A three-digit number increases by 9 if we exchange the second and third digits. The same three-digit number increases by 90 if we exchange the first and second digits. By how much will the value increase if we exchange the first and third digits?
well, the second digit must be 9 more than the third. as in 09...or, it is the complement to 9 (as in 1), as in X23,so the switched is X32 (nine more).
Complements: N,N+1 (as in 12, 23, 32, 43, and so on.
Not the second hint..the first and second:
BAx-ABx=90 so a,b must be complements also, as above.
Lets check; 123
first hint: 132-123=9
second hint:213-123=90 checks
Lets check another: 456
first hint: 465-456=9
second hint:546-456=90 checks.
so there are a multitude of answers for what the number is.
HOWEVER, the answer to the question asked is interesting.
In the first case, 321-123=198
in the second case,654-456=198
hmmmm, very interesting. Why don't you try a few more?
To solve this problem, let's break it down step by step:
Step 1: Assign variables to the unknowns.
Let's call the hundreds digit A, the tens digit B, and the units digit C.
Step 2: Create equations based on the given information.
According to the problem, when we exchange the second and third digits, the number increases by 9. This can be expressed as:
(100A + 10C + B) - (100A + 10B + C) = 9.
Similarly, when we exchange the first and second digits, the number increases by 90:
(100B + 10A + C) - (100A + 10B + C) = 90.
Step 3: Simplify and solve the equations.
Simplifying the first equation:
100A + 10C + B - 100A - 10B - C = 9,
9B - 9C = 9,
B - C = 1.
Simplifying the second equation:
100B + 10A + C - 100A - 10B - C = 90,
90B - 90A = 90,
B - A = 1.
Step 4: Equate the two differences.
Since we are asked to find the increase when the first and third digits are exchanged (A and C), we can equate the two differences:
B - C = B - A,
C = A.
Step 5: Calculate the increase.
Since C = A, exchanging the first and third digits will not change the number. Therefore, the increase will be 0.
So, by exchanging the first and third digits, the value will increase by 0.