Algebra Word Problem #4

Yellowstone National Park is a popular field trip destination. This year the senior class at High School A and the senior class at High School B both planned trips there. The senior class at High School A rented and filled 9 vans and 4 buses with 256 students. High School B rented and filled 8 vans and 8 buses with 392 students. Every van had the same number of students in it as did the buses. Find the number of students in each van and in each bus.

v = van

b = bus

$\begin{array}{lcl} A && B \\ \hline 9\text{ vans} && 8\text{ vans} \\ 4\text{ buses} && 8\text{ buses} \\ \hline =256 \text{ students} && =392 \text{ students} \end{array}$

$\begin{array}{lrclclrcl} (1)& 9v+4b &=& 256 \quad |\quad:4 && (2) & 8v+8b &=& 392 \quad |\quad : 8 \\ &2.25v+b&=&64 && & v+b &=& 49 \\ \hline \\ (1)-(2) & 2.25v+b - (v+b) &=& 64-49\\ & 2.25v+b -v-b &=& 15\\ & 1.25v &=& 15 \quad |\quad : 1.25 \\ & v &=& \frac{15}{ 1.25} \\ & \mathbf{v} &\mathbf{=}& \mathbf{12} \\ \hline \\ (2) & v+b &=& 49 \\ & 12+b &=& 49 \quad |\quad -12 \\ & b &=& 49-12\\ & \mathbf{b} &\mathbf{=}& \mathbf{37} \end{array}$

Check:

9*12 + 4*37 = 256   Okay

8*12 + 8*37 = 392   Okay

Yellowstone National Park is a popular field trip destination. This year the senior class at High School A and the senior class at High School B both planned trips there. The senior class at High School A rented and filled 9 vans and 4 buses with 256 students. High School B rented and filled 8 vans and 8 buses with 392 students. Every van had the same number of students in it as did the buses. Find the number of students in each van and in each bus.

Let the number of students in each van=V

Let the number of students in each bus=B, then we have:

9V + 4B=256 This is from School A

8V + 8B=392 This is from school B

So, the of students in each Van=12

And the number of dtudents in each Bus=37

Solve the following system:
{4 B+9 V = 256 |     (equation 1)
8 B+8 V = 392 |     (equation 2)
Swap equation 1 with equation 2:
{8 B+8 V = 392 |     (equation 1)
4 B+9 V = 256 |     (equation 2)
Subtract 1/2 × (equation 1) from equation 2:
{8 B+8 V = 392 |     (equation 1)
0 B+5 V = 60 |     (equation 2)
Divide equation 1 by 8:
{B+V = 49 |     (equation 1)
0 B+5 V = 60 |     (equation 2)
Divide equation 2 by 5:
{B+V = 49 |     (equation 1)
0 B+V = 12 |     (equation 2)
Subtract equation 2 from equation 1:
{B+0 V = 37 |     (equation 1)
0 B+V = 12 |     (equation 2)
Collect results:
Answer: | B = 37       and         V = 12