algebra help please?

Question

The sum of the ages of four daughters, Abigail, Bonnie, Cena, and Daphne is 40. The difference between the ages of the youngest, Daphne, and the oldest, Abigail, is 6. The second born, Bonnie, is 2 years younger than Abigail, and the third born, Cena, is the average age of the Daphne and Bonnie. How old is the father, who 30 years older than the youngest daughter?Please slove using one variable and explain how to slove the problem.

Thank you

Answer 1

A+B+C+D = 40
A-D=6--> D=A - 6
A-B=2 --> B= A-2
C= (B+D)/2 = (2A-8)/2 = A-4
Substituting:
A+ A-2 +A - 4 +A- 6 = 40
4A -12 = 40
4A= 52
A=13
B= A-2 = 11
D= A- 6 =7
C= (11+7)/2=9

Answer 2

The sum of the ages of four daughters, Abigail, Bonnie, Cena, and Daphne is 40
1) A + B + C + D = 40

The difference between the ages of the youngest, Daphne, and the oldest, Abigail, is 6
2) A - D = 6

The second born, Bonnie, is 2 years younger than Abigail
3) B = A - 2

the third born, Cena, is the average age of the Daphne and Bonnie
4) C = (D + B)/2

There are really four valriables here, but we can reduce the problem to one variable, D for Daphne

1) A + B + C + D = 40

From equation 2, A - D = 6, or
2a) A = D + 6

1a) D + 6 + B + C + D = 40

Substituting from equation 2a into equation 3

3) B = A - 2
B = D + 6 - 2
3a) B = D + 4

1b) D + 6 + D + 4 + C + D = 40

Substitute from equation 3a into equation 4

4) C = (D + B)/2
C = (D + D + 4)/2
C = (2D + 4)/2
C = D + 2

1c) D + 6 + D + 4 + D + 2 + D = 40
4D + 12 = 40
4D = 28
D = 7

Father is 37

Answer 3

For starters, you need to find the other daughters' ages in terms of Abigail's age:

Abigail's age = a
Daphne's age = a-6 (6-year difference between youngest & oldest)
Bonnie's age = a-2 (Bonnie is 2 years younger than Abigail)
Cena's age = [ (a-6)+(a-2) ] / 2 (average of Daphne and Bonnie)

Simplify Cena's age:

[(a-6) + (a-2)] / 2 (average of Daphne and Bonnie)
(2a - 8) / 2 (add the a's and add -2 to -6)
a -4 (divide by 2)

We know they all equal 40. Now add them together and solve for a:

a + a-2 + a-4 + a-6 = 40
4a + -2 + -4 + -6 = 40 (add the a's together)
4a - 12 = 40 (combine adding all the negative numbers into subtracting one positive number)
4a = 52 (add 12 to both sides)
a = 13 (divide both sides by 4)

Now, the ages:

Abigail = a = 13
Bonnie = a-2 = 11
Cena = a-4 = 9
Daphne = a-6 = 7

Check #1 (sum of all 4 ages = 40):

13 + 11 + 9 + 7 = 40
40 = 40 --> Checks out!!

Check #2 (Cena = average of Bonnie and Daphne)
9 = (11+7) / 2
9 = 18/2
9 = 9 --> Checks out again, all OK

Finally ... the father's age:
Daphne's age + 30
7 + 30
37

Answer 4

A + B + C + D = 40


The difference between the ages of the youngest, Daphne, and the oldest, Abigail, is 6
A - D = 6

A - B= 2 (being two years younger means that there is a difference of two years)

C = D + B /2

D = A - 6

B = A - 2

C = A - 6 + A - 2 / 2
C= 2A - 8/2
C= 2(A - 4)/2
C= A - 4

A + B + C + D = 40

(A) + (A - 2) + (A - 4) + (A - 6) = 40

4A - 12 = 40

4A = 52

A = 13

D = youngest

D = A - 6 = 7

father is 30 + D
thus father is 37

Answer 5

Let:

d = daphne

so...

abigail = d + 6

bonnie = abigail - 2 = (d + 6) -2 = d + 4

cena = (bonnie + daphne)/2 = (d + (d + 4))/2 = (2d + 4)/2 = d + 2

so the sum of the sisters is:

d + (d + 6) + (d + 4) + (d + 2) = 40

Simplify and solve (you can move the d's and numbers around because addition is commutative):

4d + 12 = 40
4d = 28
d = 7

So Daphne is 7.

Therefore,

abigail = 13
bonnie = 11
cena =9


father = d + 30
= 7 + 30
= 37

Answer 6

Another solution is to use matrix algebra.

State the equations and swapping equations as necessary:

Equation 1: A -D = 6
Equation 2: A + B + C = 40
Equation 3: A - B = 2
Equation 4: B -2C + D = 0 from: C=(D+B)/2
Equation 5: -D + E = 40

Subtract Equation 2 from equation 1 replacing equation 2:

Equation 2: B + C + D = 34

Subtract Equation 1 from Equation 3 replacing equation 3:

Equation 3: -B + D = -4
...

This manipulation is just matrix algebra:

1 0 0 -1 0 = 6
0 1 1 1 0 = 34
0 -1 0 1 0 = -4
0 1 -2 1 0 = 0
0 0 0 -1 1 = 40

Once you state the problem as a matrix you solve the matrix by hand or you can go to a matrix calculator like the one at the following site:

http://members.shaw.ca/mtx/matrix.htm

Answer 7

A=13
B=11
C=9
D=7

Father=37