Each writer, Captain Dan's Ski Lodge rents both pairs of skis and snowboards to its guests for a flat daily rate per pair of skis and a flat daily rate per snowboard. Five pairs of skis and two snowboards will cost a family $370. Three pairs of skis and four snowboards will cost a family $390. During a particularly slow season, Captain Dan announces a 10% discount on all skis and snowboards. What would be the cost of renting two pairs of skis and two snowboards if they were rented during this discount period?

Options

A. 99
B. 110
C. 198
D. 220

Answer

To solve the problem, we need to determine the cost of renting two pairs of skis and two snowboards during a discount period. We will follow these steps:

### Step 1: Set up the equations
Let:
- \( x \) = cost of one pair of skis (in dollars)
- \( y \) = cost of one snowboard (in dollars)

From the problem, we have the following two equations based on the given information:
1. \( 5x + 2y = 370 \) (Equation 1)
2. \( 3x + 4y = 390 \) (Equation 2)

### Step 2: Solve the equations
We will solve these two equations simultaneously. To eliminate \( y \), we can make the coefficients of \( y \) the same in both equations. We can multiply Equation 1 by 2:

\[ 2(5x + 2y) = 2(370) \implies 10x + 4y = 740 \quad \text{(Equation 3)} \]

Now we have:
- Equation 3: \( 10x + 4y = 740 \)
- Equation 2: \( 3x + 4y = 390 \)

### Step 3: Subtract the equations
Now we can subtract Equation 2 from Equation 3 to eliminate \( y \):

\[ (10x + 4y) - (3x + 4y) = 740 - 390 \]

This simplifies to:

\[ 7x = 350 \]

### Step 4: Solve for \( x \)
Now, divide both sides by 7:

\[ x = \frac{350}{7} = 50 \]

So, the cost of one pair of skis is \( x = 50 \) dollars.

### Step 5: Substitute \( x \) back to find \( y \)
Now we can substitute \( x \) back into one of the original equations to find \( y \). We will use Equation 1:

\[ 5(50) + 2y = 370 \]

This simplifies to:

\[ 250 + 2y = 370 \]

Subtract 250 from both sides:

\[ 2y = 120 \]

Now divide by 2:

\[ y = 60 \]

So, the cost of one snowboard is \( y = 60 \) dollars.

### Step 6: Calculate the discounted prices
Now, we need to apply the 10% discount to both prices:
- Discounted price of skis:
\[ \text{Discounted price of skis} = x - 0.1x = 50 - 5 = 45 \text{ dollars} \]

- Discounted price of snowboards:
\[ \text{Discounted price of snowboards} = y - 0.1y = 60 - 6 = 54 \text{ dollars} \]

### Step 7: Calculate the total cost for 2 pairs of skis and 2 snowboards
Now, we calculate the total cost for renting 2 pairs of skis and 2 snowboards during the discount period:

\[ \text{Total cost} = 2 \times \text{(Discounted price of skis)} + 2 \times \text{(Discounted price of snowboards)} \]
\[ = 2 \times 45 + 2 \times 54 \]
\[ = 90 + 108 = 198 \text{ dollars} \]

### Final Answer
The cost of renting two pairs of skis and two snowboards during the discount period is **198 dollars**.

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