a two digit number is such that its value is equal to four times the sum of its digits. If the digits are interchanged, the new number formed exceeds two thirds of the original number by 52. Find the number.
If the number is ab, then we are told that
10a + b = 4(a+b)
10b + a = 2/3 (10a+b)+52
Now just solve for a and b.
a=4
b=8
The original number is 48
To solve this problem, let's first analyze the given information.
Let's assume that the two-digit number is represented by AB, where A is the tens digit and B is the units digit. Therefore, the given number can be expressed as 10A + B.
According to the problem, the value of the number is equal to four times the sum of its digits. This can be expressed as:
10A + B = 4(A + B)
Simplifying the equation, we get:
10A + B = 4A + 4B
6A - 3B = 0
2A - B = 0 --- (Equation 1)
The problem also states that when the digits are interchanged, the new number formed exceeds two-thirds of the original number by 52. Mathematically, this can be represented as:
10B + A = (2/3) (10A + B) + 52
Expanding the equation and simplifying, we get:
10B + A = (20/3)A + (2/3)B + 52
10B - (2/3)B = (20/3)A - A + 52
(28/3)B = (17/3)A + 52
28B - 17A = 156 --- (Equation 2)
Now we have two equations (Equation 1 and Equation 2) with two variables (A and B) that can be solved simultaneously to find the values of A and B.
To solve the equations, we can use substitution method or elimination method.
Let's use the elimination method:
Multiplying Equation 1 by 17 and Equation 2 by 2, we get:
34A - 17B = 0 --- (Equation 3)
56B - 34A = 312 --- (Equation 4)
Adding Equation 3 and Equation 4, we can eliminate variable A:
34A - 34A - 17B + 56B = 0 + 312
39B = 312
B = 8
Substituting the value of B back into Equation 3, we can solve for A:
34A - 17(8) = 0
34A - 136 = 0
34A = 136
A = 4
Therefore, the value of A is 4 and the value of B is 8. So, the two-digit number is 48.