SOLUTION: Ten years from now, A will be twice as old as B. Five years ago, A was three times as old as B. What are the present ages of A and B?

Question 611044: Ten years from now, A will be twice as old as B. Five years ago, A was three times as old as B. What are the present ages of A and B?

Found 2 solutions by radh, Earlsdon:
Answer by radh(108) (Show Source): You can put this solution on YOUR website!
Let's map this out mathematically
Then years from now, (+10) A will be twice as old as B (2b). Five years ago, (-5) A was three times as old as B (3b).

So, that means:

and
.

Let's simplify that and move both variables on one side of the equation to get:

and

We're going to substitute this into the solver. x=b and y=a.

Solved by pluggable solver: Using Cramer's Rule to Solve Systems with 2 variables

First let . This is the matrix formed by the coefficients of the given system of equations.

Take note that the right hand values of the system are and which are highlighted here:

These values are important as they will be used to replace the columns of the matrix A.

Now let's calculate the the determinant of the matrix A to get . Remember that the determinant of the 2x2 matrix is . If you need help with calculating the determinant of any two by two matrices, then check out this solver.

Notation note: denotes the determinant of the matrix A.

---------------------------------------------------------

Now replace the first column of A (that corresponds to the variable 'x') with the values that form the right hand side of the system of equations. We will denote this new matrix (since we're replacing the 'x' column so to speak).

Now compute the determinant of to get . Once again, remember that the determinant of the 2x2 matrix is

To find the first solution, simply divide the determinant of by the determinant of to get:

So the first solution is

---------------------------------------------------------

We'll follow the same basic idea to find the other solution. Let's reset by letting again (this is the coefficient matrix).

Now replace the second column of A (that corresponds to the variable 'y') with the values that form the right hand side of the system of equations. We will denote this new matrix (since we're replacing the 'y' column in a way).

Now compute the determinant of to get .

To find the second solution, divide the determinant of by the determinant of to get:

So the second solution is

====================================================================================

Final Answer:

So the solutions are and giving the ordered pair (15, 40)

Once again, Cramer's Rule is dependent on determinants. Take a look at this 2x2 Determinant Solver if you need more practice with determinants.

So, person A is 40 and person B is 15. :)
Answer by Earlsdon(6294) (Show Source): You can put this solution on YOUR website!
You can express these two situations algebraically:
A+10 = 2(B+10) "Ten years from now (A+10), A will be twice as old as B (will be ten years from now) 2(B+10)" and
A-5 = 3(B-5) "Five years ago (A-5), A was three times as old as B (was five years ago), (3(B-5))" Solve this for A to get:
A = 3(B-5)+5 and simplify to get:
A = 3B-15+5 or
A = 3B-10 Now substitute into the first equation:
(3B-10)+10 = 2(B+10) Simplify and solve for B.
3B =2B+20
and...
A = 3B-10
A = 3(20)-10
A = 60-10

Check:
A+10 = 2(B+10)
50+10 = 2(20+10)
60 = 2(30)
60 = 60 and...
A-5 = 3(B-5)
50-5 = 3(20-5)
45 = 3(15)
45 = 45
RELATED QUESTIONS
Ten years from now, A will be twice as old as B. Five years ago, A was three times as old (answered by scott8148)
ten years from now, A will be twice as old as B. five years ago,A was three times as old... (answered by stanbon)
Ten years from now, A will be twicw as old as B. Five years ago, A was three times as old (answered by stanbon)
ten years from now,A will be twice as old B and five years ago,A was three times as old... (answered by josgarithmetic,mark_henrhiz)
ten years from now, Anna will be twice as old as Barbara. Five years ago, Anna was three... (answered by ankor@dixie-net.com)
If ten years from now, A will be twice as much as B. And five years ago, A was three... (answered by josgarithmetic)
Five years from now, A will be 4/5 as old as B; five years ago, he was 2/3 as old then.... (answered by ankor@dixie-net.com)
A is twice as old as B; ten years ago he was four times as old.what are their present... (answered by richwmiller)
A father is twice as old as his son. Ten years ago he was three times as old as his son. (answered by ankor@dixie-net.com)