Calculating Weighted Average: Worded problems

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

• Step 1: Determine the number of students with no siblings.
• Step 2: Calculate the total number of siblings.
• Step 3: Use the weighted average formula to find the average number of siblings per student.

Step 1: Determine the number of students with no siblings.

The problem provides us with the following information:
- 20 students have 2 siblings.
- 15 students have 1 sibling.
- Total number of students = 45.

The number of students with no siblings can be calculated as:

Number of students with no siblings = Total number of students - Students with 2 siblings - Students with 1 sibling.

$45 - 20 - 15 = 10$

Step 2: Calculate the total number of siblings.

The total number of siblings is calculated by summing the product of the number of siblings and the number of students in each category:

Total number of siblings = (2 siblings × 20 students) + (1 sibling × 15 students) + (0 siblings × 10 students)

$(2 \times 20) + (1 \times 15) + (0 \times 10) = 40 + 15 + 0 = 55$

Step 3: Use the weighted average formula.

The weighted average for the number of siblings per student is given by:

Average number of siblings = $\frac{\text{Total number of siblings}}{\text{Total number of students}}$

$\frac{55}{45} = \frac{11}{9} \approx 1.22$

Therefore, the students in the class have, on average, $1.22$ siblings.

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

• Identify the contribution of the aluminum component to the overall cost.

• Identify the contribution of the iron component to the overall cost.

• Combine the contributions to find the total cost of the alloy.

Let's perform the calculations:

1. Aluminum's Contribution:
The alloy is 30% aluminum, which costs $5 per 100 grams. Hence, the contribution from aluminum is:
$0.30 \times 5 = 1.5 \text{ dollars}$

2. Iron's Contribution:
The alloy is 70% iron, which costs $17 per 100 grams. Hence, the contribution from iron is:
$0.70 \times 17 = 11.9 \text{ dollars}$

3. Total Cost of the Alloy:
To obtain the total cost of 100 grams of the alloy, we add the contributions from both components:
$1.5 + 11.9 = 13.4 \text{ dollars}$

Therefore, the price of 100 grams of the alloy is $13.4 \text{ dollars}$.

To solve this problem, we'll calculate the average number of defective items per day using weighted averages:

• Step 1: Compute the total number of defective items.
  • 3 days × 4 defective items/day = 12 defective items
  • 2 days × 7 defective items/day = 14 defective items
  • 1 day × 0 defective items/day = 0 defective items
  • 10 days × 18 defective items/day = 180 defective items
• Step 2: Sum all defective items calculated: $12 + 14 + 0 + 180 = 206 \text{ defective items}$
• Step 3: Compute the total number of days: $3 + 2 + 1 + 10 = 16 \text{ days}$
• Step 4: Calculate the average number of defective items per day using: $\frac{206}{16} = 12.875$

Therefore, the average number of defective items per day is $12.875$.

To solve this problem, we need to calculate the arithmetic mean of the given dice rolls.

First, let's identify all the given numbers:
The results from the dice rolls are $4, 3, 1, 6, 5, 3, 1$.

We will follow these steps to find the average:

• Step 1: Sum all the numbers: $4 + 3 + 1 + 6 + 5 + 3 + 1$.
• Step 2: Count the total number of rolls: There are 7 numbers.
• Step 3: Divide the sum obtained in Step 1 by the count from Step 2 to determine the average.

Let's execute these steps:

Step 1: Calculate the sum of the numbers:
$4 + 3 + 1 + 6 + 5 + 3 + 1 = 23$.

Step 2: Count the number of rolls: There are $7$ rolls.

Step 3: Calculate the average:
$\text{Average} = \frac{23}{7} = 3.2857\ldots$

The average can be rounded to two decimal places, giving us approximately $3.29$.

Thus, the average number resulting from the dice rolls is $\mathbf{3.29}$.

To solve this problem, we'll proceed with the following steps:

• Step 1: Define the weights and scores for each exam.
• Step 2: Formulate the equation for the weighted average.
• Step 3: Solve the equation for $x$, the unknown score on the first exam.

Now, let's work through each step:

Step 1: We know:

• Weight of the first exam: $0.55$
• Weight of the second exam: $0.45$
• Score on the second exam: $76$
• Final average score: $84$
• Unknown score (first exam): Let it be $x$

Step 2: We use the weighted average formula:

Step 3: Now solve for $x$.

First, calculate the contribution of the second exam:

Substitute this back into the equation:

Subtract 34.2 from both sides to solve for $0.55 \times x$:

Finally, divide both sides by 0.55 to isolate $x$:

Calculate the result:

Therefore, the solution to the problem is $\mathbf{90.55}$.

Let's solve the problem step-by-step:

• First, summarize the data:

  • 2 buildings with 11 floors

  • 4 buildings with 2 floors

  • 5 buildings with 3 floors

  • Let $x$ be the number of buildings with 5 floors.

• Calculate the total number of buildings:

• $\text{Total buildings} = 2 + 4 + 5 + x = 11 + x$

• Compute the weighted sum of floors:

• $\begin{aligned} \text{Sum of floors} &= (2 \times 11) + (4 \times 2) + (5 \times 3) + (5 \times x) \\ &= 22 + 8 + 15 + 5x \\ &= 45 + 5x \end{aligned}$

• Set up the weighted average equation:

• $4.29 = \frac{45 + 5x}{11 + x}$

• Multiply both sides by $11 + x$ to eliminate the fraction:

• $4.29(11 + x) = 45 + 5x$

• Expand and solve the equation:

• $\begin{aligned} 4.29 \times 11 + 4.29x &= 45 + 5x \\ 47.19 + 4.29x &= 45 + 5x \\ 47.19 - 45 &= 5x - 4.29x \\ 2.19 &= 0.71x \end{aligned}$

• Solve for $x$:

• $x = \frac{2.19}{0.71} = 3$

Therefore, the number of buildings with 5 floors is $3$.

