This year Grandma's age and Mother's age are the same but with the digits reversed. One year ago, Grandma was twice as old as Mother. How old are Grandma and Mother today?
73 and 37
grandma's age --- 10x + y
mother's age ---- 10y + x
one year ago:
grandma : 10x+y - 1
mothers: 10y+x - 1
10x+y - 1 = 2(10y+x-1)
10x + y - 1 = 20y + 2x - 2
8x - 19y = -1
8x = 19y-1
x = (19y-1)/8
try different y's so that the numerator is a multiple of 8
e.g. when y = 3, x = 7 , the only one that works
so grandma is 73, the mother is 37
check: one year ago, grandma was 72, and mother was 36, and 72 is twice 36
To solve this problem, let's go step by step:
Step 1: Assign variables
Let's assign variables for both Grandma's age and Mother's age. Let G represent Grandma's age, and M represent Mother's age.
Step 2: Set up equations based on the given information
According to the problem, this year Grandma's age and Mother's age are the same but with the digits reversed. This means that G is the reverse of M. In terms of digits, G = 10 * (M % 10) + (M // 10).
One year ago, Grandma was twice as old as Mother. So, we can set up the equation (G - 1) = 2 * (M - 1).
Step 3: Solve the equations
Now we can solve the equations simultaneously to find the ages of Grandma and Mother.
Based on the information from step 2, we have:
G = 10 * (M % 10) + (M // 10)
(G - 1) = 2 * (M - 1)
Substituting the value of G from the first equation into the second equation, we get:
(10 * (M % 10) + (M // 10) - 1) = 2 * (M - 1)
Simplifying the equation:
10 * (M % 10) + (M // 10) - 1 = 2 * M - 2
10 * (M % 10) + (M // 10) = 2 * M - 1
Distributing the 10:
10 * (M % 10) + 10 * (M // 10) = 2 * M - 1
Combining like terms:
10 * (M % 10) + 10 * (M // 10) = 2 * M - 1
Now we can solve for M. Rewrite the equation as:
10 * (M % 10) + 10 * (M // 10) - 2 * M = -1
Simplifying further:
10 * (M % 10) - 2 * M = -10 * (M // 10) - 1
Rearranging and combining like terms:
10 * (M % 10) - 2 * M + 10 * (M // 10) = -1
Applying the floor division operator (//) and modulus operator (%):
10 * (M - 10 * (M // 10)) - 2 * M + 10 * (M // 10) = -1
10 * (M - M // 10 * 10) - 2 * M + 10 * (M // 10) = -1
Simplifying further:
10 * (M - M // 10 * 10 + M // 10) - 2 * M = -1
Combining like terms:
10 * (M + M // 10) - 2 * M = -1
Simplifying:
12 * M // 10 = -1
Now, we can solve for M:
12 * M = -10
M = -10 // 12
M = -1
Since we're talking about a person's age, we can see that a negative age is not possible. Therefore, the given problem is invalid or impossible to solve.
Please note that this analysis assumes the ages are positive whole numbers. If the problem stated that fractions or decimals were allowed, the answer might be different.