distance Spee Time v t To solve for istance, multiply both sies of the equation by t. v = t v t t v t t Substitute in the known values. 600 km/h. h 00 km Conversion Factors Many problems involve converting measurements from one unit to another. You can convert units by using an equation that shows how units are relate. For example, in.. cm relates inches an centimeters. To write a conversion factor, ivie both sies of the equation by in. in.. cm in. in.. cm/in. Because the conversion factor is equal to, you can multiply one sie of an equation by it an preserve equality. You can make a secon conversion factor by iviing both sies of the equation by. cm. in.. cm One conversion factor converts inches to centimeters an the other converts centimeters to inches. Choose the conversion factor that cancels out the unit that you have a measurement for. Convert inches to centimeters. Use to represent the unknown number of centimeters. in. 6 cm. cm. cm. cm in. Some conversions are more complicate an require multiple steps. Convert F to a Celsius temperature. The conversion formula is F ( C) F Substitute in F: F ( C) F F F C F C F C Math Skills
Data Tables Data tables help to organize ata an make it easier to see patterns in ata. If you plan ata tables before oing an experiment, they will help you recor observations in an orerly fashion. The ata table below shows Unite States immigration ata for the year 00. Always inclue units of measurement so people can unerstan the ata. Immigration to the Unite States, 00 Place of Origin Africa Asia Europe Number of Legal Immigrants,,6, Circle Graphs Use the total number to calculate percentages. For example, the percentage of immigrants from Africa in 00 was,,06, 0.0 %. Multiply each percent by 60 to fin the central angle of each wege. For Africa, the central angle is. Use a protractor to raw each central angle. Color an label the weges an finish your graph with a title. Legal Immigration to the Unite States South % Africa % North South 0, 6, Bar Graphs To make a bar graph, begin by placing category labels along the bottom axis. A an overall label for the axis Place of Origin. Decie on a scale for the vertical axis. An appropriate scale for the ata in the table is 0 to 00,000. Label the vertical axis Number of People. For each continent, raw a bar whose height correspons to the number of immigrants. You will nee to roun off the values. For example, the bar for Africa shoul correspon to,000 people. A a graph title to make it clear what the graph shows. Legal Immigration to the Unite States Data for year 00 North % Line Graphs The slope of a straight-line graph equals the rise over the run. The rise is the change in the y values an the run is the change in the x values. Using points A an B on the graph below gives Rise Run 60 Europe % Asia % Slope 0. 6 Number of People 00,000 00,000 00,000 00,000 00,000 Data for year 00 Africa Asia Europe North Place of Origin South Slope of a Straight-Line Graph y-axis 6 B Rise A Run 6 0 0 6 0 x-axis 0