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Vacation Time

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1. It was vacation time, and so I decided to visit my cousin's home. What a grand time we had! In the mornings, we both would go for a jog. The evenings were spent on the tennis court. Tiring as these activities were, we could manage only one per day, i.e., either we went for a jog or played tennis each day. There were days when we felt lazy and stayed home all day long.

Now, there were 12 mornings when we did nothing, 14 evenings when we stayed at home, and a total of 10 days when we jogged or played tennis. For how many days did I stay at my cousin's place?
Answer: 18

Solution:

Let
  • x denote the number of days we jogged in the morning and stayed at home in the evening;
  • y denote the number of days we played tennis in the evening and did nothing in the morning; and
  • z denote the number of days we neither jogged nor played tennis.

Then,

y + z = number of mornings we did nothing = 12
x + z = number of evenings we stayed at home = 14
x + y = number of days we jogged or played tennis = 10

Adding the above three equations and dividing both sides by 2 gives

x + y + z = 18

Since there are only three types of days, the total number of days I stayed with my cousin is their sum, i.e., 18.

Food for thought:

Can you solve this puzzle using a Venn diagram with two intersecting sets? Could the two sets be "days we did nothing in the morning" and "days we stayed at home in the evening"?


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