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A number 20 is divided into four parts that are in AP such that the product of the first and fourth is to the product of the second and third is 2 : 3. Find the largest part.
- a)12
- b)4
- c)8
- d)9
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
A number 20 is divided into four parts that are in AP such that the pr...
Since the four parts of the number are in AP and their sum is 20, the average of the four parts must
be 5. Looking at the options for the largest part, only the value of 8 fits in, as it leads us to think of
the AP 2, 4, 6, 8. In this case, the ratio of the product of the first and fourth (2 × 8) to the product
of the first and second (4 × 6) are equal. The ratio becomes 2:3.
be 5. Looking at the options for the largest part, only the value of 8 fits in, as it leads us to think of
the AP 2, 4, 6, 8. In this case, the ratio of the product of the first and fourth (2 × 8) to the product
of the first and second (4 × 6) are equal. The ratio becomes 2:3.
Most Upvoted Answer
A number 20 is divided into four parts that are in AP such that the pr...
To solve this question, we can start by assuming the four parts in the arithmetic progression as (a - 3d), (a - d), (a + d), and (a + 3d), where 'a' represents the second term and 'd' represents the common difference.
Let's calculate the product of the first and fourth terms:
(a - 3d)(a + 3d) = a^2 - (3d)^2
= a^2 - 9d^2
Now, let's calculate the product of the second and third terms:
(a - d)(a + d) = a^2 - d^2
Given that the ratio of the two products is 2:3, we can write the equation as:
(a^2 - 9d^2) / (a^2 - d^2) = 2/3
Cross-multiplying and simplifying the equation, we get:
3(a^2 - 9d^2) = 2(a^2 - d^2)
3a^2 - 27d^2 = 2a^2 - 2d^2
a^2 = 25d^2
Taking the square root on both sides, we get:
a = 5d
We know that the sum of the four parts is equal to 20, so:
(a - 3d) + (a - d) + (a + d) + (a + 3d) = 20
4a = 20
a = 5
Now, to find the largest part, we substitute the value of 'a' back into the equation:
(a + 3d) = 5 + 3d
To maximize the value of (a + 3d), we need to maximize 'd'. Since the common difference 'd' should be positive, the largest part is obtained when 'd' is maximum, which is when 'd' = 1.
Substituting the values into the equation, we get:
(a + 3d) = 5 + 3(1)
= 5 + 3
= 8
Therefore, the largest part is 8, which corresponds to option C.
Let's calculate the product of the first and fourth terms:
(a - 3d)(a + 3d) = a^2 - (3d)^2
= a^2 - 9d^2
Now, let's calculate the product of the second and third terms:
(a - d)(a + d) = a^2 - d^2
Given that the ratio of the two products is 2:3, we can write the equation as:
(a^2 - 9d^2) / (a^2 - d^2) = 2/3
Cross-multiplying and simplifying the equation, we get:
3(a^2 - 9d^2) = 2(a^2 - d^2)
3a^2 - 27d^2 = 2a^2 - 2d^2
a^2 = 25d^2
Taking the square root on both sides, we get:
a = 5d
We know that the sum of the four parts is equal to 20, so:
(a - 3d) + (a - d) + (a + d) + (a + 3d) = 20
4a = 20
a = 5
Now, to find the largest part, we substitute the value of 'a' back into the equation:
(a + 3d) = 5 + 3d
To maximize the value of (a + 3d), we need to maximize 'd'. Since the common difference 'd' should be positive, the largest part is obtained when 'd' is maximum, which is when 'd' = 1.
Substituting the values into the equation, we get:
(a + 3d) = 5 + 3(1)
= 5 + 3
= 8
Therefore, the largest part is 8, which corresponds to option C.
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A number 20 is divided into four parts that are in AP such that the product of the first and fourth is to the product of the second and third is 2 : 3. Find the largest part.a)12b)4c)8d)9Correct answer is option 'C'. Can you explain this answer?
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