aminah thinks of a 4-digit number. what is the number?
every digit is different.
the sum of the thousands digit and the tens digit is 10.
the sum of all the digits is 16.
the difference between the hundreds digit and the ones digit is 6.
the difference between the thousands digit and the hundreds digit is 3.
the number is greater than 8000
The digits are a b c d
a = 8 or 9
a+c = 10
a+b+c+d = 16
now provide the other two equations and solve things.
To find the 4-digit number Aminah is thinking of, let's go through the given clues step-by-step:
1. The sum of the thousands digit and the tens digit is 10.
Let's assume the thousands digit is X and the tens digit is Y.
So, X + Y = 10.
2. The sum of all the digits is 16.
Since we have X and Y from the previous clue, let's assume the units digit is Z and the hundreds digit is W.
So, X + Y + Z + W = 16.
3. The difference between the hundreds digit and the ones digit is 6.
This means W - Z = 6.
4. The difference between the thousands digit and the hundreds digit is 3.
This means X - W = 3.
5. The number is greater than 8000.
Since the thousands digit is X and it cannot be zero, it must be greater than 8.
Now, let's solve the system of equations:
From clues 1 and 2, we can rewrite the equation as:
X + Y + Z + W = 16
X + Y = 10
Simplifying the system, we get:
Z + W = 6
X - W = 3
Y = 10 - X
From clue 4, we can rewrite X as:
X = W + 3
Substituting this value of X into the equation for Y from clue 1:
Y = 10 - (W + 3)
Y = 7 - W
Now, substitute these values for Y and X into the second equation:
Z + W = 6
(W + 3) - W + Z = 6
3 + Z = 6
Z = 6 - 3
Z = 3
Now, substituting this value of Z into the equation for Y:
Y = 7 - W
Y = 7 - (W + 3)
Y = 4 - W
Lastly, substitute these values for Y and Z into the first equation:
X + Y = 10
(W + 3) + (4 - W) = 10
7 = 10
Since the equation is not true, there are no values that satisfy all the given clues. It seems there is an inconsistency or contradiction in the clues provided.
To find the 4-digit number Aminah is thinking of, we can follow these steps:
Step 1: Determine the thousands digit.
- Since the number is greater than 8000, the thousands digit must be 8 or 9.
- We know that the difference between the thousands digit and the hundreds digit is 3, so the possibilities for the thousands digit are 8 and 9.
- Let's assume the thousands digit is 8 initially.
Step 2: Determine the hundreds digit.
- We know that the difference between the hundreds digit and the ones digit is 6, and the difference between the thousands digit and the hundreds digit is 3.
- If the thousands digit is 8, the hundreds digit can be 5 (8 - 3).
- Since every digit is different, the hundreds digit cannot be 8.
- Therefore, the hundreds digit is 5.
Step 3: Determine the tens digit.
- The sum of the thousands digit and the tens digit is 10.
- If the thousands digit is 8 and the hundreds digit is 5, the sum of the thousands and tens digit would be 13. This is not 10, so the thousands digit cannot be 8.
- Therefore, the thousands digit must be 9.
Step 4: Determine the ones digit.
- The sum of all the digits is 16, and we already have the thousands, hundreds, and tens digits.
- So, the ones digit can be found by subtracting the sum of the thousands, hundreds, and tens digits from 16.
- 16 - (9 + 5) = 16 - 14 = 2
Therefore, the number that Aminah is thinking of is 9582.