Question

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?

Answer 1

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.