Sum of the digits of a two-digit number is 9. When we interchange the digits, it is found that the resulting new number is greater than the original number by 27. What is the two-digit number?
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Answers
Explanation:
Let the digits of the original number be x and y
Hence, the original number is 10x + y (Assuming x to be the ten's digit and y to be the one's digit)
After interchanging the digits the new number will be 10y + x (After reversing, y becomes the ten's digit and x becomes the one's digit)
Condition 1: Sum of the digits is 9 ⇒ x + y = 9 --------- equation (i)
Condition 2: The number obtained by interchanging the digits is 27 greater than the earlier number.
⇒ New number = 27 + original number
⇒ New number = 27 + original number⇒ 10y + x = 27 + (10x + y)
⇒ New number = 27 + original number⇒ 10y + x = 27 + (10x + y)⇒ 10y + x = 27 + 10x + y
⇒ New number = 27 + original number⇒ 10y + x = 27 + (10x + y)⇒ 10y + x = 27 + 10x + y⇒ 10y - y + x - 10x = 27
⇒ New number = 27 + original number⇒ 10y + x = 27 + (10x + y)⇒ 10y + x = 27 + 10x + y⇒ 10y - y + x - 10x = 27⇒ 9y - 9x = 27
⇒ New number = 27 + original number⇒ 10y + x = 27 + (10x + y)⇒ 10y + x = 27 + 10x + y⇒ 10y - y + x - 10x = 27⇒ 9y - 9x = 27⇒ y - x = 3 -------- equation
(ii)By adding equation(i) and
equation(ii):
⇒ New number = 27 + original number⇒ 10y + x = 27 + (10x + y)⇒ 10y + x = 27 + 10x + y⇒ 10y - y + x - 10x = 27⇒ 9y - 9x = 27⇒ y - x = 3 -------- equation (ii)By adding equation(i) and equation(ii):x + y + y - x = 9 + 3
⇒ New number = 27 + original number⇒ 10y + x = 27 + (10x + y)⇒ 10y + x = 27 + 10x + y⇒ 10y - y + x - 10x = 27⇒ 9y - 9x = 27⇒ y - x = 3 -------- equation (ii)By adding equation(i) and equation(ii):x + y + y - x = 9 + 3⇒ 2y = 12
⇒ New number = 27 + original number⇒ 10y + x = 27 + (10x + y)⇒ 10y + x = 27 + 10x + y⇒ 10y - y + x - 10x = 27⇒ 9y - 9x = 27⇒ y - x = 3 -------- equation (ii)By adding equation(i) and equation(ii):x + y + y - x = 9 + 3⇒ 2y = 12⇒ y = 6
⇒ New number = 27 + original number⇒ 10y + x = 27 + (10x + y)⇒ 10y + x = 27 + 10x + y⇒ 10y - y + x - 10x = 27⇒ 9y - 9x = 27⇒ y - x = 3 -------- equation (ii)By adding equation(i) and equation(ii):x + y + y - x = 9 + 3⇒ 2y = 12⇒ y = 6From equation(i): x + 6 = 9
⇒ New number = 27 + original number⇒ 10y + x = 27 + (10x + y)⇒ 10y + x = 27 + 10x + y⇒ 10y - y + x - 10x = 27⇒ 9y - 9x = 27⇒ y - x = 3 -------- equation (ii)By adding equation(i) and equation(ii):x + y + y - x = 9 + 3⇒ 2y = 12⇒ y = 6From equation(i): x + 6 = 9 ⇒ x = 9 - 6 = 3
⇒ New number = 27 + original number⇒ 10y + x = 27 + (10x + y)⇒ 10y + x = 27 + 10x + y⇒ 10y - y + x - 10x = 27⇒ 9y - 9x = 27⇒ y - x = 3 -------- equation (ii)By adding equation(i) and equation(ii):x + y + y - x = 9 + 3⇒ 2y = 12⇒ y = 6From equation(i): x + 6 = 9 ⇒ x = 9 - 6 = 3⇒ x = 3 and y = 6
⇒ New number = 27 + original number⇒ 10y + x = 27 + (10x + y)⇒ 10y + x = 27 + 10x + y⇒ 10y - y + x - 10x = 27⇒ 9y - 9x = 27⇒ y - x = 3 -------- equation (ii)By adding equation(i) and equation(ii):x + y + y - x = 9 + 3⇒ 2y = 12⇒ y = 6From equation(i): x + 6 = 9 ⇒ x = 9 - 6 = 3⇒ x = 3 and y = 6⇒ The required number is 10x + y = 10 × 3 + 6 = 30 + 6 = 36
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Answer
Hence the number is 36
Explanation:
The sum of the two digits = 9
On interchanging the digits, the resulting new number is greater than the original number by 27.
Let us assume the digit of units place = x
Then the digit of tens place will be = (9 – x)
Thus the two-digit number is 10(9 – x) + x
Let us reverse the digit
the number becomes 10x + (9 – x)
As per the given condition
10x + (9 – x) = 10(9 – x) + x + 27
⇒ 9x + 9 = 90 – 10x + x + 27
⇒ 9x + 9 = 117 – 9x
On rearranging the terms we get,
⇒ 18x = 108
⇒ x = 6
So the digit in units place = 6
Digit in tens place is
⇒ 9 – x
⇒ 9 – 6
⇒ 3
Hence the number is 36
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