TOPICS: DECIMALS RATIO & PROPORTION

Transcription

1 MAT018B TOPICS: DECIMALS RATIO & PROPORTION 4.1 Decimal Notation, Order, and Rounding: I. Place value There is no oneths! Look at the pattern of pairings below. Hundreds tens ones tenths hundredths Three Hundred Forty- Seven And Nineteen Hundredths II. Decimal- fractional conversion. A. Decimal fraction: count # of digits right of point and place over power of 10 with that # of zeroes then try to reduce = Two two zeroes B. Fraction decimal: Generally divide up but for now use shortcut for powers of ten. You want 1 downstairs so insert decimal points and then move them evenly to get 1 downstairs. Then get rid of the 1. = = = three zeros III. To decide which decimal number is larger, compare place-value by place-value from left to right until tie is broken < tie breakers! IV. Rounding, as before, involves underlining the place-value digit requested and looking right. See 1.4. round to nearest hundredth

2 4.2 Addition and Subtraction I. To add or subtract, line up vertically on decimal point and carry or borrow as necessary. You may need to insert zeroes II. To solve equations (two ways). A. Use related sentence. w+67.8= w= B. Do opposite below on both sides a la algebra. w = = w = Multiplication: I. The Rules!! Step 1: Line numbers up vertically, flush right. Step 2: Multiply ignoring points. Step 3: Count # of digits right of point(s). Step 4: Count in that number of digits from the right on your product to place point II. Powers of ten tricks: {Don t use this until you ve gotten really sick of multiplying these out!} A. Multiplying by 10,100,1000,, count # of zeroes and move point right that many = 30 two B. Multiplying by 0.1, 0.01, 0.001, count # of digits right of point and move it left that many times = III. Converting. A. $/ : Try a proportion if confused. EX What is 34 cents in dollars? Ex =$0.34 B. 2.7 million means (2.7) (1,000,000). 2

3 4.4 Division I. The Rules, Step 1 Set up with Step 2 If need be, move point in outside # all the way right Step 3 Move point in inside # that # right as well. (Add zeroes if necessary). Step 4 Place point on top directly over inside point. Don t move it again! Step 5 Division as usual II. Powers of ten tricks: These tricks are basically the opposites of those in 4.3. Again, don t use them until you re sick of doing it the long way. III. Solving equations again: two ways. A. Related sentence 4.2x = 14 x = B. Opposite on both sides a la algebra IV. P.E.M.D.A.S. again! 4.5 Converting from fraction notation to decimal notation I. The Rules: A. The honest way again involves a proportion with 1 downstairs. write as a decimal: = 8x = 3, x = B. The trick: just divide up II. Sometimes division finishes and sometimes it repeats et cetera 3

4 Put a bar over the fewest possible number of digits. 0.3 (not 0.33) III. Rounding uses the same rules as 4.1. (You may have to expand a repeating decimal to see what s going on). round 1.17 to the nearest thousandth IV. When calculating with a mixture of fractions and decimals, try converting all fractions to decimal because decimals are easier to work with. If anything repeats, convert all too fractional form (putting decimals over one if necessary). 4.6 Estimating Multiple choice answers can be found by process of elimination! 4.7 Applications and Problem Solving I. As always Please Let Frankie Sing! Or Pictures, Labels, Formulas, Substitutions! Also, if you see a rate ( per, each, etc.) try setting up a proportion! 5.1 Introduction to Ratios: I. A ratio shows a relationship between two numbers. 5 to 7 can be written as or as 5:7 II. Simplified fractions don t have fractions or decimals within them and they are reduced. 1.2 to 8.6; = = 5.2 Rates and Unit Price I. A rate is a ratio that has different units on top, from those on the bottom (Often you divide up and attach the units with a slash. Ex 18.5 miles/hour) II. Unit price is the amount of money for one. (That s what unit means.) The safest way to calculate it is again using a proportion! What is unit price if 14 oz for 42? = = = 3 III. Often a one is implied. Ex miles per hour means miles per one hour. 5.3 Proportions I. Proportions are very powerful, and we have a deep natural understanding of them. Artists consider them constantly. II. A proportion is two ratios equal to each other. (This can be flipped in lots of ways and still be true.) 4

5 III. To determine whether two pairs of numbers are proportional, the quick way is to cross multiply and make sure the products are equal. 3, 6, and 1, = 6 1 yes IV. Solving Equations A. In algebra we clear denominators. The following trick is a version of that. We call it cross multiplying. (It is not cross canceling which is something done when multiplying fractions!) = = x = 12 Cross Multiply Solve 5.4 Applications of Proportions Step 1: Read! Often two numbers are related in one sentence. Step 2: Write these numbers as a rate so include units. Step 3: Set this equal to another empty fraction. Step 4: Reread! Put x for unknown and number in final position using units consistently. Step 5: Cross multiply and solve. Ex Three gallons to cover 1275 ft 2. How much covered with 7 gallons? 5.5 Geometric Applications I. Similar triangles (and other shapes) have different sizes but the same shape because sides are proportional. (All of trigonometry is based on this!) Ex 3 5 is proportional to Because = 12 II. Many proportions can be formed using the sides of these similar triangles. 5