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Let a two digit number be k times the sum of its digits. If the number formed by interchanging the digits is m times the sum of the digits, then the value of m is
- a)9 - k
- b)10 - k
- c)11 - k
- d)k - 1
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
Let a two digit number be k times the sum of its digits. If the number...
Let the unit and ten’s digits be x and y respectively
The number is k times the sum of digits
10x + y = k(x + y)
(10 - k)x = (k - 1)y ---- (1)
Reversed number is m times the sum of digits
10y + x = m(x + y)
(1 - m)x = (m - 10)y ---- (2)
Divide equation 2 by 1
(10 - k) / (1 - m) = (k - 1) / (m - 10)
(10 - k) (m - 10) = (1 - m) (k - 1)
10m + 10k - 100 - km = -1 + m + k - km
10m - m + 10k - k = 100 - 1
9(m + k) = 99
m + k = 11
∴ m = 11 - k
The number is k times the sum of digits
10x + y = k(x + y)
(10 - k)x = (k - 1)y ---- (1)
Reversed number is m times the sum of digits
10y + x = m(x + y)
(1 - m)x = (m - 10)y ---- (2)
Divide equation 2 by 1
(10 - k) / (1 - m) = (k - 1) / (m - 10)
(10 - k) (m - 10) = (1 - m) (k - 1)
10m + 10k - 100 - km = -1 + m + k - km
10m - m + 10k - k = 100 - 1
9(m + k) = 99
m + k = 11
∴ m = 11 - k
Most Upvoted Answer
Let a two digit number be k times the sum of its digits. If the number...
Given:
Let the two-digit number be represented as 10x + y, where x and y are the digits of the number.
The sum of the digits can be represented as x + y.
The number formed by interchanging the digits is represented as 10y + x.
To find:
The value of m, which represents the number formed by interchanging the digits divided by the sum of the digits.
Solution:
Step 1: Express the given conditions mathematically.
The given number is k times the sum of its digits:
10x + y = k(x + y) ...(1)
The number formed by interchanging the digits is m times the sum of the digits:
10y + x = m(x + y) ...(2)
Step 2: Simplify the equations.
From equation (1), we can expand and simplify:
10x + y = kx + ky
9x = (k - 1)y ...(3)
From equation (2), we can expand and simplify:
10y + x = mx + my
9y = (m - 1)x ...(4)
Step 3: Solve the equations.
We have two equations with two unknowns, (x and y). We can solve these equations simultaneously.
From equation (3), we can express y in terms of x:
y = (9x)/(k - 1) ...(5)
Substituting equation (5) into equation (4), we can solve for x:
9(9x)/(k - 1) = (m - 1)x
81x = (k - 1)(m - 1)x
81 = (k - 1)(m - 1)
Step 4: Find the value of m.
Since the equation is satisfied for all values of x, we can equate the coefficients of x on both sides of the equation.
k - 1 = 9 ...(6)
m - 1 = 9 ...(7)
From equation (6), k = 10.
From equation (7), m = 10 - 1 = 9.
Therefore, the value of m is 9.
Hence, the correct option is (a) 9 - k.
Let the two-digit number be represented as 10x + y, where x and y are the digits of the number.
The sum of the digits can be represented as x + y.
The number formed by interchanging the digits is represented as 10y + x.
To find:
The value of m, which represents the number formed by interchanging the digits divided by the sum of the digits.
Solution:
Step 1: Express the given conditions mathematically.
The given number is k times the sum of its digits:
10x + y = k(x + y) ...(1)
The number formed by interchanging the digits is m times the sum of the digits:
10y + x = m(x + y) ...(2)
Step 2: Simplify the equations.
From equation (1), we can expand and simplify:
10x + y = kx + ky
9x = (k - 1)y ...(3)
From equation (2), we can expand and simplify:
10y + x = mx + my
9y = (m - 1)x ...(4)
Step 3: Solve the equations.
We have two equations with two unknowns, (x and y). We can solve these equations simultaneously.
From equation (3), we can express y in terms of x:
y = (9x)/(k - 1) ...(5)
Substituting equation (5) into equation (4), we can solve for x:
9(9x)/(k - 1) = (m - 1)x
81x = (k - 1)(m - 1)x
81 = (k - 1)(m - 1)
Step 4: Find the value of m.
Since the equation is satisfied for all values of x, we can equate the coefficients of x on both sides of the equation.
k - 1 = 9 ...(6)
m - 1 = 9 ...(7)
From equation (6), k = 10.
From equation (7), m = 10 - 1 = 9.
Therefore, the value of m is 9.
Hence, the correct option is (a) 9 - k.
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Let a two digit number be k times the sum of its digits. If the number formed by interchanging the digits is m times the sum of the digits, then the value of m isa)9 - kb)10 - kc)11 - kd)k - 1Correct answer is option 'C'. Can you explain this answer?
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