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Sample Question 6:
Step 1: Let's call the two digits a and b. In this context, a and b are place values in our two-digit numbers.
Step 2: Set up the equation for the given condition, which is \(5a + 5b = 10a + b - 10b - a\), or \(9a - 9b\), since the difference between a two-digit number and the number obtained by reversing its digits is 5 times the sum of the digits of either number.
Step 3: Simplify this equation to get \(2a = 7b\).
Step 4: From this equation, extract the values of a and b. We find that \(a = 7\) and \(b = 2\) since b and a are both less than 10.
Step 5: These values of a and b give us our two-digit number and its reverse, which are 72 and 27, respectively.
Step 6: Finally, find the sum of these two numbers which equals 99.
So, the answer is \(\boxed{\textbf{(D) }99}\).
The difference between a two-digit number and the number obtained by reversing its digits is \(5\) times the sum of the digits of either number. What is the sum of the two digit number and its reverse?
\(\textbf{(A) }44\qquad \textbf{(B) }55\qquad \textbf{(C) }77\qquad \textbf{(D) }99\qquad \textbf{(E) }110\)
Answer Keys
Question 6: DSolutions
Question 6Step 1: Let's call the two digits a and b. In this context, a and b are place values in our two-digit numbers.
Step 2: Set up the equation for the given condition, which is \(5a + 5b = 10a + b - 10b - a\), or \(9a - 9b\), since the difference between a two-digit number and the number obtained by reversing its digits is 5 times the sum of the digits of either number.
Step 3: Simplify this equation to get \(2a = 7b\).
Step 4: From this equation, extract the values of a and b. We find that \(a = 7\) and \(b = 2\) since b and a are both less than 10.
Step 5: These values of a and b give us our two-digit number and its reverse, which are 72 and 27, respectively.
Step 6: Finally, find the sum of these two numbers which equals 99.
So, the answer is \(\boxed{\textbf{(D) }99}\).