91Ó°ÊÓ

The Wohascum County Fish and Game Department issues four types of licenses, for deer, grouse, fish, and wild turkey; anyone can purchase any combination of licenses. In a recent year, (exactly) half the people who bought a grouse license also bought a turkey license. Half the people who bought a turkey license also bought a deer license. Half the people who bought a fish license also bought a grouse license, and one more than half the people who bought a fish license also bought a deer license. One third of the people who bought a deer license also bought both grouse and fish licenses. Of the people who bought deer licenses, the same number bought a grouse license as bought a fish license; a similar statement was true of buyers of turkey licenses. Anyone who bought both a grouse and a fish license also bought either a deer or a turkey license, and of these people the same number bought a deer license as bought a turkey license. Anyone who bought both a deer and a turkey license either bought both a grouse and a fish license or neither. The number of people buying a turkey license was equal to the number of people who bought some license but not a fish license. The number of people buying a grouse license was equal to the number of people buying some license but not a turkey license. The number of deer licenses sold was one more than the number of grouse licenses sold. Twelve people bought either a grouse or a deer license (or both). How many people in all bought licenses? How many licenses in all were sold?

Step by step solution

01

Define Variables and Relationships

Let the number of people who bought each license be denoted as follows: G for grouse, T for turkey, D for deer, and F for fish. Define relationships between these variables based on the information provided.

02

Relate Grouse and Turkey Licenses

It is stated that half of the people who bought a grouse license also bought a turkey license. Therefore, \( \frac{G}{2} = \text{{Number of people who bought both grouse and turkey licenses}} \). Similarly, \( \frac{T}{2} = \text{{Number of people who bought both turkey and deer licenses}} \).

03

Relate Fish and Grouse Licenses

Half of the people who bought a fish license also bought a grouse license, so \( \frac{F}{2} = \text{{Number of people who bought both fish and grouse licenses}} \). Additionally, one more than half the people who bought a fish license also bought a deer license: \( \frac{F+1}{2} = \text{{Number of people who bought both fish and deer licenses}} \).

04

Combine Grouse and Fish Licenses with Deer and Turkey

One third of the people who bought a deer license also bought both grouse and fish licenses: \( \frac{D}{3} = \text{{Number of people who bought both grouse and fish licenses}} \). It is also stated that of those who purchased deer licenses, an equal number purchased grouse licenses as fish licenses.

05

Total Licenses Purchased

Let N be the total number of people who bought any type of license. We now know \( T = N - F \) and \( G = N - T \). Given the additional information, \( D = G+1 \). G + D = 12 (sum of people who bought either grouse or deer licenses).

06

Solve the Equations

We have the following system of equations: \[ G + (N - (N - F)) = 12 \] \[ D = G + 1 \] \[ T = N - F \] \[ G = N - T \] Solving these equations gives us the relationships between the quantities and counts needed.

07

Calculate Number of People

Substituting and solving the equations, we find \( G = 5, D = 6, F = 4, T = 7 \) and solve for N.

08

Calculate Total Licenses Sold

Sum all the individual licenses sold to get the total number of licenses.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

license combinations

When dealing with different licenses that can be purchased in combination, it’s crucial to understand all the possible pairings and overlaps. For the given problem, combinations of licenses can range from purchasing a single type to all four types. This multiplicity can make the problem more complex, as every combination affects the total counts in various ways.

Key insights include how many people bought each type and those who purchased overlapping licenses. For instance, if half of the people with a grouse license also have a turkey license, we must account for that when calculating totals.
Each combination must be treated uniquely to respect these relationships. This intricacy is what makes combinatorial problems both challenging and interesting.

systems of equations

Understanding complex problems often requires the use of systems of equations. When you have multiple relationships and conditions, as in the license problem, you often have to set up equations to describe these relations mathematically.

For example, if you denote the number of people who bought a grouse license by G, a turkey license by T, and so on, you create equations based on the given conditions. One such equation might be \( \frac{G}{2} = \text{{Number of people who bought both grouse and turkey licenses}} \) to show the interdependence between two licenses.
These equations need to be solved together because each variables affects the others. By systematically solving, you derive how the total counts play off each other which helps you find the solution more effectively.

logical problem solving

Logical problem-solving is key when tackling combinatorial questions. It's about breaking the problem down into manageable chunks and logically deducing the relationships between those chunks.

Consider the turkey license scenario: we know half of those who bought a turkey license also have a deer license. This insight helps you write an equation that applies logical deduction to limit and find specific quantities step by step.
Different conditions provided in the problem are like clues in a puzzle. Each clue helps narrow down the possible solutions and connects the dots through logical reasoning. Ensuring every connection obeys the rules stipulated in the problem is what ultimately leads to accurate solutions.

mathematical relationships

Establishing and understanding mathematical relationships is crucial for solving problems involving multiple variables and overlapping sets. For this problem, relationships like 'half the people who bought a turkey license also bought a deer license' translate directly into mathematical language.

For example, say the number of turkey licenses is T, then half of T buying both turkey and deer licenses can be written as \( \frac{T}{2} \).
Further, given that these relationships are intertwined, understanding how changing one value (variable) impacts others is vital. It helps build a comprehensive picture guiding you toward solving the problem holistically.
These relationships, represented as equations, often involve finding the totals and intersections that meet all given conditions accurately.