A tour guide offers two types of guided tours: a walking tour and a trolley tour. On Satur… — SAT

A tour guide offers two types of guided tours: a walking tour and a trolley tour. On Saturday, the guide led $3$ walking tours and $2$ trolley tours, which lasted a total of $8$ hours. On Sunday, the guide led $2$ walking tours and $4$ trolley tours, which lasted a total of $12$ hours. If each walking tour has a duration of $w$ hours and each trolley tour has a duration of $t$ hours, what is the total duration, in hours, of $1$ walking tour and $1$ trolley tour?

A
2.0
3.5Answer
C
4.0
D
4.25

Answer

The correct answer is 3.5.

The correct answer is

3.5

. The scenario can be represented by the system of linear equations

3w + 2t = 8

and

2w + 4t = 12

. Multiplying the first equation by

2

gives

6w + 4t = 16

. Subtracting the second equation from this yields

4w = 4

, which simplifies to

w = 1

. Substituting

w = 1

into the first equation yields

3(1) + 2t = 8

, which simplifies to

2t = 5

t = 2.5

. The sum of the durations of one walking tour and one trolley tour is

w + t = 1 + 2.5 = 3.5

hours.

Step-by-Step Solution

Represent the given scenarios with a system of linear equations using the variables

w

(duration of a walking tour) and

t

(duration of a trolley tour).

The system of equations is:
1)

3w + 2t = 8

2w + 4t = 12

This translates the word problem into mathematical equations.

Solve the system of equations for

w

and

t

. For example, multiply the first equation by

2

to align the

t

coefficients, and then subtract the second equation from it.

Multiplying the first equation by

2

gives

6w + 4t = 16

.
Subtracting the second equation (

2w + 4t = 12

) from this yields

4w = 4

, which simplifies to

w = 1

This eliminates the variable

t

to find the value of

w

Substitute

w = 1

back into one of the original equations to solve for

t

Using the first equation:

3(1) + 2t = 8 \implies 3 + 2t = 8 \implies 2t = 5 \implies t = 2.5

This determines the value of the second variable,

t

Calculate the sum of

w

and

t

to find the total duration of

1

walking tour and

1

trolley tour.

w + t = 1 + 2.5 = 3.5

This directly answers the question asked in the problem stem.

Key Concept

Solving a system of linear equations to find a linear combination of variables

Estimated Time:1m 30s

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Key Concept

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