In 2006, the digits in Perry's age were the same as the digits in Laura's age, but in reverse order.
In 2005, Perry was twice Laura's age.
What is the difference between Perry's and Laura's ages?
assume their ages were 2-digit numbers
in 2006:
Perry's age : 10x + y
Laura's age : 10y + x
in 2005:
Perry's age : 10x + y - 1
Laura's age : 10y + x- 1
10y+x-1 = 2(10x+y-1)
10y +x-1 = 20x+2y-2
19x -8y = 1
x = (8y+1)/19
or
y = (19x-1)/8
so 19x-1 has to be divisible by 8
of course we only have to try y's from 1 to 9 , possible values of (19y-1)/8 are:
1,18/8 --- 2, 37/8 ----- 3, 7 ---- ahhhh
sure enough , when y = 3, x = 7
So in 2006 Perry is 37 and Laura is 73
and the difference of their ages in 2006 is 36
By trial and error:
Perry's age = 73
Laura's age = 37
Difference of their ages = 36 years
Checking:
73-1 = 2(37-1)
72 = 2(36)
72 = 72
Great answer Reiny but line 8 is the wrong way round and should read
P=2*L
10x+y-1=2(10y+x-1)
but it doesn't change the answer
To solve this question, we need to use mathematical equations. Let's break it down step by step:
Step 1: Understand the given information.
In 2006, the digits in Perry's age were the same as the digits in Laura's age, but in reverse order. This means that the two numbers involved are palindromic, i.e., they read the same forwards and backward.
Step 2: Set up the equations.
Let's assume Perry's age in 2006 is represented by the palindrome XY (where X and Y are digits), and Laura's age is represented by YX (i.e., the reverse of Perry's age).
Step 3: Express the problem in the form of an equation.
In 2005, Perry was twice Laura's age, so we can set up the equation:
XY - 1 = 2(YX - 1)
Perry's age minus one (2006 - 2005 = 1) should equal twice Laura's age minus one.
Step 4: Simplify the equation.
XY - 1 = 2YX - 2
Step 5: Rearrange the equation to isolate one of the variables.
XY - 2YX = -2 + 1
XY - 2YX = -1
Step 6: Factor out one of the variables for substitution.
Y(X - 2) = -1
Step 7: Solve for the remaining variable.
(X - 2) = -1 / Y
Step 8: Test possibilities for Y.
Since Y must be a digit (between 0 and 9), we can test all possible values for Y. For each value of Y, we substitute it into the equation (X - 2) = -1 / Y and solve for X.
If we start with Y = 1, then (X - 2) = -1 / 1 -> (X - 2) = -1 -> X =1.
But 11 is not a suitable age as it is a double-digit number and contradicts the given information that the digits in their ages are the same.
We continue testing this procedure for all possible values of Y.
Step 9: Find the suitable ages for Perry and Laura.
After substituting all possible single-digit values of Y, we find that Y = 3 is a suitable value. Substituting Y = 3 into the equation (X - 2) = -1 / Y, we get (X - 2) = -1 / 3 -> (X - 2) = -1/3 -> X = 1.
Thus, Perry's age in 2006 (XY) is 13, and Laura's age (YX) is 31.
Step 10: Calculate the difference between their ages.
The difference between Perry's and Laura's ages is:
Difference = Perry's age - Laura's age
Difference = 13 - 31
Difference = -18
Therefore, the difference between Perry's and Laura's ages is 18.