Mike and Paul went shopping together with a total sum of $90. Mike spent twice as much as Paul. The amount Paul had left was $9 more than what he had spent. He had twice as much money left as Mike.
(a) How much money did Paul spend?
(b) How much money did Mike have at first?
Let's assume that Paul spent x dollars.
According to the question, Mike spent twice as much as Paul, so Mike spent 2x dollars.
The amount of money Paul had left was $9 more than what he had spent, so Paul had x + $9 dollars left.
He had twice as much money left as Mike, so x + $9 = 2(2x) = 4x.
=> 4x = x + $9
=> 4x - x = $9
=> 3x = $9
=> x = $9/3 = $3
Therefore, Paul spent $3.
To find out how much money Mike had at first, we need to add the money he spent to the money he had left.
Mike had 2x = 2 * $3 = $6 left.
So, he spent $90 - $6 = $84.
Therefore, Mike had $84 at first. Answer: \boxed{84}.
Let's solve the problem step by step:
Step 1: Assign variables.
Let's use the variable "P" to represent the amount Paul spent and "M" to represent the amount Mike spent.
Step 2: Translate the given information into equations.
The problem states that Mike spent twice as much as Paul. So we have the equation: M = 2P.
The problem also states that the amount Paul had left was $9 more than what he had spent. So we have the equation: Paul's left money = P + $9.
Finally, the problem mentions that Paul had twice as much money left as Mike. So we have the equation: Paul's left money = 2 * (Mike's left money).
Step 3: Set up the equations.
We have three equations to work with:
M = 2P
Paul's left money = P + $9
Paul's left money = 2 * (Mike's left money)
Step 4: Solve the equations.
Since we know that Mike and Paul went shopping together with a total of $90, we can set up another equation: M + P = $90.
From the equation M = 2P, we can substitute 2P for M in the equation M + P = $90:
2P + P = $90
3P = $90
P = $90 / 3
P = $30
Now we can substitute the value of P into the equation M = 2P:
M = 2 * $30
M = $60
Step 5: Answer the questions.
(a) Paul spent $30.
(b) Mike had $60 at first.
Let's solve this step-by-step:
Step 1: Let's represent the amount of money Paul spent as "x".
Step 2: According to the problem, Mike spent twice as much as Paul. So, Mike spent 2x.
Step 3: The total sum of money spent by both Mike and Paul was $90. Therefore, the equation becomes:
x + 2x = 90
Step 4: Simplify the equation:
3x = 90
Step 5: Divide both sides of the equation by 3:
x = 30
(a) Paul spent $30.
Step 6: According to the problem, the amount Paul had left was $9 more than what he spent. So, the money left with Paul is x + $9, which is 30 + $9 = $39.
Step 7: It is given that Paul had twice as much money left as Mike. So, Mike had $39 ÷ 2 = $19.50 left.
Step 8: The problem asks for the amount of money Mike had at first. To find this, we need to add Mike's spending and the money he had left. Therefore, Mike had 2x + $19.50 = 2(30) + $19.50 = $60 + $19.50 = $79.50.
(b) Mike had $79.50 at first.