if three numbers are in the ratio 1:2:3 and the sum of their squares is = 224.Then the sum and difference between the squares of greatest and least numbers
Answers
Ratio of three positive numbers = 1/2 : 1/3 : 1/4
LCM of 2, 3 and 4 = 12
1/2 : 1/3 : 1/4 = 6 : 4 : 3
Let the numbers be 6x, 4x and 3x.
⇒ (6x)2 + (4x)2 + (3x)2 = 244
⇒ 36x2 + 16x2 + 9x2 = 244
⇒ 61x2 = 244
⇒ x2 = 4
⇒ x = ±2
The numbers are positive.
Therefore, three numbers are (6 x 2), (4 x 2), (3 x 2) i.e. 12, 8 and 6
Thanks
(i.) Therefore the sum of the squares of the greatest number and the least number is 160.
(ii.) Therefore the difference between the squares of the greatest number and the least number is 128.
Given:
The ratio of 3 numbers = 1: 2: 3
The sum of their squares = 224
To Find:
(i.) The sum of the squares of the greatest number and the least number.
(ii.) The difference between the squares of the greatest number and the least number.
Solution:
The given question can be solved as shown below.
Given that,
The ratio of 3 numbers =a: b: c = 1: 2: 3
The sum of their squares = a² + b² + c² = 224
Let a = k; b = 2k; c = 3k
The sum of the squares,
⇒ a² + b² + c² = 224
⇒ k² + ( 2k )² + ( 3k )² = 224
⇒ k² + 4k² + 9k² = 224
⇒ 14k² = 224
⇒ k² = 224/14 = 16 ⇒ k = 4
Then the numbers, a = k = 4; b = 2k = 8; c = 3k = 12
⇒ a = 4; b = 8; c = 12
Least number = a = 4
Greatest number = c = 12
(i.) The sum of the squares of the greatest number and the least number:
⇒ a² + c² = 4² + 12² = 16 + 144 = 160
Therefore the sum of the squares of the greatest number and the least number is 160.
(ii.) The difference between the squares of the greatest number and the least number:
⇒ c² - a² = 144 - 16 = 128
Therefore the difference between the squares of the greatest number and the least number is 128.
(i.) Therefore the sum of the squares of the greatest number and the least number is 160.
(ii.) Therefore the difference between the squares of the greatest number and the least number is 128.
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