Alicia and Neha 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 X be the amount Neha had at first.
After Alicia spent 1 3 of her money, she had X - X/3 = 2/3*X left.
After Neha spent $10, she had X - $10 left.
2/3*X = 1/2*(X - $10)
2/3*X = 1/2*X - 1/2*$10
2/3*X - 1/2*X = - 1/2*$10
4/6*X - 3/6*X = -1/2*$10
X/6 = -$10/2
X/6 = -$5
X = -$5 * 6
X = -$30
Neha had -$30 at first. Answer: \boxed{30}.
Let's calculate step by step:
Let's assume that Neha had x dollars initially.
After Alicia spent 1/3 of her money, Alicia had 2/3 of her money left.
Given that Neha spent $10, the amount of money Neha had left was x - 10.
According to the problem, Alicia's money left is 1/2 of Neha's money left.
So, we can write the equation:
2/3 * Alicia's initial money = 1/2 * (Neha's initial money - $10)
Now, let's use the given information that the total amount they had was $66.
We know that Alicia and Neha had a total of $66.
So, we can write the equation:
Alicia's initial money + Neha's initial money = $66
Let's solve these equations step by step:
2/3 * Alicia's initial money = 1/2 * (Neha's initial money - $10)
Multiplying both sides by 3 to get rid of the fraction:
2 * Alicia's initial money = 3/2 * (Neha's initial money - $10)
Dividing both sides by 2:
Alicia's initial money = 3/4 * (Neha's initial money - $10)
Now, let's substitute Alicia's initial money with (66 - Neha's initial money) in the above equation:
66 - Neha's initial money = 3/4 * (Neha's initial money - $10)
Expanding the brackets:
66 - Neha's initial money = 3/4 * Neha's initial money - 3/4 * $10
Simplifying:
66 - Neha's initial money = 3/4 * Neha's initial money - 3 * $10/4
66 - Neha's initial money = 3/4 * Neha's initial money - $30/4
Common denominators:
66 - Neha's initial money = 3/4 * Neha's initial money - 30/4
Converting whole numbers:
(264 - 4 * Neha's initial money) / 4 = 3 * Neha's initial money - 30/4
Multiplying both sides by 4:
264 - 4 * Neha's initial money = 12 * Neha's initial money - 30
Bringing like terms to one side:
12 * Neha's initial money + 4 * Neha's initial money = 264 + 30
16 * Neha's initial money = 294
Dividing both sides by 16:
Neha's initial money = 294 / 16
Neha's initial money = $18.375
Since Neha cannot have a fraction of a dollar, we can round Neha's initial money to the nearest whole number:
Neha's initial money = $18
Therefore, Neha had $18 at first.
To find out how much Neha had at first, let's break down the problem step by step:
Let's assume the amount of money Alicia had at first was 'A' dollars and the amount of money Neha had at first was 'N' dollars.
According to the given information, the total amount of money they had together was $66. So, we can create an equation:
A + N = 66 ...equation (1)
Now, let's move on to the information about how much money Alicia and Neha had left after spending.
After Alicia spent 1/3 of her money, she had (2/3)A dollars left.
Neha spent $10, so she had (N - 10) dollars left.
According to the second part of the problem, the amount of money Alicia had left was 1/2 of the amount of money Neha had left. Therefore, we can create another equation:
(2/3)A = (1/2)(N - 10) ...equation (2)
Now, we can solve this system of equations (equations 1 and 2) to find the values of A and N.
Let's solve equation (2) for A:
Multiply both sides of equation (2) by 3/2:
(2/3)A * (3/2) = (1/2)(N - 10) * (3/2)
A = (3/4)(N - 10) ...equation (3)
Substitute equation (3) into equation (1):
(3/4)(N - 10) + N = 66
(3/4)N - 30/4 + 4/4N = 66
(7/4)N - 30/4 = 66
Multiply both sides by 4 to get rid of the fraction:
7N - 30 = 264
7N = 264 + 30
7N = 294
N = 294/7
N = 42
So Neha had $42 at first.
Therefore, Neha had $42 at first.