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The sum of two numbers is 80 and their HCF and LCM are 4 and 364 respectively. The sum of the reciprocals of two numbers
- a)5/91
- b)19/140
- c)80/361
- d)None of these
Correct answer is option 'A'. Can you explain this answer?
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The sum of two numbers is 80 and their HCF and LCM are 4 and 364 respe...
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The sum of two numbers is 80 and their HCF and LCM are 4 and 364 respe...
Given:
The sum of two numbers is 80.
The highest common factor (HCF) of the two numbers is 4.
The least common multiple (LCM) of the two numbers is 364.
To Find:
The sum of the reciprocals of the two numbers.
Solution:
Let the two numbers be a and b.
Step 1: Finding the Numbers
We know that the sum of two numbers is 80.
So, we can write the equation as:
a + b = 80 ...(1)
Step 2: Finding the HCF and LCM
We know that the HCF of the two numbers is 4 and the LCM is 364.
The product of two numbers is equal to the product of their HCF and LCM.
So, we have:
a * b = 4 * 364
a * b = 1456 ...(2)
Step 3: Solving the Equations
Now, we have two equations:
a + b = 80 ...(1)
a * b = 1456 ...(2)
We can solve these equations simultaneously to find the values of a and b.
By substituting the value of b from equation (1) into equation (2),
a * (80 - a) = 1456
80a - a^2 = 1456
a^2 - 80a + 1456 = 0
Solving this quadratic equation, we get:
(a - 16)(a - 64) = 0
So, either a - 16 = 0 or a - 64 = 0.
If a - 16 = 0, then a = 16.
If a - 64 = 0, then a = 64.
Therefore, the two numbers are 16 and 64.
Step 4: Finding the Sum of the Reciprocals
Now, we need to find the sum of the reciprocals of the two numbers.
The reciprocal of a number x is 1/x.
So, the sum of the reciprocals of the two numbers is:
1/16 + 1/64
Taking the LCM of the denominators, we get:
4/64 + 1/64 = 5/64
Therefore, the sum of the reciprocals of the two numbers is 5/64, which is equivalent to option A.
The sum of two numbers is 80.
The highest common factor (HCF) of the two numbers is 4.
The least common multiple (LCM) of the two numbers is 364.
To Find:
The sum of the reciprocals of the two numbers.
Solution:
Let the two numbers be a and b.
Step 1: Finding the Numbers
We know that the sum of two numbers is 80.
So, we can write the equation as:
a + b = 80 ...(1)
Step 2: Finding the HCF and LCM
We know that the HCF of the two numbers is 4 and the LCM is 364.
The product of two numbers is equal to the product of their HCF and LCM.
So, we have:
a * b = 4 * 364
a * b = 1456 ...(2)
Step 3: Solving the Equations
Now, we have two equations:
a + b = 80 ...(1)
a * b = 1456 ...(2)
We can solve these equations simultaneously to find the values of a and b.
By substituting the value of b from equation (1) into equation (2),
a * (80 - a) = 1456
80a - a^2 = 1456
a^2 - 80a + 1456 = 0
Solving this quadratic equation, we get:
(a - 16)(a - 64) = 0
So, either a - 16 = 0 or a - 64 = 0.
If a - 16 = 0, then a = 16.
If a - 64 = 0, then a = 64.
Therefore, the two numbers are 16 and 64.
Step 4: Finding the Sum of the Reciprocals
Now, we need to find the sum of the reciprocals of the two numbers.
The reciprocal of a number x is 1/x.
So, the sum of the reciprocals of the two numbers is:
1/16 + 1/64
Taking the LCM of the denominators, we get:
4/64 + 1/64 = 5/64
Therefore, the sum of the reciprocals of the two numbers is 5/64, which is equivalent to option A.
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