Each of Alice and Beatrice has their birthday on the same day. In 8 years' time, Alice will be twice as old as Beatrice. Ten years ago, the sum of their ages was 21. How old is Alice now?
present time:
Alice --- a
Beatrice --- b
10 years ago:
a-10 + b-10 = 21
a+b = 41 **
in 8 year's time:
a+8 = 2(b+8)
a+8 = 2b + 16
a - 2b = 8 ***
subtract *** from **
3b = 33
b = 11 , then a = 30
State your conclusions
Let's solve this step-by-step.
Let's assume Alice's current age as "A" and Beatrice's current age as "B".
According to the information given:
1. Each of Alice and Beatrice has their birthday on the same day.
2. In 8 years' time, Alice will be twice as old as Beatrice.
3. Ten years ago, the sum of their ages was 21.
Let's write equations based on the given information:
Equation 1: Each of Alice and Beatrice has their birthday on the same day.
A = B
Equation 2: In 8 years' time, Alice will be twice as old as Beatrice.
A + 8 = 2(B + 8)
Equation 3: Ten years ago, the sum of their ages was 21.
(A - 10) + (B - 10) = 21
Now, let's solve these equations to find the values of A and B.
Using Equation 1:
A = B
Substituting this into Equation 2:
A + 8 = 2(B + 8)
B + 8 + 8 = 2B + 16
B + 16 = 2B + 16
B - 2B = 16 - 16
-B = 0
B = 0
Using Equation 3:
(A - 10) + (B - 10) = 21
(A - 10) + (0 - 10) = 21
(A - 10) - 10 = 21
A - 20 = 21
A = 21 + 20
A = 41
Therefore, Alice's current age is 41.
To solve this problem, we need to set up equations based on the given information. Let's denote Alice's current age as A and Beatrice's current age as B.
From the first statement, we know that Alice and Beatrice share the same birthday, so their current ages are the same: A = B.
In 8 years' time, Alice will be twice as old as Beatrice. So, in 8 years, Alice's age will be A + 8, and Beatrice's age will be B + 8. Therefore, we can write the equation: A + 8 = 2(B + 8).
Also, ten years ago, the sum of their ages was 21. So, ten years ago, Alice's age was A - 10, and Beatrice's age was B - 10. Hence, we can write the equation: (A - 10) + (B - 10) = 21.
Now, we can solve these equations simultaneously to find the values of A and B.
From the equation A = B, we can substitute B with A in the other equations: A + 8 = 2(A + 8).
Let's solve this equation:
A + 8 = 2A + 16
A - 2A = 16 - 8
-A = 8
A = -8
However, we cannot have a negative age, so this solution is not valid. It means that there is no solution to this problem, which suggests that the given information might be contradictory or incorrect.
Please double-check the problem or provide more information if necessary.
so put the words into math.
a+8 = 2(b+8)
a-10 + b-10 = 21
so, a=30