Lisa spends `(3)/(8)` of the monthyl paycheck on rent and `(5)/(12)` on food. Her roommate, Carrie who earns twice as much as Lisa, spends `(1)/(4)` of her monthyl paycheck on rent and `(1)/(2)` on food. If the two women decide to donate the remainder of their money to charity each month, what fraction of their combined monthly income will they donate ?

Answer

To solve the problem step by step, we will follow the reasoning outlined in the video transcript.

### Step 1: Define Lisa's Salary
Let \( X \) be Lisa's monthly salary.

### Step 2: Calculate Lisa's Expenditures
Lisa spends:
- On rent: \( \frac{3}{8}X \)
- On food: \( \frac{5}{12}X \)

### Step 3: Calculate Lisa's Remaining Amount
To find out how much Lisa has left after her expenditures, we subtract her spending from her salary:
\[ \text{Remaining amount for Lisa} = X - \left(\frac{3}{8}X + \frac{5}{12}X\right) \]

### Step 4: Find a Common Denominator
To combine the fractions \( \frac{3}{8}X \) and \( \frac{5}{12}X \), we need a common denominator. The least common multiple of 8 and 12 is 24.

- Convert \( \frac{3}{8} \) to \( \frac{9}{24} \)
- Convert \( \frac{5}{12} \) to \( \frac{10}{24} \)

Now we can add them:
\[ \frac{3}{8}X + \frac{5}{12}X = \frac{9}{24}X + \frac{10}{24}X = \frac{19}{24}X \]

### Step 5: Calculate Remaining Amount for Lisa
Now, substituting back into the remaining amount:
\[ \text{Remaining amount for Lisa} = X - \frac{19}{24}X = \frac{5}{24}X \]

### Step 6: Define Carrie's Salary
Since Carrie earns twice as much as Lisa, let \( Y = 2X \).

### Step 7: Calculate Carrie's Expenditures
Carrie spends:
- On rent: \( \frac{1}{4}Y = \frac{1}{4}(2X) = \frac{1}{2}X \)
- On food: \( \frac{1}{2}Y = \frac{1}{2}(2X) = X \)

### Step 8: Calculate Carrie's Remaining Amount
Now, we calculate how much Carrie has left after her expenditures:
\[ \text{Remaining amount for Carrie} = Y - \left(\frac{1}{4}Y + \frac{1}{2}Y\right) \]
Substituting \( Y = 2X \):
\[ = 2X - \left(\frac{1}{4}(2X) + \frac{1}{2}(2X)\right) = 2X - \left(\frac{1}{2}X + X\right) = 2X - \frac{3}{2}X = \frac{1}{2}X \]

### Step 9: Combine Remaining Amounts
Now we combine the remaining amounts for both Lisa and Carrie:
\[ \text{Total remaining for charity} = \frac{5}{24}X + \frac{1}{2}X \]
Convert \( \frac{1}{2}X \) to have a common denominator of 24:
\[ \frac{1}{2}X = \frac{12}{24}X \]
So,
\[ \text{Total remaining for charity} = \frac{5}{24}X + \frac{12}{24}X = \frac{17}{24}X \]

### Step 10: Calculate Combined Monthly Income
The combined income of Lisa and Carrie is:
\[ \text{Combined income} = X + Y = X + 2X = 3X \]

### Step 11: Calculate the Fraction Donated
Now, we find the fraction of their combined income that they donate:
\[ \text{Fraction donated} = \frac{\text{Total remaining for charity}}{\text{Combined income}} = \frac{\frac{17}{24}X}{3X} \]
This simplifies to:
\[ = \frac{17}{24} \div 3 = \frac{17}{72} \]

### Final Answer
The fraction of their combined monthly income that Lisa and Carrie will donate is \( \frac{17}{72} \).

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