SOLUTION: Find the six-digit number in which the first digit is one less than the second digit and half the third; the fourth digit is one more than the third and is also the sum of the firs

Question 214070: Find the six-digit number in which the first digit is one less than the second digit and half the third; the fourth digit is one more than the third and is also the sum of the first and second digits; and the fifth and sixth digits make a number that is the sum of the first four digits. The sum of all the digits is 19.
Answer by ankor@dixie-net.com(22740) (Show Source): You can put this solution on YOUR website!
Write an equation for each statement
:
Find the six-digit number
u, v, w, x, y, z
:
in which the first digit is one less than the second digit
u = v-1
:
and half the third;
u = .5w
:
the fourth digit is one more than the third
x = w+1
:
and is also the sum of the first and second digits;
x = u + v
:
the fifth and sixth digits make a number that is the sum of the first four digits.
10y + z = u + v + w + x
:
The sum of all the digits is 19.
u + v + w + x + y + z = 19
:
We can replace the 1st 4 digits with 10y + z
10y + z + y + z = 19
11y + 2z = 19
:
From this we can write an equation that has only one positive integer solution
2z = 19-11y
z =
y = 1
z =
z =
z = 4
:
u + v + w + x + y + z = 19
Replace y and z
u + v + w + x + 1 + 4 = 19
u + v + w + x = 19 - 5
u + v + w + x = 14
:
Replace u with .5w, replace x with w+1
.5w + v + w + (w+1) = 14
2.5w + v = 14 - 1
2.5w + v = 13
:
We know that u = v-1 and we know that u = .5w, therefore
.5w = v - 1
or
v = .5w + 1
Replace v with (.5w+1)
2.5w + .5w + 1 = 13
3w = 13 - 1
3w = 12
w = 4
:
Find v
v = .5(4) + 1
v = 3
:
So far we have:
v=3, w=4, y=1, z=4
find x
x = w+1
x = 4+1
x = 5
:
find u
u = .5w
u = .5(4)
u = 2
:
The number: 234514
:
Check this:
2 + 3 + 4 + 5 + 1 + 4 = 19

RELATED QUESTIONS
What is a six digit number in which the first digit is 1 more than the third and the... (answered by ankor@dixie-net.com)

I am thinking of a five-digit number where the fifth digit is one-half the fourth digit... (answered by Edwin McCravy,solver91311)

QUiZ TiME I am thinking of a five digit number, where the fifth digit is one-half the... (answered by Edwin McCravy)

Find a six-digit number in which the first digit is two less than the second. The... (answered by Mathtut)

Find a six digit number in which the first digit is three less than the second and five... (answered by Boreal)

all of six digit numbers in which the first digit is one less than the second, the third... (answered by greenestamps)

Find a six-digit number in which the first digit is one less than the third, the third... (answered by Edwin McCravy)

Find a five-digit number in which the first and last digits are the same and the first... (answered by ankor@dixie-net.com)

Find a five-digit number in which the first digit equals the sum of the fourth and fifth... (answered by stanbon,solver91311)