Find the four-digit number described by these three clues.
1) The first digit is one-third of the third digit.
2) The fourth digit is twice the second digit.
3) The first and third digits add up to twice the sum of the second and fourth digits.
The first digit has to be 1,2,or 3, it can't be anything else.
posibilities:
1x3x
2x6x
3x9x
The second digit can be 1,2,3,4
x2x4
x3x6
x4x8
The final clue lets you determine which digits it is.
To find the four-digit number described by the clues, let's break down the information and solve step by step.
Let's represent the four-digit number as ABCD, where A represents the first digit, B represents the second digit, C represents the third digit, and D represents the fourth digit.
Now, let's look at the clues one by one:
1) The first digit is one-third of the third digit.
This clue can be represented as: A = (1/3)C
2) The fourth digit is twice the second digit.
This clue can be represented as: D = 2B
3) The first and third digits add up to twice the sum of the second and fourth digits.
This clue can be represented as: A + C = 2(B + D)
Now that we have represented the clues mathematically, let's substitute the values from the given information into the equations.
From clue 1: A = (1/3)C
From clue 2: D = 2B
From clue 3: A + C = 2(B + D)
Now, we can solve the equations simultaneously to find the values of A, B, C, and D.
Let's start by substituting the values from clues 1 and 2 into clue 3:
(1/3)C + C = 2(B + 2B)
Simplifying, we have:
(4/3)C = 6B
Now, let's solve for C in terms of B:
C = (6B * 3) / 4
C = (18B) / 4
C = (9B) / 2
Now, we have C in terms of B. Let's substitute this expression into the equation from clue 2:
D = 2B
Since D is also equal to twice the second digit, we can substitute the expression (9B/2) for C:
D = 2(9B/2)
Simplifying:
D = 9B
Now, we have D in terms of B. Let's substitute this expression into the equation from clue 3:
A + C = 2(B + D)
A + (9B/2) = 2(B + 9B)
A + (9B/2) = 2(10B)
Simplifying:
A + (9B/2) = 20B
To make solving easier, let's multiply the entire equation by 2:
2A + 9B = 40B
Rearranging the equation:
2A = 40B - 9B
Simplifying:
2A = 31B
Now, we have A in terms of B. Let's substitute this expression into the equation from clue 1:
A = (1/3)C
(31B) = (1/3)C
To solve for C, multiply both sides by 3:
C = 93B
Now, we know that A = 31B and C = 93B. Let's substitute these values into the equation from clue 3:
A + C = 2(B + D)
31B + 93B = 2(B + 9B)
Solving for B:
124B = 20B
B = 0
Since B is 0, let's substitute this value into our expressions for A and C:
A = 31(0) = 0
C = 93(0) = 0
Finally, let's substitute the values A = 0, B = 0, C = 0 into the expression for D:
D = 9B = 9(0) = 0
Therefore, the four-digit number described by the clues is 0000.