Patrick saved 7/25 of his salary. He gave 1/2 of the remainder to his mother, and spent the rest. He has $1080 left. How much was Patrick's salary?
saved: 7/25, leaving 18/25
gave 1/2 * 18/25 = 9/25, leaving 9/25
so,
9/25 x = 1080
find x
Let's break down the problem step by step to find the solution.
1. Patrick saved 7/25 of his salary.
So, the amount he saved can be represented as 7/25 * salary.
2. The remainder after saving is given to his mother.
This means the amount he gave to his mother can be represented as 1/2 * (salary - 7/25 * salary).
3. He spent the rest of the money, which is $1080.
Therefore, the amount Patrick spent can be represented as [(salary - 7/25 * salary) - 1/2 * (salary - 7/25 * salary)].
4. According to the problem, the amount he spent is $1080.
So we can set up the equation:
[(salary - 7/25 * salary) - 1/2 * (salary - 7/25 * salary)] = $1080.
Now let's solve this equation step by step.
Step 1:
Combine like terms within the parentheses on the left side of the equation.
[(18/25 * salary) - 1/2 * (18/25 * salary)] = $1080
Step 2:
Multiply the fractions within the parentheses by the common denominator (25) to get rid of the denominators.
[(18 * 1 * salary) - (1/2 * 18 * salary)] / 25 = $1080
Step 3:
Multiply and simplify within the parentheses.
[(18 * salary) - (9 * salary)] / 25 = $1080
Step 4:
Combine like terms in the numerator.
(9 * salary) / 25 = $1080
Step 5:
Multiply both sides of the equation by 25 to isolate the salary.
9 * salary = $1080 * 25
Step 6:
Divide both sides of the equation by 9 to solve for the salary.
salary = ($1080 * 25) / 9
Calculating this gives us:
salary = $3,000
Therefore, Patrick's salary was $3,000.
To find out Patrick's salary, we will work backwards using the information given in the problem.
Let's start by figuring out how much money Patrick had left after giving 1/2 of the remainder to his mother. We know he had $1080 left, so let's call this amount "x".
If he had given 1/2 of the remainder to his mother, the remaining amount would be 1/2 of "x". So, we can express this as (1/2) * x.
Now, let's determine the amount Patrick had before giving the remaining half to his mother. Since he spent the rest, the amount before giving half to his mother would be twice the remaining amount, which can be expressed as 2 * (1/2) * x.
Finally, we know that Patrick saved 7/25 of his salary. So, the amount before giving half to his mother would be equal to 7/25 of his salary. We can express this as (7/25) * S, where S represents his salary.
Putting it all together, the equation becomes:
(7/25) * S = 2 * (1/2) * x
Simplifying this equation:
(7/25) * S = x
We also know that x = $1080, so we can substitute this value back into the equation:
(7/25) * S = $1080
To solve for S (Patrick's salary), we'll divide both sides of the equation by (7/25):
S = ($1080) / (7/25)
Now, let's find the value of S:
S = ($1080) * (25/7)
S ≈ $3878.57
Therefore, Patrick's salary was approximately $3,878.57.