alex has twice as much money as jennifer. jennifer has $6 less than shannon. together they have $54. How much money does each have?
a = 2j
j = s-6
a+j+s = 54
2(s-6) + (s-6) + s = 54
4s = 72
s = 18
j = 12
a = 24
To solve this problem, let's break it down step by step:
Step 1: Assign variables to the unknowns
Let's use the variables:
- Let x be the amount of money Shannon has.
- Let y be the amount of money Jennifer has.
- Let 2y be the amount of money Alex has (since Alex has twice as much as Jennifer).
Step 2: Translate the given information into equations
From the problem statement, we have the following information:
- Alex has twice as much money as Jennifer: 2y.
- Jennifer has $6 less than Shannon: x - $6.
- Together, they have $54: 2y + (x - $6) = $54.
Step 3: Solve the equation
Now we can solve the equation we formed in step 2.
Combining like terms, we get:
2y + x - $6 = $54.
Next, let's isolate the variables:
2y + x = $60.
Step 4: Use the given relationship between the variables
We know that Alex has twice as much money as Jennifer:
2y = 2(y).
Step 5: Apply substitution
Substituting 2y with y + $6 (since Jennifer has $6 less than Shannon), we get:
y + $6 + x = $60.
Step 6: Simplify and solve for y
Simplifying the equation, we have:
y + x = $54.
Now we have a system of two equations:
{ 2y + x = $60,
y + x = $54. }
Subtracting the second equation from the first, we get:
2y + x - (y + x) = $60 - $54,
y = $6.
Step 7: Substitute the value of y back into an equation to find x
Using the second equation, we can substitute the value of y:
$6 + x = $54.
Simplifying the equation:
x = $54 - $6,
x = $48.
Step 8: Calculate the amounts for each person
We have found that Shannon (x) has $48, Jennifer (y) has $6, and Alex (2y) has twice as much as Jennifer, which equals 2($6) = $12.
So, Shannon has $48, Jennifer has $6, and Alex has $12.
Therefore, Shannon has $48, Jennifer has $6, and Alex has $12.