Math  /  Algebra

QuestionYour roommate, Gianna, offered to buy groceries for you and your other roommate. The total bill was $77.80\$ 77.80. She forgot to save the individual receipts but remembered that your groceries were $0.05\$ 0.05 cheaper than half of her groceries, and that your other roommate's groceries were $1.90\$ 1.90 more than your groceries. How much was each of your share of the groceries?

Studdy Solution

STEP 1

1. The total grocery bill was \$77.80.
2. Your groceries were \$0.05 cheaper than half of Gianna's groceries.
3. Your other roommate's groceries were \$1.90 more than your groceries.
4. We need to find the cost of groceries for you, Gianna, and your other roommate.

STEP 2

1. Define variables for the cost of groceries for each person.
2. Write equations based on the problem description.
3. Solve the system of equations to find each person's share.

STEP 3

Define variables for the cost of groceries for each person.
Let: - x x be the cost of your groceries. - y y be the cost of Gianna's groceries. - z z be the cost of your other roommate's groceries.

STEP 4

Write equations based on the problem description.
From the problem, we have the following relationships:
1. The total cost equation: $ x + y + z = 77.80 \]

2. Your groceries were \0.05cheaperthanhalfofGiannasgroceries:0.05 cheaper than half of Gianna's groceries: x = \frac{y}{2} - 0.05 \]
3. Your other roommate's groceries were \1.90morethanyourgroceries:1.90 more than your groceries: z = x + 1.90 \]

STEP 5

Solve the system of equations to find each person's share.
First, substitute the expression for x x from the second equation into the third equation: z=(y20.05)+1.90z = \left(\frac{y}{2} - 0.05\right) + 1.90 Simplify: z=y2+1.85z = \frac{y}{2} + 1.85
Now substitute both expressions for x x and z z into the total cost equation: (y20.05)+y+(y2+1.85)=77.80\left(\frac{y}{2} - 0.05\right) + y + \left(\frac{y}{2} + 1.85\right) = 77.80
Combine like terms: y+y+1.80=77.80y + y + 1.80 = 77.80
Simplify: 2y+1.80=77.802y + 1.80 = 77.80
Subtract 1.80 from both sides: 2y=76.002y = 76.00
Divide by 2: y=38.00y = 38.00
Now substitute y=38.00 y = 38.00 back into the equations for x x and z z : x=38.0020.05=19.000.05=18.95x = \frac{38.00}{2} - 0.05 = 19.00 - 0.05 = 18.95
z=38.002+1.85=19.00+1.85=20.85z = \frac{38.00}{2} + 1.85 = 19.00 + 1.85 = 20.85
The cost of groceries for each person is: - Your groceries: 18.95 \boxed{18.95} - Gianna's groceries: 38.00 \boxed{38.00} - Your other roommate's groceries: 20.85 \boxed{20.85}

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