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

• Step 1: Set up the equation using the weighted average formula.
• Step 2: Solve the equation to find the number of 3 km runs.

Now, let's work through each step:
Step 1: The average distance is given by $5.92 = \frac{\text{Total Distance}}{\text{Total Runs}}$.
Let $x$ be the number of times Andrea runs 3 km. The total number of runs is $7 + x$, because she runs 8 km 7 times.
The total distance she runs is $8 \times 7 + 3 \times x = 56 + 3x$.
Using the weighted average formula, we have:
$\frac{56 + 3x}{7 + x} = 5.92$

Step 2: Solve the equation for $x$:
Multiply both sides by $7 + x$ to clear the fraction:
$56 + 3x = 5.92(7 + x)$
Expand the right side:
$56 + 3x = 41.44 + 5.92x$
Rearranging gives:
$56 - 41.44 = 5.92x - 3x$
$14.56 = 2.92x$
Divide both sides by 2.92 to solve for $x$:
$x \approx \frac{14.56}{2.92} \approx 5$

Therefore, Andrea runs a distance of $3$ km $5$ times.

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

• Step 1: Set up the equation using the weighted average formula
• Step 2: Solve the equation for $x$
• Step 3: Calculate the percentage of blue and red paint

Now, let's work through each step:

Step 1: The equation for the weighted average is:
$C_p = x \cdot C_b + (1-x) \cdot C_r$

Substituting the given values:
$91 = x \cdot 95 + (1-x) \cdot 70$

Step 2: Simplify and solve for $x$:
$91 = 95x + 70 - 70x$
Combine terms:
$91 = 25x + 70$
Subtract 70 from both sides:
$21 = 25x$
Divide by 25:
$x = \frac{21}{25} = 0.84$

Step 3: $x = 0.84$ means 84% of the paint is blue:
The percentage of red paint is $1 - x = 0.16$ or 16%.

Therefore, the solution to the problem is blue: 84% red: 16%.

To solve this problem, let's calculate step by step:

• Determine the total number of parks: Since all parks combined plant an average of 24.2 trees each, let's assume there are $n$ parks. This implies a total of $24.2n$ trees.
• Calculate trees in known parks: The first two parks have 19 trees each, thus total $2 \times 19 = 38$ trees. The next three parks have 28 trees each, amounting to $3 \times 28 = 84$ trees. Together, they plant $38 + 84 = 122$ trees.
• Set up the equation: Let $x$ be the number of parks that planted 24 trees. Then the rest of the parks account for $24x$ trees. Our equation becomes:

• Express $n$ in terms of $x$:
• Since $n = 2 + 3 + x$ (two parks with 19 trees, three with 28, and the ones we are solving for), we substitute this into the equation:

• Solve for $x$:

Therefore, $\boxed{5}$ parks planted exactly 24 trees. This confirms with answer choice (3).

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

• Step 1: Identify the number of bags and compute the total number of marbles.
• Step 2: Apply the formula for the average and solve for the unknown.
• Step 3: Solve the equation to determine the number of bags containing 9 marbles.

Let's work through these steps:

Step 1: The problem details that the average number of marbles per bag is 9.64. We need to find the total number of bags. Let $n$ be the number of bags that contain 9 marbles. Calculating the total number of bags is necessary as a starting point.

Given bags and their marbles:
- 1 bag containing 18 marbles
- 2 bags containing 12 marbles each: Total $= 2 \times 12 = 24$ marbles
- 3 bags containing 7 marbles each: Total $= 3 \times 7 = 21$ marbles

Total marbles in known bags = $18 + 24 + 21 = 63$ marbles

Step 2: Use the total average calculation to find the number of bags:

Total marbles = $63 + 9n$

The number of total bags is $1 + 2 + 3 + n = 6 + n$.

Thus, by the average formula:

Step 3: Solve for $n$.

Subtract $57.84$ from both sides:

Divide by 0.64 to solve for $n$:

This result implies rounding is needed, thus $n = 8$ bags.

Therefore, the solution to the problem is $8$ bags.

To solve for the number of dumbbells weighing 7.1 kg, we will leverage the weighted average given by the problem:

• Calculate weights of known dumbbells:
  • Total weight of 3 dumbbells at 4.5 kg each: $3 \times 4.5 = 13.5$ kg
  • Total weight of 4 dumbbells at 5.2 kg each: $4 \times 5.2 = 20.8$ kg
• Let $x$ be the number of dumbbells weighing 7.1 kg. Therefore, the remaining $10 - x$ dumbbells weigh 3.8 kg each.
• Calculate total weight: $13.5 + 20.8 + 7.1x + 3.8(10 - x) = 5.22 \times 17$ Simplifying the right, we have: $13.5 + 20.8 + 7.1x + 38 - 3.8x = 88.74$ $7.1x - 3.8x + 13.5 + 20.8 + 38 = 88.74$ $3.3x + 72.3 = 88.74$ $3.3x = 88.74 - 72.3$ $3.3x = 16.44$ $x = \frac{16.44}{3.3} = 5$

Therefore, the number of dumbbells weighing 7.1 kg is $5$.

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'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.