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Academic Decathlon 2004 Logic Quiz

Motor club survey (700 points)

On a survey in a motor club, the members were asked whether they had a Mercedes, a Ford or a Jaguar. The following was gleaned from the survey: 35 people said that they had a Mercedes and a jaguar; 47 said they had a Ford and a jaguar; 30 said they had a Ford and a Mercedes; 21 said they had all three cars. The analyst concluded also that they had registered 113 Jaguars, 120 Mercedes and 77 Fords (no one owns more than one of any type).

Plot the numbers on the circles provided to answer the following questions:

A) How many people owned a Ford but nothing else?

B) How many people owned a Jaguar and a Ford but not a Mercedes?

C) How many people owned a Mercedes or a Ford?

D) How many owned a Ford or a Jaguar?

E) How many did not own a Ford?

F) How many owned a jaguar and a Mercedes but not a Ford?

G) How many owned a Mercedes or a Jaguar but not a Ford?

Answer:

First, plot the number of people who have all three cars:

Now, fill in the numbers of people who own two cars. 30 people said they own a Ford and a Mercedes. You might ask yourself if these 30 people are in addition to the 21 who own all three cars or whether they include the 21 all-car-owners. You could try it both ways, and then you'll discover a contradiction in one of them. It turns out that the 30 people who own both a Ford and a Mercedes include the 21 who own all three cars. So 30-21=9 people own just a Ford and a Mercedes.

Similarly, 47-21=26 people own just a Ford and a Jaguar,
and 35-21=14 people own just a Mercedes and a Jaguar.

There are 113 Jaguars, and 26+21+14=61 people own a Jaguar and some other car. So 113-61=52 people own just a Jaguar.

There are 120 Mercedes, and 9+21+14=44 people own a Mercedes and some other car. So 120-44=76 people own just a Mercedes.

There are 77 Fords, and 26+21+9=56 people own a Ford and some other car. So 77-56=21 people own just a Ford.

A) How many people owned a Ford but nothing else?
You can read this right off the Venn diagram: 21.

B) How many people owned a Jaguar and a Ford but not a Mercedes?
Another one right off the diagram: 26.

C) How many people owned a Mercedes or a Ford?
For this one, you could add all six numbers from the diagram, but a quick look at question D tells you there's an easier way. First, add up all the numbers on the diagram to see how many people were surveyed.

21+52+76+14+9+26+21=219 total people surveyed.

Now the people who owned a Mercedes or a Ford are the total minus those who owned just a Jaguar. 219-52=167.

D) How many owned a Ford or a Jaguar?
The total minus the number owning just a Mercedes is 219-76=143.

E) How many did not own a Ford?
The numbers that lie outside the Ford circle are 52+14+76=142.

F) How many owned a Jaguar and a Mercedes but not a Ford?
Read it right off the diagram: 14.

G) How many owned a Mercedes or a Jaguar but not a Ford?
Is this question different from E? I don't think so. 142.

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