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Two numbers are in the ratio 4 : 7. If their HCF is 26, then the sum of these two numbers will be: (SSC Sub. Ins. 2018)
- a)364
- b)286
- c)338
- d)312
Correct answer is option 'B'. Can you explain this answer?
Most Upvoted Answer
Two numbers are in the ratio 4 : 7. If their HCF is 26, then the sum o...
Ratio and HCF
Given:
- Two numbers are in the ratio 4 : 7
- HCF of the two numbers is 26
To find:
- Sum of the two numbers
Solution:
Let the two numbers be 4x and 7x (as they are in the ratio 4:7)
The HCF of these two numbers is given as 26. Therefore, 26 is a factor of both the numbers.
So, we can write 4x = 26a and 7x = 26b, where a and b are co-prime (as HCF of a and b is 1)
We need to find the sum of the two numbers, which is 4x + 7x = 11x
Now, let's substitute the values of 4x and 7x in terms of a and b:
4x = 26a
7x = 26b
Multiplying the first equation by 7 and the second equation by 4, we get:
28x = 182a
28x = 104b
As a and b are co-prime, there exists a positive integer k such that a = 104k and b = 182k.
Substituting these values in the above equations:
28x = 182a = 182(104k) = 18928k
28x = 104b = 104(182k) = 18928k
Therefore, we have:
28x = 18928k
Simplifying this equation, we get:
x = 676k
So, the two numbers are:
4x = 4(676k) = 2704k
7x = 7(676k) = 4732k
The sum of the two numbers is:
4x + 7x = 2704k + 4732k = 7436k
Since k is a positive integer, we need to find the smallest value of k such that the sum of the two numbers is a multiple of 26.
7436k = 26(286k)
Therefore, the sum of the two numbers is 26 times 286k, which is a multiple of 26.
Hence, the correct answer is option B) 286.
Given:
- Two numbers are in the ratio 4 : 7
- HCF of the two numbers is 26
To find:
- Sum of the two numbers
Solution:
Let the two numbers be 4x and 7x (as they are in the ratio 4:7)
The HCF of these two numbers is given as 26. Therefore, 26 is a factor of both the numbers.
So, we can write 4x = 26a and 7x = 26b, where a and b are co-prime (as HCF of a and b is 1)
We need to find the sum of the two numbers, which is 4x + 7x = 11x
Now, let's substitute the values of 4x and 7x in terms of a and b:
4x = 26a
7x = 26b
Multiplying the first equation by 7 and the second equation by 4, we get:
28x = 182a
28x = 104b
As a and b are co-prime, there exists a positive integer k such that a = 104k and b = 182k.
Substituting these values in the above equations:
28x = 182a = 182(104k) = 18928k
28x = 104b = 104(182k) = 18928k
Therefore, we have:
28x = 18928k
Simplifying this equation, we get:
x = 676k
So, the two numbers are:
4x = 4(676k) = 2704k
7x = 7(676k) = 4732k
The sum of the two numbers is:
4x + 7x = 2704k + 4732k = 7436k
Since k is a positive integer, we need to find the smallest value of k such that the sum of the two numbers is a multiple of 26.
7436k = 26(286k)
Therefore, the sum of the two numbers is 26 times 286k, which is a multiple of 26.
Hence, the correct answer is option B) 286.
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Community Answer
Two numbers are in the ratio 4 : 7. If their HCF is 26, then the sum o...
Let the two numbers are 4x and 7x.
H. C. F. of 4x and 7x = x.
Now, x = 26.
So, two numbers are 4 × 26 and 7 × 26.
Sum of two numbers = 4 × 26 + 7 × 26 = 11 × 26 = 286.
H. C. F. of 4x and 7x = x.
Now, x = 26.
So, two numbers are 4 × 26 and 7 × 26.
Sum of two numbers = 4 × 26 + 7 × 26 = 11 × 26 = 286.
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