Alicia and nela had a total of $66. After Alicia spent 1 3 of her money and neha spent $10 the amount of money alicia had left was 1 2 of the amount of money neha had left. How much did neha have at first?
Let's say Alicia had x dollars.
After Alicia spent 1/3 of her money, she had x - (1/3)x = (2/3)x dollars left.
After Neha spent $10, Alicia had (2/3)x - 10 dollars left.
The amount of money Neha had left is 1/2 * ((2/3)x - 10).
Since Alicia and Neha had a total of $66, we can write the equation (2/3)x - 10 + 1/2 * ((2/3)x - 10) = 66.
Multiplying through the parentheses, we get (2/3)x - 10 + (1/2) * (2/3)x - (1/2)*10 = 66.
Multiplying further, we get (2/3)x - 10 + (1/3)x - 5 = 66.
Combining like terms, we get (5/3)x - 15 = 66.
Adding 15 to both sides, we get (5/3)x = 81.
Dividing both sides by (5/3), we get x = (3/5)*81 = <<(3/5)*81=48.6>>48.6 dollars.
Therefore, Neha had $48.6 at first. Answer: \boxed{48.6}.
Let's assume the amount of money Alicia had initially is "A" dollars and the amount of money Neha had initially is "N" dollars.
According to the given information, the total amount of money they had together is $66:
A + N = $66.
After Alicia spent 1/3 of her money, she had 2/3 of her initial amount of money left:
Alicia's money left = (2/3) * A.
After Neha spent $10, the amount of money she had left is:
Neha's money left = N - $10.
It is also given that the amount Alicia had left was 1/2 of the amount Neha had left:
(2/3) * A = (1/2) * (N - $10).
Now, we can solve the equations to find the values of A and N.
From the first equation, we have:
A = $66 - N.
Substituting this value of A in the second equation, we get:
(2/3) * ($66 - N) = (1/2) * (N - $10).
Now we can solve this equation for N:
(2/3) * $66 - (2/3) * N = (1/2) * N - $5.
(Simplifying the equation)
$44 - (2/3) * N = (1/2) * N - $5.
(Simplifying further)
(2/3) * N + (1/2) * N = $44 + $5.
(Making a common denominator)
(4/6) * N + (3/6) * N = $49.
(Simplifying again)
(7/6) * N = $49.
Now we can solve for N:
N = ($49 * 6) / 7.
Calculating this, we find:
N = $42.
Therefore, Neha had $42 at first.
To solve this problem, we need to set up equations based on the given information and then solve them.
Let's start by assigning variables to the unknown quantities. Let:
- Alicia's initial amount of money be represented by A,
- Neha's initial amount of money be represented by N.
We know that the sum of their money is $66, so we can write the equation:
A + N = 66 ----(1)
Now, let's analyze the information about Alicia's remaining money after spending 1/3 of it.
Alicia has 2/3 of her initial money left, which can be written as:
2/3 * A
And Neha's remaining money after spending $10 can be represented by:
N - 10
Given that Alicia had 1/2 of Neha's remaining money, we can write the equation:
2/3 * A = 1/2 * (N - 10) ----(2)
Now we have a system of two equations (equations 1 and 2). We can solve this system to find the values of A and N.
First, let's simplify equation (2):
2/3 * A = 1/2 * N - 5
To eliminate the fractions, we can multiply both sides of equation (2) by 6 to get rid of the denominators:
4A = 3(N - 10)
4A = 3N - 30 ----(3)
We now have a system of equations 1 and 3:
A + N = 66 ----(1)
4A = 3N - 30 ----(3)
We can now solve this system of equations using substitution or elimination.
Let's use the substitution method to solve for N:
From equation (1), we can rewrite it as:
N = 66 - A
Substituting N in equation (3) with 66 - A, we have:
4A = 3(66 - A) - 30
4A = 198 - 3A - 30
4A + 3A = 168
7A = 168
A = 168/7
A = 24
Now, substituting the value of A = 24 into equation (1):
24 + N = 66
N = 66 - 24
N = 42
Therefore, Neha initially had $42.