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 be reversed the number increases the original by 198 find the original number
I coudnt solve.
I coudnt solve. Couldn't solve for abc
To find the original number, we need to break down the problem into steps.
Let's assume the three-digit number is represented as ABC, where A is the hundreds digit, B is the tens digit, and C is the units digit.
Step 1: Sum of the digits
The sum of the digits is given as 20, so we can write the following equation:
A + B + C = 20
Step 2: Middle digit in terms of the other two digits
The middle digit (B) is equal to one-fourth the sum of the other two digits (A and C). We can express this as:
B = 1/4 * (A + C)
Step 3: Reversing the number
When we reverse the number ABC, we get CBA. The new number is increased by 198, so we can write the following equation:
(100 * C + 10 * B + A) - (100 * A + 10 * B + C) = 198
Now, let's solve the equations to find the original number:
From Step 1, we have:
A + B + C = 20 --(1)
From Step 2, we have:
B = 1/4 * (A + C) --(2)
From Step 3, we have:
(100 * C + 10 * B + A) - (100 * A + 10 * B + C) = 198
Simplifying:
100C + 10B + A - 100A - 10B - C = 198
99C - 99A = 198
C - A = 2 --(3)
Now let's solve the equations simultaneously.
From equation (3), we can conclude that C = A + 2.
Substituting this value in equation (2), we have:
B = 1/4 * (A + A + 2)
B = 1/4 * (2A + 2)
B = 1/2 * (A + 1)
Substituting the values of B and C in equation (1), we have:
A + 1/2 * (A + 1) + A + 2 = 20
Simplifying:
2.5A + 3/2 = 20
2.5A = 20 - 3/2
2.5A = 17.5
A = 17.5 / 2.5
A = 7
Substituting the value of A into C = A + 2, we have:
C = 7 + 2
C = 9
Finally, substituting the values of A and C into B = 1/2 * (A + 1), we have:
B = 1/2 * (7 + 1)
B = 1/2 * 8
B = 4
Therefore, the original three-digit number is 749.
let the digit be abc
a+b+c=20
b=1/4(a+c)
100a+10b+c=100b+10c+b+198
now solve for abc and good luck