Ruby had 5 times as much money as Sally. Ruby spent $80 and Sally had 3/7 as much as Ruby in the end. How much did Ruby have at first?
Use the fraction given to cut the model drawing into equal parts for Ruby and Sally to get the answer $150
Ruby had $150 at first.
Let's use a step-by-step approach to solve the problem.
Step 1: Let's assume that Ruby had x dollars at first.
Step 2: According to the problem, Ruby had 5 times as much money as Sally. Therefore, Sally had (1/5)x dollars at first.
Step 3: Ruby spent $80, so Ruby had x - $80 dollars remaining.
Step 4: Sally had 3/7 as much as Ruby in the end. Therefore, Sally had (3/7)(x - $80) dollars.
Step 5: According to the problem, Sally had (1/5)x dollars at first. So, we can set up the following equation:
(1/5)x = (3/7)(x - $80)
Step 6: Let's solve the equation for x to find out the value of x, which represents the amount of money Ruby had at first.
First, let's remove the fractions by multiplying both sides of the equation by the denominators:
(7/5)x = (3/7)(x - $80)
Simplifying the equation:
7 * 7x = 5 * 3(x - $80)
49x = 15(x - $80)
Expanding:
49x = 15x - $1200
Subtracting 15x from both sides:
49x - 15x = - $1200
Combining like terms:
34x = - $1200
Dividing both sides by 34:
x = (- $1200) / 34
Calculating x:
x ≈ - $35.29
Since money cannot be negative, it means our assumption for x was incorrect.
Step 7: Let's try a different assumption for x. Let's assume that Ruby had y dollars at first.
Step 8: According to the problem, Ruby had 5 times as much money as Sally. Therefore, Sally had (1/5)y dollars at first.
Step 9: Ruby spent $80, so Ruby had y - $80 dollars remaining.
Step 10: Sally had 3/7 as much as Ruby in the end. Therefore, Sally had (3/7)(y - $80) dollars.
Step 11: According to the problem, Sally had (1/5)y dollars at first. So, we can set up the following equation:
(1/5)y = (3/7)(y - $80)
Step 12: Let's solve the equation for y to find out the value of y, which represents the amount of money Ruby had at first.
First, let's remove the fractions by multiplying both sides of the equation by the denominators:
(7/5)y = (3/7)(y - $80)
Simplifying the equation:
7 * 7y = 5 * 3(y - $80)
49y = 15(y - $80)
Expanding:
49y = 15y - $1200
Subtracting 15y from both sides:
49y - 15y = - $1200
Combining like terms:
34y = - $1200
Dividing both sides by 34:
y = (- $1200) / 34
Calculating y:
y ≈ - $35.29
Again, we obtained a negative value for y, which means our assumption for y was also incorrect.
Step 13: Since both assumptions for the initial amount of money are incorrect, we need to rethink our approach.
Let's try a different approach:
Let's assume that Sally had y dollars at first.
According to the problem, Ruby had 5 times as much money as Sally. Therefore, Ruby had 5y dollars at first.
Ruby spent $80, so Ruby had 5y - $80 dollars remaining.
Sally had 3/7 as much as Ruby in the end. Therefore, Sally had (3/7)(5y - $80) dollars.
According to the problem, Sally had y dollars at first. So, we can set up the following equation:
y = (3/7)(5y - $80)
Let's solve the equation for y to find out the value of y, which represents the amount of money Sally had at first.
First, let's remove the fraction by multiplying both sides of the equation by 7:
7y = 3(5y - $80)
Expanding:
7y = 15y - 3($80)
Distributing:
7y = 15y - $240
Subtracting 15y from both sides:
7y - 15y = - $240
Combining like terms:
-8y = - $240
Dividing both sides by -8:
y = (- $240) / -8
Calculating y:
y = $30
Step 14: Now that we know y, which represents the amount of money Sally had at first, we can find the amount of money Ruby had at first by multiplying y by 5.
Ruby had 5 * $30 = $150 at first.
Therefore, Ruby had $150 at first.