A three-digit number has the following properties:
1. When we change the units digit with the tens digit, it increases by 18 units;
2. When we change the tens digit with the hundreds digit, it increases by 180 units.
3. How many units will this number increase if we exchange the units digit with the hundreds digit?
1. When we change the units digit with the tens digit, it increases by 18 units;
2. When we change the tens digit with the hundreds digit, it increases by 180 units.
3. How many units will this number increase if we exchange the units digit with the hundreds digit?
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The number will increase by 396 units.
Let's assume that the three-digit number is ABC. Swapping the units digit with the tens digit, we get ACB.
1. We have the information that ABC = ACB + 18. Note that this expression can be written as follows:
100A + 10B + C = 100A + 10C + B + 18
9B - 9C = 18
B - C = 2
B = 2 + C.
2. Now, let's change the tens digit to the hundreds digit. So we have: BAC. Also, ABC = BAC + 180. In the same way as we did before, we get:
100A + 10B + C = 100B + 10A + C + 180
90A - 90B = 180
A - B = 2.
Realize that B = 2 + C. Then:
A - (2 + C) = 2
A - 2 - C = 2
A - C = 4.
3. If we change the units digit with the hundreds digit, the number will be CBA = 100C + 10B + A. Subtracting ABC - CBA, we find:
ABC - CBA = 100A + 10B + C - (100C + 10B + A) = 100A + 10B + C - 100C - 10B - A = 99A - 99C = 99(A - C) = 99.4 = 396.