SOLUTION: Solve algebraically using one variable: The first digit in a four digit number is four more than the second and the second is three less than the third. The fourth digit is one les

Question 84151: Solve algebraically using one variable: The first digit in a four digit number is four more than the second and the second is three less than the third. The fourth digit is one less than the third and the sum of the four digits is 21. What is the number? (HINT: first decide which digit you should call 'x'.)

Found 2 solutions by venugopalramana, bucky:
Answer by venugopalramana(3286) (Show Source): You can put this solution on YOUR website!
Solve algebraically using one variable: The first digit in a four digit number is four more than the second and the second is three less than the third. The fourth digit is one less than the third and the sum of the four digits is 21. What is the number? (HINT: first decide which digit you should call 'x'.)
LET III DIGIT = X
II DIGIT = X-3
I DIGIT =X-3+4=X+1
IV DIGIT = X-1
SUM OF ALL DIGITS =X+X-3+X+1+X-1=21
4X=24
X=6
HENCE THE NUMBER IS
7365
Answer by bucky(2189) (Show Source): You can put this solution on YOUR website!
Given: an unknown four-digit number.
.
Call the first digit x. Do this because the second digit is given in terms of the first.
And the third digit is in terms of the second which is based on the first. And the third
is based on the second which depends on the first. And the fourth is based on the third,
which is based on the second, which depends on the first. Therefore, every one of the
four digits is based on the first digit in some way. So if you can find the unknown
first digit, you should be able to find all the other digits.
.
This first digit is 4 more than the second digit. This means that if you took 4 away from
the first digit the answer would be the second digit. Therefore, you can write the second
digit as x - 4
.
The second digit is 3 less than the third. This means that if you add 3 to the second digit,
the answer would be the third digit. Since the second digit is x - 4, adding 3 to it results
in (x - 4) + 3 = x - 4 + 3 = x - 1. Therefore, the third digit is represented as x - 1.
.
Finally, the fourth digit is 1 less than the third. That means that if we subtract
1 from the third digit the result is the fourth digit. Since the third digit is x - 1,
taking 1 away from it gives (x - 1) - 1 = x - 1 - 1 = x - 2.
.
Let's summarize to this point:
.
First digit = x
Second digit = x - 4
Third digit = x - 1
Fourth digit = x - 2
.
The problem says that the sum of these digits is 21. Write this in equation form:
.
x + (x - 4) + (x - 1) + (x - 2) = 21
.
Since each set of parentheses is preceded by a plus sign, you can just remove the
parentheses without any changes to the terms inside each set. (I just put the parentheses
in the equation to separate each of the four numbers.) So remove the parentheses to get:
.
x + x - 4 + x - 1 + x - 2 = 21
.
Adding all the x terms together reduces the equation to:
.
4x - 4 - 1 - 2 = 21
.
Now add the three negative numbers together to get:
.
4x - 7 = 21
.
Get rid of the -7 on the left side by adding +7 on the left. But if you do that you also
have to add +7 to the right side to keep the equation in balance. Adding +7 to both
sides makes the equation become:
.
4x = 28
.
Finally solve for x by dividing both sides of the equation by 4 which is the multiplier
of the x. When you divide both sides of the equation by 4 it becomes:
.
x = 7
.
Since x is the first digit of the 4 digits, you can now solve for the other three
digits.
.
You know that the second digit is represented by x - 4. Substituting 7 for x, the second
digit becomes 7 - 4 and the answer to that is 3. So the second digit is 3.
.
The third digit is represented by x - 1. Substituting 7 for x results in 7 - 1 which
simplifies to 6. So the third digit is 6.
.
Finally the fourth digit is represented by x - 2. Substituting 7 for x gives you 7 - 2 which
results in the fourth digit being 5.
.
The number is 7365.
.
Let's check it against the problem statement.
.
Is the first digit 4 more than the second digit? Yes, 7 is 4 more than 3.
.
Is the second digit 3 less than the third digit? Yes, 3 is 3 less than 6.
.
Is the fourth digit 1 less than the third digit? Yes, 5 is 1 less than 6.
.
Finally, is the sum of the 4 digits equal to 21? Yes, 7 + 3 + 6 + 5 = 21
.
Everything checks, so the answer of 7365 is correct.
.
Hope this helps you to understand the problem a little better.

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