The sum of the digits of a three digit number is 20.The middle digit is equal to one fourth the sum of the other two. If the order of the digits is reversed the number increases by 198. Find original number
To solve this problem, let's break it down step by step.
Step 1: Represent the three-digit number
Let's represent the three-digit number using the digits of hundreds, tens, and ones. We'll call the hundreds digit "A", the tens digit "B", and the ones digit "C". So the original number can be represented as ABC.
Step 2: Form equations based on the given information
We are given two conditions:
1. The sum of the digits is 20.
2. The middle digit is equal to one-fourth the sum of the other two.
Using these conditions, we can form equations as follows:
Equation 1: A + B + C = 20 (sum of the digits)
Equation 2: B = (A + C) / 4 (middle digit is one-fourth the sum of the other two)
Step 3: Reverse the digits and form a new number
If we reverse the order of the digits - ABC becomes CBA - the number increases by 198. Mathematically, we can represent this as:
New number = Original number + 198
So the new number is CBA, and the original number is ABC.
Step 4: Substitute variables and simplify equations
Let's substitute the reversed digits into our equations.
Equation 1: C + B + A = 20
Equation 2: B = (C + A) / 4
And substituting the new number:
CBA = ABC + 198
Step 5: Solve the equations
We have three variables (A, B, and C) and three equations, so we can solve for them simultaneously. Let's solve these equations to find the values of A, B, and C.
From Equation 1, we can express A in terms of C:
A = 20 - B - C
Substituting this expression in Equation 2:
B = (C + (20 - B - C)) / 4
4B = C + 20 - B - C
5B = 20
B = 4
Now, substituting B = 4 in Equation 1:
A + 4 + C = 20
A + C = 16
From the condition that the new number (CBA) is the original number (ABC) reversed plus 198:
100C + 10B + A = 100A + 10B + C + 198
99C = 99A + 198
C = A + 2
We can substitute this expression in the equation A + C = 16:
A + (A + 2) = 16
2A + 2 = 16
2A = 14
A = 7
Now we can find C using C = A + 2:
C = 7 + 2
C = 9
So, the original number (ABC) is 794.
If the digits are xyz, then we are told
x+y+z = 20
y = (x+z)/4
100z+10y+x = 198 + 100x+10y+z
Now just solve for x,y,z