Math / Algebra

Question5) The senior classes at High School A and High School B planned separate trips to New York City. The senior class at High School A and High School B planned separes with 814 students. High School B rented School A rented and filled 13 vans and 14 buses willed 14 vans and 12 buses with 732 students. Each van and each bus carried the same num and filled 14 vans and 12 buses with 732 students. Each van and each bus.

Studdy Solution

STEP 1

What is this asking? Two high schools rented vans and buses, and we need to figure out how many students fit in each van and each bus, knowing that each vehicle carries the same number of students at both schools. Watch out! Don't mix up the number of vans and buses between the two schools!

STEP 2

1. Set up equations
2. Solve for the number of students in each van and bus

STEP 3

Let $v$ be the number of students in each van, and $b$ be the number of students in each bus.

STEP 4

High School A had $\textbf{13}$ vans and $\textbf{14}$ buses with a total of $\textbf{814}$ students.
So, our equation is $13v + 14b = 814$ .
This tells us that $\textbf{13}$ times the number of students in a van plus $\textbf{14}$ times the number of students in a bus equals the total number of students from High School A.

STEP 5

High School B had $\textbf{14}$ vans and $\textbf{12}$ buses with a total of $\textbf{732}$ students.
So, our equation is $14v + 12b = 732$ .
This tells us that $\textbf{14}$ times the number of students in a van plus $\textbf{12}$ times the number of students in a bus equals the total number of students from High School B.

STEP 6

We'll use the elimination method.
Let's multiply the first equation (High School A's equation) by $\textbf{12}$ and the second equation (High School B's equation) by $\textbf{14}$ .
We do this so that the coefficients of $b$ will be the same, allowing us to eliminate the $b$ variable.

STEP 7

Multiplying High School A's equation by $\textbf{12}$ gives us $12 \cdot (13v + 14b) = 12 \cdot 814$ , which simplifies to $156v + 168b = 9768$ . Multiplying High School B's equation by $\textbf{14}$ gives us $14 \cdot (14v + 12b) = 14 \cdot 732$ , which simplifies to $196v + 168b = 10248$ .

STEP 8

Now, subtract the modified High School A equation from the modified High School B equation: $(196v + 168b) - (156v + 168b) = 10248 - 9768$ This simplifies to $40v = 480$ .
Notice how the $b$ terms add to zero, leaving us with only the $v$ variable!

STEP 9

Divide both sides of $40v = 480$ by $\textbf{40}$ to find $v$ : $v = \frac{480}{40} = 12$ So, there are $\textbf{12}$ students in each van!

STEP 10

Substitute $v = 12$ back into either of the original equations.
Let's use High School A's equation: $13v + 14b = 814$ . Substituting $v = 12$ gives us $13 \cdot 12 + 14b = 814$ , which simplifies to $156 + 14b = 814$ . Subtract $\textbf{156}$ from both sides: $14b = 814 - 156$ , so $14b = 658$ . Divide both sides by $\textbf{14}$ : $b = \frac{658}{14} = 47$ . So, there are $\textbf{47}$ students in each bus!

STEP 11

There are $\textbf{12}$ students in each van and $\textbf{47}$ students in each bus.

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Studdy Solution

STEP 1

What is this asking? Two high schools rented vans and buses, and we need to figure out how many students fit in each van and each bus, knowing that each vehicle carries the same number of students at both schools. Watch out! Don't mix up the number of vans and buses between the two schools!

STEP 2

1. Set up equations
2. Solve for the number of students in each van and bus

STEP 3

Let $v$ be the number of students in each van, and $b$ be the number of students in each bus.

STEP 4

STEP 5

STEP 6

We'll use the elimination method.
Let's multiply the first equation (High School A's equation) by $\textbf{12}$ and the second equation (High School B's equation) by $\textbf{14}$ .
We do this so that the coefficients of $b$ will be the same, allowing us to eliminate the $b$ variable.

STEP 7

Multiplying High School A's equation by $\textbf{12}$ gives us $12 \cdot (13v + 14b) = 12 \cdot 814$ , which simplifies to $156v + 168b = 9768$ . Multiplying High School B's equation by $\textbf{14}$ gives us $14 \cdot (14v + 12b) = 14 \cdot 732$ , which simplifies to $196v + 168b = 10248$ .

STEP 8

Now, subtract the modified High School A equation from the modified High School B equation: $(196v + 168b) - (156v + 168b) = 10248 - 9768$ This simplifies to $40v = 480$ .
Notice how the $b$ terms add to zero, leaving us with only the $v$ variable!

STEP 9

Divide both sides of $40v = 480$ by $\textbf{40}$ to find $v$ : $v = \frac{480}{40} = 12$ So, there are $\textbf{12}$ students in each van!

STEP 10

STEP 11

There are $\textbf{12}$ students in each van and $\textbf{47}$ students in each bus.

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Studdy Solution

STEP 1

What is this asking? Two high schools rented vans and buses, and we need to figure out how many students fit in each van and each bus, knowing that each vehicle carries the same number of students at both schools. Watch out! Don't mix up the number of vans and buses between the two schools!

STEP 2

1. Set up equations2. Solve for the number of students in each van and bus

STEP 3

Let vvv be the number of students in each **van**, and bbb be the number of students in each **bus**.

STEP 4

STEP 5

STEP 6

We'll use the **elimination method**.Let's multiply the first equation (High School A's equation) by 12\textbf{12}12 and the second equation (High School B's equation) by 14\textbf{14}14.We do this so that the coefficients of bbb will be the same, allowing us to eliminate the bbb variable.

STEP 7

STEP 8

STEP 9

Divide both sides of 40v=48040v = 48040v=480 by 40\textbf{40}40 to find vvv: v=48040=12v = \frac{480}{40} = 12v=40480​=12 So, there are 12\textbf{12}12 students in each **van**!

STEP 10

STEP 11

There are 12\textbf{12}12 students in each van and 47\textbf{47}47 students in each bus.

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1. Set up equations
2. Solve for the number of students in each van and bus

Let $v$ be the number of students in each van, and $b$ be the number of students in each bus.

We'll use the elimination method.
Let's multiply the first equation (High School A's equation) by $\textbf{12}$ and the second equation (High School B's equation) by $\textbf{14}$ .
We do this so that the coefficients of $b$ will be the same, allowing us to eliminate the $b$ variable.

Divide both sides of $40v = 480$ by $\textbf{40}$ to find $v$ : $v = \frac{480}{40} = 12$ So, there are $\textbf{12}$ students in each van!

There are $\textbf{12}$ students in each van and $\textbf{47}$ students in each bus.