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Chapter 4

Chapter 4. Solving and Applying Proportions. 4-1 Ratio and Proportion. All ratios must compare numbers with the SAME units (so you may have to convert) Ratios compare 2 numbers, so if it is improper do NOT convert to a mixed number

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Chapter 4

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  1. Chapter 4 Solving and Applying Proportions

  2. 4-1 Ratio and Proportion • All ratios must compare numbers with the SAME units (so you may have to convert) • Ratios compare 2 numbers, so if it is improper do NOT convert to a mixed number • All rates must compare numbers with DIFFERENT units (i.e. miles per hour) • A unit rate is a type of rate where the denominator is one (i.e. how many miles per 1 gallon) • Ex1. Find the unit price of grapes if you can buy 32 ounces of grapes for $3.96

  3. When converting from one unit to another (known as unit analysis or dimensional analysis) you must multiply by fractions that are equivalent to 1 in order to maintain the integrity of the ratio • Bob drives 292.5 miles in 6.5 hours. How many feet per second would this be? • When you set to ratios equal to one another, that is a proportion • Proportions are solved using the means-extremes property (cross-multiply) • When you set up a proportion, make sure the units line up appropriately

  4. Proportions can be written like or a:b = c:d • You must show work on multi-step proportions • Solve each proportion • Ex2. Ex3. • Ex4. Complete the following statement: $56/day = _____/week • See page 748 for a list of conversions you will need

  5. 4-2 Proportions and Similar Figures • Similar figures are just shrunken or enlarged versions of one another • If figures are similar, corresponding angles are congruent (the same) and corresponding sides are proportional • Symbol for “is congruent to” is ≅ • Symbol for “is similar to” is ~ • When you are naming similar figures, the corresponding angles must be in the same order • ∆ABC ~ ∆DEF means that <A ≅ <D, <B ≅ <E and <C ≅ <F

  6. See ∆ABC and ∆FGH at the top of page 190 • A scale drawing is in direct proportion to the original object • Ex1. Solve for h • Ex2. The scale of a map is 1 in : 20 miles. If two cities are 3.5 inches apart on the map, how far apart are they in real life?

  7. 4-3 Proportions and Percent Equations • You can solve percent questions using proportions or equations (follow the directions for each question) • Solve using a proportion: or • To solve using an equation: % • of = is or % • whole = part • To use equations, you must convert the percent to a decimal • Ex1. Find 40% of 200 using an equation

  8. Ex2. Use a proportion to find: 39 is what percent of 158? • Know what is in the chart on page 199 • Ex3. 85% of a school’s population has been surveyed. If there were 324 students surveyed, how many total students are there at that school? • Ex4. What is 475% as a decimal? • Ex5. What is .38% as a decimal? • Always check the reasonableness of your answer before moving on

  9. 4-4 Percent of Change • Find the percent of change by dividing the amount of change by the original amount • If the amount is increasing, it is percent of increase and if the amount is decreasing, it is the percent of decrease • Ex1. A TV was originally $350, but it went on sale for $295. What was the percent of decrease (nearest tenth)? • The greatest possible error is the maximum a measurement is allowed to be off and still be acceptable (.5 the measuring unit)

  10. If you are measuring to the nearest whole inch, the greatest possible error would be half of an inch, but if you are measuring to the nearest half of an inch, then the greatest possible error would be .25 inches (.5 inches • .5) • Ex2. You measure a room and find the dimensions to be 15 feet by 10 feet. Use the greatest possible error to find the maximum and minimum possible areas. • To find the percent error, divide the greatest possible error by the measurement • Ex3. You measure a friend’s height to be 72.5 inches. What is the percent of error?

  11. 4-5 Applying Ratios to Probability • Write “the probability of an event” as P(event) • Using a deck of cards, if you want to know the probability of drawing a king, write it as P(king) • To find the theoretical probability, divide the number of favorable outcomes by the total number of possible outcomes (could be fraction, decimal or percent) • Ex1. In a standard 52-card deck, find P(queen) • A probability of 0 means that it CANNOT happen • A probability of 1 (or 100%) means that it must happen

  12. Complementary events are two events that have no elements in common and together they make up every possible outcome • Unlike theoretical probability, experimental probability is based on what actually happened during an experiment (divide the number of times the event happened by the number of times the experiment was run) • Ex2. A quality control inspector looked at 250 bolts. Of those, 14 were unacceptable. What is the probability of choosing a bad bolt if you were to chose one at random from this group? • To run an acceptable experiment, you must pre-determine the number of trials you will run

  13. 4-6 Probability of Compound Events • Independent events are those events that do NOT affect one another, even though they might be completed one after the other • To find the probability of compound independent events, find the probability of each event and multiply the two together • Ex1. Suppose you roll a 6-sided die twice. The first time you are hoping to roll a 5, and the second time you want to roll a number less than 3. What is the probability? • Dependent events are those events that do affect one another

  14. To find the probability of dependent events, you must adjust the probability of the second event: P(A) · P(B after A) • Ex2. Tiles containing the letters A L G E B R A I C are placed into a bag, find the probability of randomly choosing an A followed by a R, without replacement.

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