Slide 1:
Ratio & Proportion By: Mr. Emerson R. Responzo
Faculty
University of La Salette
Malvar, Santiago City Slide 2:
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. Slide 3:
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). Slide 4:
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) Slide 5:
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 Slide 6:
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 Slide 7:
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 Slide 8:
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 Slide 9:
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 Slide 10:
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 Slide 11:
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 Slide 12:
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 Slide 13:
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 Slide 14:
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 Slide 15:
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 Slide 16:
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 Slide 17:
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? Slide 18:
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? Slide 19:
A. Express in RATIO, and simplify whenever possible: BC : AB
ED : BC
AE : AD
AC : AD
AB : AD QUIZ Slide 20:
B. In this figure, if KJ/KF = GI/GF, find IJ if the perimeter of the triangle FIJ is 40. Slide 21:
Thank you and Good bye…