Ann is older than Betty. Their ages in years are such that twice the square of Betty's age subtracted from the square of Ann's age gives a number equal to 6 times the difference of their ages. Given also that the sum of their ages is equal to 5 times the difference of their ages, find the age in years if each of the sisters.
This question is actually from my exercise book. The answer shows:
Let Ann be A and Betty be B
A^2-2B^2= 6(A-B)
A+B=5(A-B)=
5A-5B
I don't understand the part where they wrote (a+b)=5(a-b), could someone explain it to me. Much appreciated!
It is just a matter of translating the English to Math.
Going with your definitions of A and B
"the sum of their ages" ---> A + B
"the difference of their ages" --> A-B
"is equal to 5 times" ---> = 5(....
(the sum of their ages) is equal to 5 times the difference of their ages
--> A+B = 5(A-B)
or
A+B = 5A - 5B
6B = 4A
B = (2/3)A or 2A/3
Now do the same analysis for the first part of the question.
so back in A^2 - 2B^2 = 6(A-B) = 6A - 6B
A^2 - 2(2A/3)^2 = 6A - 6(2A/3)
A^2 - 8A^2 /9 = 6A - 4A = 2a
times 9 to clear fractions
9A^2 - 8A^2 = 18A
A^2 - 18A = 0
A(A-18) = 0
A = 0 or A = 18 , A=0 clearly does not work
So Ann is 18 and Betty is 12
check:
square of Ann's age = 324
Twice the square of Betty's age = 288
that difference is 36
6 times the difference of their ages = 6(18-12) = 36
both results are 36, so my answer is correct
Thanks
In order to solve this problem, we need to set up a system of equations based on the given information.
Let's assign variables:
Age of Ann = A
Age of Betty = B
According to the problem, we have two conditions:
1) "Twice the square of Betty's age subtracted from the square of Ann's age gives a number equal to 6 times the difference of their ages":
A^2 - 2B^2 = 6(A - B)
2) "The sum of their ages is equal to 5 times the difference of their ages":
A + B = 5(A - B)
Now let's go step by step to explain how they got (A + B) = 5(A - B):
First, let's multiply out 5(A - B) to get:
5(A - B) = 5A - 5B
Now, we can rewrite the second equation as follows:
A + B = 5A - 5B
To simplify further, we can rearrange the equation by subtracting A from both sides:
B = 4A - 5B
Finally, by adding 5B to both sides, we get:
A + B = 5A
So, in summary, they wrote (A + B) = 5(A - B) by multiplying out 5(A - B) and rearranging the equation.
Now you have the system of equations:
A^2 - 2B^2 = 6(A - B)
A + B = 5A - 5B
You can solve this system to find the age of Ann (A) and Betty (B).
Certainly! Let's break it down step by step.
The problem statement says that the sum of Ann's age (A) and Betty's age (B) is equal to 5 times the difference of their ages (A - B). We can express this mathematically as:
A + B = 5(A - B)
Now let's simplify the equation:
A + B = 5A - 5B
The right side of the equation states that we have 5 times the difference of their ages. If we distribute this 5 to both terms inside the parentheses, we get 5A - 5B. Therefore, the equation can be simplified to:
A + B = 5A - 5B
This equation represents the sum of their ages being equal to 5 times the difference of their ages.