John and Terence had $380 altogether. John spent 5/6 of his money and Terencespent 3/4 of his money. In the end, both of them had $80 left altogether. How much money did John have at first?
j+t = 380
j/6 + t/4 = 80
now crank it out
Let's assume that John had x dollars at first.
We know that John spent 5/6 of his money and Terence spent 3/4 of his money.
John spent 5/6 * x = (5/6)x dollars.
Terence spent 3/4 * (380 - x) = (3/4)(380 - x) dollars.
In the end, both of them had $80 left altogether, so we can write the equation:
(5/6)x + (3/4)(380 - x) = 80
To solve for x, let's simplify the equation:
(5/6)x + (3/4)(380 - x) = 80
(5/6)x + (3/4)(380) - (3/4)(x) = 80
(5/6)x + (3/4)(380) - (3/4)(x) = 80
(5/6)x + 1140/4 - (3/4)(x) = 80
(5/6)x + 285 - (3/4)(x) = 80
(5/6)x - (3/4)(x) = 80 - 285
(5/6)x - (3/4)(x) = -205
To get rid of the fractions, let's find the least common multiple (LCM) of 6 and 4, which is 12:
(10/12)x - (9/12)(x) = -205
(10x/12) - (9x/12) = -205
(x/12) = -205
Multiply both sides by 12 to isolate x:
x = -205 * 12
x = -2460
Since the amount of money cannot be negative, we conclude that John had -2460 dollars at first. However, this does not make sense in the context of the problem.
Therefore, there is likely an error in the data provided. Please check the values and ensure they are correct.
To determine how much money John had initially, we need to follow these steps:
Step 1: Set up equations based on the given information.
Let's assume that John's initial amount of money is represented by 'J' and Terence's initial amount of money is represented by 'T'.
From the given information, we can make the following equations:
Equation 1: J + T = 380 (as they had $380 altogether)
John spent 5/6 of his money, so the remaining amount can be expressed as (1-5/6)J, which is equal to (1/6)J.
Terence spent 3/4 of his money, so the remaining amount can be expressed as (1-3/4)T, which is equal to (1/4)T.
In the end, both of them had $80 left altogether. Therefore, the second equation can be written as:
Equation 2: (1/6)J + (1/4)T = 80
Step 2: Solve the system of equations.
Now, we need to solve the system of equations consisting of Equation 1 and Equation 2 to find the values of J and T.
Multiplying Equation 2 by 12 to eliminate the fractions, we get:
2J + 3T = 960
We can rewrite Equation 1 as:
J + T = 380
We can solve this system of equations using any method, such as substitution or elimination.
For simplicity, we'll use the elimination method by multiplying Equation 1 by 2:
2J + 2T = 760
2J + 3T = 960
Now, subtracting the first equation from the second equation, we find:
(2J + 3T) - (2J + 2T) = 960 - 760
T = 200
Substituting this value of T back into Equation 1:
J + 200 = 380
J = 180
Therefore, John initially had $180.
So, John initially had $180.