Find the greater of two numbers whose arithmetic mean is 20 and whose geometric mean is 16
Answers
Answer:
32
Step-by-step explanation:
Given:- Arithmetic mean of two numbers = 20
Geometric mean = 16.
To Find:- Greater of the two number that satisfies the above conditions.
Solution:-
Let the two numbers be a and b.
Arithmetic mean is average of the numbers.
Formula to calculate average of two number = (a + b)/2
Formula to calculate geometric mean of two number = √ab
According to the question,
(a + b)/2 = 20
⇒ a + b = 20 × 2
⇒ a + b = 40
⇒ a = 40 - b ---------- ( 1 )
From the question, we can also write
√ab = 16
After squaring both sides, we get
⇒ ( √ab )² = 16²
⇒ ab = 16 × 16
⇒ ab = 256
⇒ (40 - b) b = 256 [ From equation ( 1 ) ]
⇒ 40b - b² = 256
⇒ b² - 40b + 256 = 0
⇒ b² - (32 + 8)b + 256 = 0
⇒ b² - 32b - 8b + 256 = 0
⇒ b(b - 32) - 8(b -32) = 0
⇒ (b - 32)(b - 8) = 0
⇒ b = 32, 8
∴ a = 40 - 32 or a = 40 - 8
= 8 or 32.
Therefore the two numbers whose arithmetic mean is 20 and geometric mean is 16 are 32 and 8.
The greater of two numbers is 32.
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