Rama was getting some items for the new school year. First, he bought some textbooks with $8 more than 1/3 of his money. Next, Next, he bought his stationery with $12.20 less than 1/2 of his remaining money. Lastly, he bought some school socks with $2.80 more than 1/2 of the money left. Then, he had $15.40 with him. How
much money did he have at first?
Let's work backwards from the final amount of money Rama had.
When he had $15.40 left, he used some of that money to buy the school socks. We know that the socks cost $2.80 more than 1/2 of the money Rama had left. So we can write an equation:
$15.40 - socks cost = 1/2 remaining money + $2.80
Simplifying this equation:
socks cost = $15.40 - 1/2 remaining money - $2.80
Now let's consider the stationery. Before buying the socks, Rama had a certain amount of money left after buying the textbooks. He used some of that money to buy the stationery. We know that the stationery cost $12.20 less than 1/2 of the remaining money. So we can write another equation:
remaining money after textbooks = stationery cost + 1/2 remaining money + $12.20
Simplifying this equation:
stationery cost = remaining money after textbooks - 1/2 remaining money - $12.20
Finally, we know that Rama bought the textbooks with $8 more than 1/3 of his money. Let's call the amount of money he had at first "x". Then we can write a third equation:
x = textbook cost + 1/3x + $8
Simplifying this equation:
textbook cost = x - 1/3x - $8
Now we have three equations:
socks cost = $15.40 - 1/2 remaining money - $2.80
stationery cost = remaining money after textbooks - 1/2 remaining money - $12.20
textbook cost = x - 1/3x - $8
We can use these equations to solve for x, the amount of money Rama had at first.
First, we can use the textbook equation to find the cost of the textbooks:
textbook cost = x - 1/3x - $8
textbook cost = 2/3x - $8
Next, we can use the stationery equation to find the remaining money after buying the textbooks:
stationery cost = remaining money after textbooks - 1/2 remaining money - $12.20
stationery cost = 1/2 remaining money - $12.20
remaining money after textbooks = stationery cost + 1/2 remaining money + $12.20
remaining money after textbooks = 1/2 remaining money + $12.20 + 1/2(2/3x - $8)
remaining money after textbooks = 1/2 remaining money + 1/3x - $2.40
Now we can use the socks equation to find the remaining money after buying the stationery:
socks cost = $15.40 - 1/2 remaining money - $2.80
socks cost + 1/2 remaining money = $12.60
remaining money after stationery = $12.60 - socks cost
remaining money after stationery = $12.60 - ($15.40 - 1/2 remaining money - $2.80)
remaining money after stationery = $12.60 - $15.40 + 1/2 remaining money + $2.80
remaining money after stationery = 1/2 remaining money - $0.80
Now we can combine the equations for remaining money after textbooks and remaining money after stationery:
remaining money after textbooks = 1/2 remaining money + 1/3x - $2.40
1/2 remaining money - $0.80 = 1/2 remaining money + 1/3x - $2.40
1/3x = $1.60
x = $4.80
Therefore, Rama had $4.80 at first.
Let's solve this problem step by step.
Let's assume Rama's initial amount of money as "x".
Step 1: Rama bought some textbooks with $8 more than 1/3 of his money.
The amount spent on textbooks = 1/3 x + $8
Step 2: Rama bought his stationery with $12.20 less than 1/2 of his remaining money.
After buying textbooks, Rama's remaining money = x - (1/3 x + $8) = 2/3 x - $8
The amount spent on stationery = 1/2(2/3 x - $8) - $12.20
Step 3: Rama bought some school socks with $2.80 more than 1/2 of the money left.
After buying stationery, Rama's remaining money = (2/3 x - $8) - (1/2(2/3 x - $8) - $12.20) = $15.40
The amount spent on school socks = 1/2[(2/3 x - $8) - (1/2(2/3 x - $8) - $12.20)] + $2.80
Now, we can set up the equation and solve for x:
x - (1/3 x + $8) - 1/2(2/3 x - $8) + $12.20 + 1/2[(2/3 x - $8) - (1/2(2/3 x - $8) - $12.20)] + $2.80 = $15.40
Solving this equation will give us the value of x, which is the initial amount of money Rama had.