The sum of the 2digits of a 2 digit number is 15. If th number formed by reversing the digits is less than the orginal number by 27.Find the orginal number
Let the number be ab
a+b=15
10b+a = 10a+b - 27
a + b = 15
-9a + 9b = -27
a + b = 15
a - b = 3
2a = 18
a=9
b=6
check: 69 = 96 - 27
Answering by logic and mental calculations:
Since the order of digits is not important (they will be reversed anyway), the only two-digit combinations that add up to 15 are (8,7) and (6,9).
(8,7) does not work because 87-78=9. Now try 96-69=27, which is obviously the answer.
When we have a two digit number, it can be expressed as :
10 a + b
In this case :
10 a + b = orginal number
The sum of the 2 digits is 15
a + b = 15 Subtract a to both sides
a - a + b = 15 - a
b = 15 - a
If you reverse the digits of 10 a + b, it will be :
10 b + a
10 b + a = orginal number - 27 Add 27 to both sides
10 b + a + 27 = orginal number - 27 + 27
10 b + a + 27 = orginal number
10( 15 - a ) + a + 27 = orginal number
150 - 10 a + a + 27 = orginal number
177 - 9 a = orginal number
Also
10 a + b = orginal number
10 a + 15 - a = orginal number
9 a + 15 = orginal number
177 - 9 a = 9 a + 15 Subtract 15 to both sides
177 - 9 a - 15 = 9 a + 15 - 15
162 - 9 a = 9 a Add 9 a to both sides
162 - 9 a + 9 a = 9 a + 9 a
162 = 18 a Divide both sides by 18
162 / 18 = a
9 = a
a = 9
b = 15 - a
b = 15 - 9 = 6
orginal number = 10 a + b = 10 * 9 + 6 = 90 + 6 = 96
To find the original number, we can form an equation based on the given information. Let's denote the tens digit as 'x' and the ones digit as 'y'.
According to the problem, the sum of the two digits is 15, so we can write the equation: x + y = 15.
We are also given that when the digits are reversed, the new number is 27 less than the original number. We can express this as: 10y + x = (10x + y) - 27.
Let's rearrange the second equation to make x the subject:
10y + x = 10x + y - 27
9y - 9x = -27
y - x = -3 (by dividing both sides by 9)
Now, we have a system of two equations:
x + y = 15
y - x = -3
To solve this system of equations, we can use the method of substitution or elimination. Let's use the method of elimination:
Add the first equation to the second equation:
(x + y) + (y - x) = 15 + (-3)
2y = 12
y = 6
Substitute the value of y back into the first equation:
x + 6 = 15
x = 9
Therefore, the original number is 96.