ratio and proportion

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gives examples, discussion for good for daily teaching

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Ratio & Proportion By: Mr. Emerson R. Responzo Faculty University of La Salette Malvar, Santiago City

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In your Physical Education class, you have learned that a basketball team is made up of 12 players. But not all of them are allowed to play inside the court at the same time. How many of them could play at a time, and how many are at the “bench”? * In a basketball team, only 5 of them must be inside the court at a time (others play only in a substitution basis—depending upon the coach), others remain seated at the bench. In symbols, it is 5 : 7 –meaning, THE RATIO OF THE IN-COURT PLAYERS TO THE BENCHED PLAYERS IS 5 TO 7.

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5 : 7 is an example of a RATIO—the quotient of x divided by y, where y is NOT equal to zero. A RATIO can be written as: x to y x : y x/y ? All of these ratios are read “x to y” (sometimes x is to y).

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To simplify a ratio, divide out the common factors. Example: Simplify 24 : 36 9 : 24 3a : 15 2 : 3 (common factor: 12) 3 : 8 (common factor : 3) a : 5 (common factor: 3)

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B A D C 16 8 70 Use PARALLELOGRAM ABCD to express the ratio in simplest form. AB to BC BC : AD m?A : m?D 8 : 16 or 1 : 2 16 : 16 or 1 :1 70 : 110 or 7 : 11

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RATIO is also used to find out the distribution of values of a whole into parts. EXAMPLE: The measures of two complementary angles are in the ratio 2 : 3. Find their measures. SOLUTION: Since 2 to 3 Is the ratio, let 2x and 3x represent the actual angle measures. Thus, 2x + 3x = 90 5x = 90 x = 18 1st angle= 2x 2nd angle= 3x = 2(18) = 36 = 2(18) = 54

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The Ratio of two supplementary angles is 3 to 7. SOLUTION: Let 3x = 1st angle 7x = 2nd angle 3x + 7x = 180 10x = 180 x = 18 = 3(18) = 54 = 7(18) = 126

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The Ratio of two complementary angles is 1 to 5. SOLUTION: Let x = 1st angle 5x = 2nd angle x + 5x = 90 6x = 90 x = 15 = 1(15) = 15 = 5(15) = 75

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Do this: The measures of the acute angles of a right triangle are in the ratio 15:3. Find their actual measures. SOLUTION: Let 15x = 1st acute angle 3x = 2nd acute angle 15x + 3x = 90 18x = 90 X = 5 = 15(5) = 75 = 3(5) = 15

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The measures of the acute angles of a right triangle are in the ratio 6:3. Find their actual measures. SOLUTION: Let 6x = 1st acute angle 3x = 2nd acute angle 6x + 3x = 90 9x = 90 X = 10 = 6(10) = 60 = 3(10) = 30

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Another: Find the measures of the angles of a triangle that are in the ratio 2 : 5 : 8. SOLUTION: Let 2x = 1st angle 5x = 2nd angle 8x = 3rd angle 2x + 5x + 8x = 180 15x = 180 x = 12 = 2(12) = 24 = 5(12) = 60 = 8(12) = 96

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Another: The measures of the angles of a triangle are in the ratio 1 : 2 : 3. SOLUTION: Let x = 1st angle 2x = 2nd angle 3x = 3rd angle 1x + 2x + 3x = 180 6x = 180 x = 30 = 1(30) = 30 = 2(30) = 60 = 3(30) = 90

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Find the measures of the angles of a triangle that are in the ratio 2: 7 : 9. SOLUTION: Let 2x = 1st angle 7x = 2nd angle 9x = 3rd angle 2x + 7x + 9x = 180 18x = 180 x = 10 = 2(10) = 20 = 7(10) = 70 = 9(10) = 90

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PROPORTION means the equality of 2 ratios. In symbols, it is a/b = c/d or 1st term 2nd term 3rd term 4th term Note: b and d must not be zero a:b = c:d

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The product of the means equals the product of the extremes. OR Example: find x, x : 5 = 15 : 25 a:b = c:d 25x = 5(15) 25x = 75 x = 3 ad = bc

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Identify the MEANS and the EXTREMES and look for the missing term (x). 1. 3 : x = 9 : 21 2. (x+2) : 8 = (3x – 7) : 16 9x = 63 x = 7 8(3x-7) = 16(x+2) 24x-56 = 16x+32 24x-16x = 32+56 8x = 88 x = 11 2(90-x) = 8x 180 – 2x = 8x -2x-8x = -180 -10x = -180 x = 18

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A Ratio ? is the quotient of x divided by y, where y is not equal to zero. ? can be written as x to y, x : y, or x/y. ? is simplified by dividing out the common factors. What is a ratio?

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A Proportion = is the equality of 2 ratios. = can be written as a : b = c : d or a/b = c/d, where a and d are the extremes and b and c are the means, with b and d not equal to zero. = has extremes product equal to the means product What is a proportion?

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A. Express in RATIO, and simplify whenever possible: BC : AB ED : BC AE : AD AC : AD AB : AD QUIZ

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B. In this figure, if KJ/KF = GI/GF, find IJ if the perimeter of the triangle FIJ is 40.

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Thank you and Good bye…