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A two-digit number is 3 more than six times the sum of its digits. If 18 is added to the number obtained by interchanging by interchanging the digits, we get the original number. Find the number.
- a)25
- b)45
- c)75
- d)None of these
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
A two-digit number is 3 more than six times the sum of its digits. If...
Let the tens and the units digits in the number be x and y, respectively. So, the number may be written as 10x + y. According to the given condition. 10x + y = 6(x + y) + 3 ⇒ 4x - 5y = 3
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(i) If we interchange the digits of Original number then we get new number, i.e 10y + x
According to the given condition 10y + x + 18 = 10x + y ⇒ 9x - 9y = 18 ⇒ x — y = 2
(ii) Now multiplying equation (ii) by 4. we get, 4x - 4y = 8 ....(iii) On subtracting (iii)
from (i). we get, 4x - 5y - (4x - 4y) = 3 - 8 A y = 5 On substituting y = 5 in (ii). we get, x -
5 = 2 ⇒ x = 7 A number is 10x + y = 10(7) + 5 = 75
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A two-digit number is 3 more than six times the sum of its digits. If...
Understanding the Problem
We have a two-digit number which we can represent as 10a + b, where a is the tens digit and b is the units digit. The problem gives us two conditions to work with.
Condition 1: Number Relation
The two-digit number is 3 more than six times the sum of its digits.
- This can be expressed mathematically as:
10a + b = 6(a + b) + 3
Condition 2: Interchanging Digits
If we interchange the digits, the new number becomes 10b + a. The problem states that adding 18 to this new number gives us the original number.
- This can be expressed as:
10b + a + 18 = 10a + b
Solving the Equations
1. From Condition 1:
- Rearranging gives us:
10a + b - 6a - 6b = 3
4a - 5b = 3 (Equation 1)
2. From Condition 2:
- Rearranging gives us:
10b + a - 10a - b = -18
9b - 9a = -18
b - a = -2 (Equation 2)
Finding Values of a and b
From Equation 2, we can express b in terms of a:
- b = a - 2
Substituting b in Equation 1:
- 4a - 5(a - 2) = 3
- 4a - 5a + 10 = 3
- -a + 10 = 3
- a = 7
Now, substituting a back to find b:
- b = 7 - 2 = 5
Final Calculation
Thus, the original two-digit number is:
- 10a + b = 10(7) + 5 = 75
Answer
The number is 75, which is option 'C'.
We have a two-digit number which we can represent as 10a + b, where a is the tens digit and b is the units digit. The problem gives us two conditions to work with.
Condition 1: Number Relation
The two-digit number is 3 more than six times the sum of its digits.
- This can be expressed mathematically as:
10a + b = 6(a + b) + 3
Condition 2: Interchanging Digits
If we interchange the digits, the new number becomes 10b + a. The problem states that adding 18 to this new number gives us the original number.
- This can be expressed as:
10b + a + 18 = 10a + b
Solving the Equations
1. From Condition 1:
- Rearranging gives us:
10a + b - 6a - 6b = 3
4a - 5b = 3 (Equation 1)
2. From Condition 2:
- Rearranging gives us:
10b + a - 10a - b = -18
9b - 9a = -18
b - a = -2 (Equation 2)
Finding Values of a and b
From Equation 2, we can express b in terms of a:
- b = a - 2
Substituting b in Equation 1:
- 4a - 5(a - 2) = 3
- 4a - 5a + 10 = 3
- -a + 10 = 3
- a = 7
Now, substituting a back to find b:
- b = 7 - 2 = 5
Final Calculation
Thus, the original two-digit number is:
- 10a + b = 10(7) + 5 = 75
Answer
The number is 75, which is option 'C'.
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A two-digit number is 3 more than six times the sum of its digits. If 18 is added to the number obtained by interchanging by interchanging the digits, we get the original number. Find the number.a)25b)45c)75d)None of theseCorrect answer is option 'C'. Can you explain this answer?
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