A number consist of 3 digits.The middle digit is 0 and the sum of the first and last digits is 13.If the digits are reversed the number is increased by 297.Find the original number.
let the appearance of the number be a0b
the value of the number is 100a + b
the value of the number reversed is 100b + a
so 100b + a - (100a + b) = 297
99b - 99a = 297
11b - 11a = 33
but we also know that a+b = 13 or a = 13-b
by substitution,
11b - 11(13-b) = 33
22b = 176
b = 8 , then a = 5
finish it up.
Let's solve this step-by-step:
Step 1: Let's assume the first digit of the number is x and the last digit is y. Since the middle digit is 0, we can write the number as 100x + 10(0) + y, which simplifies to 100x + y.
Step 2: We are given that the sum of the first and last digits is 13, so we can write the equation as x + y = 13.
Step 3: We are also given that when the digits are reversed, the number is increased by 297. When the digits are reversed, the new number can be written as 100y + 10(0) + x, which simplifies to 100y + x.
Step 4: We can write the equation for the reversed number as 100y + x = 100x + y + 297.
Step 5: Simplifying the equation, we get 99y - 99x = 297.
Step 6: Dividing both sides of the equation by 99, we get y - x = 3.
Step 7: We can now solve the system of linear equations by performing substitution. Using the value of x + y = 13 from Step 2, we can rewrite the equation y - x = 3 as y = 3 + x.
Step 8: Substituting the value of y from Step 7 into the equation x + y = 13, we get x + (3 + x) = 13.
Step 9: Simplifying, we get 2x + 3 = 13.
Step 10: Subtracting 3 from both sides, we get 2x = 10.
Step 11: Dividing both sides by 2, we get x = 5.
Step 12: Substituting the value of x into the equation y = 3 + x from Step 7, we get y = 3 + 5.
Step 13: Simplifying, we get y = 8.
Step 14: Therefore, the original number is 100x + y, which is 100(5) + 8.
Step 15: Simplifying, we get the original number as 500 + 8.
Step 16: Therefore, the original number is 508.
To solve this problem, let's break it down step by step:
Step 1: Let's assign variables for the first and last digits of the number. Let's call them "A" and "C". Since the middle digit is 0, it does not affect the sum of the first and last digits.
Step 2: We know that the sum of the first and last digits is 13, so we can write the equation:
A + C = 13
Step 3: We are also given that if the digits are reversed, the number is increased by 297. This means that the original number minus the reversed number equals 297. We can write the equation:
(100A + 10B + C) - (100C + 10B + A) = 297
Step 4: Simplifying the equation from step 3, we get:
(100A + 10B + C) - (100C + 10B + A) = 297
99A - 99C = 297
Dividing both sides by 99, we get:
A - C = 3
Step 5: Now we have two equations:
A + C = 13 (from step 2)
A - C = 3 (from step 4)
Step 6: We can solve these two equations simultaneously to find the values of A and C. Adding the two equations, we get:
2A = 16
Dividing both sides by 2, we find:
A = 8
Step 7: Plugging the value of A back into one of the equations, we find:
8 + C = 13
Subtracting 8 from both sides, we find:
C = 5
Step 8: Finally, we can form the original number using the values of A and C. The original number is:
ABC = 850