Alice writes an addition problem, adding two 2-digit counting numbers to get a sum of 134.
Bob erases the tens digit of one of Alice's addends, adds again, and gets a sum of 84.
What is the smallest possible value for the larger of Alice's two addends?
To find the smallest possible value for the larger addend, we need to consider the constraints given in the problem.
Let's assume the two addends Alice wrote are represented by the variables A and B, where A is the larger addend.
According to the problem:
1. The sum of the two addends is 134: A + B = 134.
2. After erasing the tens digit of one addend, the sum becomes 84: (A/10) + B = 84.
To find the smallest possible value for A, we need to find a combination of A and B that satisfies both equations.
First, let's solve the second equation for B:
B = 84 - (A/10).
Now substitute this value of B in the first equation:
A + (84 - A/10) = 134.
Simplifying the equation:
10A + 840 - A = 1340,
9A = 500,
A = 500/9.
Since A must be a two-digit number, we can consider the next smallest integer, which is 55. But let's check if this value satisfies all conditions:
B = 84 - (55/10),
B = 84 - 5.5,
B = 78.5.
Since the sum of two integers cannot end in a decimal, 55 is not a valid value for A.
The next possible value is 60. Let's check if it satisfies the conditions:
B = 84 - (60/10),
B = 84 - 6,
B = 78.
This gives us a valid solution where A = 60 and B = 78. Therefore, the smallest possible value for the larger of Alice's two addends is 60.