There is a number between 10 and 100. The number is 8 times the sum of its digit.If 45 is subtracted from it , the result is number formed by interchanging its digit.Find the number.
soln
Let the number be x , let the sumb be y
8x= y
so 8x/y - 45 = yx
I think you will have better luck working with the digits.
Let the number have the digits xy
So, its value is 10x+y. Now we have
10x+y = 8(x+y)
10x+y-45 = 10y+x
Now you can easily discover that the number is 72
To solve this problem, we need to follow a step-by-step approach:
1. Let's assume the number as a two-digit number with digits a and b, where a is the tens digit and b is the ones digit. Therefore, the number can be represented as 10a + b.
2. According to the given conditions, the number is 8 times the sum of its digits. So, we can write the equation as:
10a + b = 8(a + b)
3. Simplifying the equation:
10a + b = 8a + 8b
2a = 7b
4. From the equation 2a = 7b, we can see that 'a' has to be a multiple of 7 and 'b' has to be a multiple of 2 for the equation to be true. Let's consider the possible values for 'a' and 'b' that satisfy this condition.
a = 7 and b = 2 (which gives 14 = 14)
a = 14 and b = 4 (which gives 28 = 28)
a = 21 and b = 6 (which gives 42 = 42)
a = 28 and b = 8 (which gives 64 = 64)
a = 35 and b = 10 (which is not possible as b cannot be 10)
5. Out of these possible values, we need to find the number for which "when 45 is subtracted from it, the result is a number formed by interchanging its digits." Let's check this for each case.
For a = 7 and b = 2:
(10a + b) - 45 = (10b + a)
(10(7) + 2) - 45 = (10(2) + 7)
72 - 45 = 27
This is not true, so this case is not valid.
For a = 14 and b = 4:
(10a + b) - 45 = (10b + a)
(10(14) + 4) - 45 = (10(4) + 14)
144 - 45 = 54
This is not true, so this case is not valid.
For a = 21 and b = 6:
(10a + b) - 45 = (10b + a)
(10(21) + 6) - 45 = (10(6) + 21)
216 - 45 = 162
This is not true, so this case is not valid.
For a = 28 and b = 8:
(10a + b) - 45 = (10b + a)
(10(28) + 8) - 45 = (10(8) + 28)
288 - 45 = 82
This is true, so this is our solution.
6. Therefore, the number that satisfies all the given conditions is 28.
In this way, we have found the number by analyzing the given conditions step by step.