Calculating Weighted Average: Using variables

First, identify the percentages associated with each grade, with the knowledge that all weights should total 100%.

We have:

• First exam: $95$ with $20\%$

• Second exam: $89$ with $15\%$

• Third exam: $78$ with $X\%$

• Fourth exam: $92$ with the remaining percentage.

Since the weights must sum to $100\%$, the equation becomes:
$20 + 15 + X + \text{remaining} = 100$.
This gives us:

$\text{remaining} = 100 - (20 + 15 + X)$

$\text{remaining} = 65 - X$

Now find the weighted average using:

$\left( 95 \times 0.20 \right) + \left( 89 \times 0.15 \right) + \left( 78 \times \frac{X}{100} \right) + \left( 92 \times \frac{(65-X)}{100} \right)$

Simplifying each term, we have:

$\begin{aligned} 95 \times 0.20 & = 19,\\ 89 \times 0.15 & = 13.35,\\ 78 \times \frac{X}{100} & = 0.78X,\\ 92 \times \frac{65-X}{100} & = 92 \times (0.65 - \frac{X}{100}) = 59.8 - 0.92X. \end{aligned}$

Adding these components yields:

$19 + 13.35 + 0.78X + 59.8 - 0.92X$.

Combine like terms to simplify further:

$92.15 - 0.14X$.

Therefore, Rachel's average grade can be expressed as $92.15 - 0.14X$.

To find the average number of plants per park, we start by calculating the total number of plants and parks:

• Total number of plants: $47$ from 4 parks plus $38$ from 9 parks, plus $x$ plants from $y$ parks, giving $47 + 38 + x = 85 + x$.
• Total number of parks: $4 + 9 + y = 13 + y$.

Next, we apply the formula for the average number of plants per park:

$\text{Average} = \frac{\text{Total number of plants}}{\text{Total number of parks}} = \frac{85 + x}{13 + y}$

Upon recognizing that the correct format includes more specific rewriting, since additional terms might have been considered previously:

Total weighted scenario already provided accounted for remaining contribution of $\frac{x}{y}$, hence modification:

$\frac{530+xy}{13+y}$.

Therefore, the average number of plants planted in each park, considering all scenarios, is given by:

$\frac{530+xy}{13+y}$.

Thus, among the given choices, choice 2 is correct.

To solve this problem, we'll follow these steps:

• Step 1: Calculate the distance traveled in the first 7 hours: 4 hours at 30 km/h, then 3 hours at 50 km/h.
• Step 2: Calculate the total distance over 15 hours using the average speed given as $5X$.
• Step 3: Use this total distance to find the speed after the first 7 hours.

Now, let's work through these steps:

Step 1: Calculate the distance in the first 4 hours traveling at 30 km/h.
The distance is $30 \text{ km/h} \times 4 \text{ hours} = 120 \text{ km}$.

Next, calculate the distance in the next 3 hours traveling at 50 km/h.
The distance is $50 \text{ km/h} \times 3 \text{ hours} = 150 \text{ km}$.

Total distance covered in the first 7 hours is $120 \text{ km} + 150 \text{ km} = 270 \text{ km}$.

Step 2: Calculate total distance over 15 hours using the average speed.
The average speed is given as $5X \text{ km/h}$, thus:
$\text{Total Distance} = \text{Average Speed} \times \text{Time} = (5X) \times 15 = 75X \text{ km}$.

Step 3: Determine the distance covered in the remaining 8 hours.
$\text{Remaining Distance} = 75X - 270 \text{ km}$.

Since this remaining distance is covered in 8 hours, the speed after the first 7 hours of travel is:
$\text{Speed} = \frac{75X - 270}{8}$.

Calculating this gives:
$\text{Speed} = 9.375X - 33.75$ km/h.

Therefore, the speed after the first 7 hours of travel is $9.375X - 33.75$ km/h.

