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Chapter 5—Ratios and Proportions 5.1 Ratios and Rates Terms to Know: Ratio: Rate: A comparison of two A ratio of quantities quantities using with different units; e.g. division—part to whole 30 miles or part to part 2 hours E.g.: 3 girls and 4 boys Also must be in lowest Written in different terms or cannot be left “complex”—e.g. ways: 3 , 3 to 4, 3:4 4 1 Should always be in c 15 miles or 2 lowest terms, or 4 fl oz 3 hour simplified Unit Rate: A rate with a denominator of 1—e.g. 2.5 miles = .5 miles 5 minutes 1 minutes Finding ratios and rates from a written statement There are 30 students in the class—18 female and 12 male. The subway car travels at 4.5 miles in 5 minute. *May need to convert between units—e.g. 2.5 miles to ___ miles 5 minutes ___ hour Finding ratios and rates from a table Pounds per box: Boxes 0 1 2 3 Pounds 0 30 60 90 Notebooks 0 5 10 15 Cost (dollars) 0 9.45 18.90 28.35 Cost per notebook: *Complex fraction—at least one fraction or decimal in the numerator, denominator, or both; e.g. 1 or .72 ; you may need to simplify complex fractions when finding ratios 4 6 1 2 and rates *When dealing with unit cost, the unit of $ is on top—e.g. $25.50 5 hours Ratio tables—find and organize equivalent ratios and rates—look at how the table is 1 organized; order (least to greatest in the independent variable (x) Finding a rate from a graph—(follow the following steps) 1. Choose and interpret a point on the line (pick a friendly point)—what does the ordered pair represent? Point æ 1 1 ö indicates what? ç , ÷ è2 4ø 1 2. Solve for the rate— y - value ; or 4 x - value ¼ ÷ ½ what is the rate? 1 2 Some things you need to know how to do when working with ratios and rates: Simplify a fraction (a ratio can be simplified just like you would a fraction) Convert between units—unit conversion—e.g. inches to feet or hours to minutes Additional Notes: 2 5.2 Proportions Terms to Know: Equivalent ratios Ratios of equal “value”—e.g. 3 1 5 = = 6 2 10 Proportion An equation stating that two ratios are equivalent Proportional relationship Two quantities that form a proportion are proportional; when these two quantities are proportional, they form a proportional relationship Cross-product In the proportion a c products = b d a • c = b • d are the crossproducts. The crossproducts of a proportion are equal (if not, the ratios are NOT proportional) You can use the Multiplication Property of Equality to show that crossproducts are = Determining whether ratios form a proportion Compare cross-products—e.g. 25 , 15 20 12 Using “compatible numbers”—e.g. 4 , 18 6 24 *Compare Unit Rates—e.g. Determining whether two quantities are proportional Given the two ratios—e.g. 6 , 8 or 34 6.8 , 4 12 50 10 From a (ratio) table—e.g. 3 From a written statement—e.g. 28 points in 3 games and 112 points in 12 games Additional Notes: 5.2 (Extension)—Graphing Proportional Relationships Terms or concepts you should know: (x, y)—ordered pair; represents a position on the number line origin—the point on a coordinate grid (plane) where both the x-and y-value are zero; (0, 0)—think of it as the “center of the grid” proportional relationship (or direct proportion)—the graph of the points fall in a straight line and travels through the origin; represents a constant rate of change; on the table or graph of a proportional relationship, you can find the unit rate from the point (1, y) Constant rate of change Shown in a table and graph: 4 From a table of values—How can I determine whether two quantities are proportional? The ratio of each y must be proportional—meaning they are equivalent ratios x One of the ordered pairs MUST be (0, 0)—meaning it travels through the origin— you may need to continue the table o o The relationships may not be proportional, but they are linear—why? You could always graph the points and connect through them to see if the line would travel through the origin From a graph—How do I determine whether two quantities are proportional by looking at a graph? Additional Notes: 5.3 Writing Proportions & 5.4 Solving Proportions Proportions can be written: For your information: Rows go « and columns go Using columns: 5 Using rows: Solving Proportions Using “mental math”—(Section 5.3)—one strategy in solving a proportion; numbers b 2.6 in the proportion are compatible numbers—e.g. 5 = 25 30 1.5 9 y or Using the Multiplication Property of Equality—multiply by the reciprocal e.g. 3 4 = u 6 Using Cross-Products Property—cross-product and you’ll have an equation to solve k 8 6 16 8 4 = 11 ( y + 2) 5.4 27 7 h *The parentheses in the third problem are important! Why? Unit Conversion: Write and solve a proportion to solve the statement 42 in. ______ cm 12.6 kg ______ lb 3 oz ______ g *There is a conversion chart in the back of the book Remember: Percent (%) means “per 100” or x ; 75% = 75 100 100 5.5 Slope What do you think of when you hear slope? Steepness. How steep is the slope of Jack Frost Mountain? Some mountains are steeper than others; some travel upward left to right while some travel downward left to right—some don’t have any steepness! Slope The rate of change between any two points on a line It is the measure of the steepness of the line To find the slope of a line, find the ratio of the change in y (vertical change) to the change in x (horizontal change) slope = y - change x - change Sometimes you’ll hear it as a ratio of rise to run especially when referring to a graph Later one in the year, you will use the variable m to refer to the slope of a line You should be able to not only identify the slope, but interpret its meaning (tell me what the slope represents—e.g. 6 Solving for the slope of a line given two points: Given the two points: (–1, –1), (2, 5) From a table of values: Lawns Mowed, x Earnings, y ($) 3 45 6 90 9 135 12 180 From a graph: 7 Things to remember: Tables can be horizontal or vertical—vertical tables are “easier” to use and identify ordered pairs The line of a graph can go up left to right, down left to right, or be “flat”—no steepness Interpreting the meaning of the slope (steepness) of a line is more important than solving for it The higher the slope, the steeper the line When graphing points on a coordinate grid, (x, y) Additional Notes: 5.6 Direct Variation (In words) Direct variation: Two quantities x and y show direct variation when y = kx, where k is a number and k ≠ to 0; k is called the constant of proportionality (or constant of variation) (On a graph) The graph of y = kx is a straight line with a slope of k that passes through the origin; the two quantities that show direct variation are in a proportional relationship o The ratio of each y will be proportional, meaning each ratio is an x equivalent ratio o Look for (0, 0)—if it’s in the table or a point on the graph of the line, the two quantities show direct variation Tell whether x and y show direct variation—explain your reasoning Real-World Applications • Graph the data—tell whether x and y are directly proportional—power washer paint sprayer Time (minutes)—x Area (ft2)—y .5 10 1 20 1.5 30 2 40 8 Write an equation that represents the line—y = kx where k is the constant of proportionality Use the equation to answer questions How much area can be painted in 30 minutes? One hour? Use values to find the constant of proportionality—told that the values vary directly y = 14; x = 16 k = _____________ equation relating x and y _________________ Things to remember: Remember to scale your graph appropriately—label each axis with a title Title the graph x is the independent variable while y is the dependent variable Use a ruler to draw STRAIGHT lines!!! 9