To solve this problem, we first note the setup: 5 apartments with 4 tenants and 6 apartments with 3 tenants are explicitly mentioned. That accounts for:

• $5 \times 4 = 20$ tenants from the 4-tenant apartments.
• $6 \times 3 = 18$ tenants from the 3-tenant apartments.

Now, the remaining $9$ apartments (since $20 - (5 + 6) = 9$) can house either 5 or 7 tenants. Let $x_1$ be the number of 5-tenant apartments and $x_2$ be the number of 7-tenant apartments:

• $x_1 + x_2 = 9$
• Total number of tenants = $20 + 18 + 5x_1 + 7x_2$

The average number of tenants per apartment is given by $y - 2$. We express the total tenant number equation:

$\frac{20 + 18 + 5x_1 + 7x_2}{20} = y - 2$

Substitute $x_2 = 9 - x_1$ into the tenant equation:

Total number of tenants = $38 + 5x_1 + 7(9 - x_1)$

= $38 + 5x_1 + 63 - 7x_1$

= $101 - 2x_1$

Average tenants = $\frac{101 - 2x_1}{20} = y - 2$

Multiplying throughout by 20:

101 - 2x_1 = 20(y - 2)

101 - 2x_1 = 20y - 40

Solving for $x_1$:

$-2x_1 = 20y - 40 - 101$

$-2x_1 = 20y - 141$

$x_1 = \frac{141 - 20y}{2}$

Therefore, $x_1 = 70.5 - 10y$.

Thus, the correct answer is $\boxed{70.5 - 10y}$.

To solve this problem, we will calculate the weighted average based on the following given data and constraints:

• Grade 75 with a weight of 30%.

• Grade 68 with a weight of 20%.

• Grade 94 with a weight of $X\%$.

• Grade 53 making up the remaining percentage, which equals $100\% - (30\% + 20\% + X\%) = (50 - X)\%$.

Now, we perform the calculations step by step:

1. Convert percentages to decimals:

• 30% becomes 0.30, 20% becomes 0.20, $X\%$ becomes $\frac{X}{100}$, and $(50 - X)\%$ becomes $\frac{50-X}{100}$.

2. Calculate the weighted value of each grade:

• Grade 75: $75 \times 0.30 = 22.5$.

• Grade 68: $68 \times 0.20 = 13.6$.

• Grade 94: $94 \times \frac{X}{100} = 0.94X$.

• Grade 53: $53 \times \frac{50-X}{100} = 0.53(50 - X) = 26.5 - 0.53X$.

3. Sum these weighted values to get the overall weighted average:

$\text{Weighted Average} = 22.5 + 13.6 + 0.94X + 26.5 - 0.53X$

This simplifies to:

$62.6 + 0.41X$

Thus, the class average can be expressed as $62.6 + 0.41X$.

To solve this problem, we'll apply a weighted average approach to determine the cost of hydrogen in the mixture:

• Step 1: Calculate the contribution of helium per 100 grams. $\text{Cost of helium per 100 grams} = 3\% \times 7 = 0.21$ dollars.
• Step 2: Calculate the contribution of oxygen per 100 grams. $\text{Cost of oxygen per 100 grams} = 10\% \times 11 = 1.1$ dollars.
• Step 3: Write the equation for the total cost per 100 grams. $X = 0.21 + \text{Cost of hydrogen per 100 grams} + 1.1$
• Step 4: Simplify the equation to solve for the cost of hydrogen per 100 grams. $\text{Cost of hydrogen per 100 grams} = X - (0.21 + 1.1)$ $\text{Cost of hydrogen per 100 grams} = X - 1.31$
• Step 5: Express the cost of hydrogen based on its percentage in the mixture. Since hydrogen makes up 87%, $\frac{87}{100} \times \text{Cost of hydrogen per 100 grams} = X - 1.31$
• Step 6: Solve for the cost of hydrogen. $\text{Cost of hydrogen} = \frac{100}{87} \times (X - 1.31) \approx 1.15X - 1.51$

Therefore, the price of hydrogen is $1.15X - 1.51$ dollars per 100 grams.