Brain Teasers
The Tourist Group
A tourist group of 86 people broke up into two groups. The smaller group went to the beach, and the bigger group went to see a water mill. The following statements are all true. How many people were in each group?
a) A quarter of the people in one of the groups is more than the number of people in the other group.
b) One of the groups contains 6 times as many women as men.
c) One of the groups contains only married couples.
d) The number of people in one of the groups is divisible by 4.
e) One of the groups contains 8 times as many people who are younger than 50 as those who are at least 50 years old.
a) A quarter of the people in one of the groups is more than the number of people in the other group.
b) One of the groups contains 6 times as many women as men.
c) One of the groups contains only married couples.
d) The number of people in one of the groups is divisible by 4.
e) One of the groups contains 8 times as many people who are younger than 50 as those who are at least 50 years old.
Answer
The first thing is to make this question an equation. The equation is x+y=86. x is the larger group and y is the smaller group. So, y=86-x. So, x+86-x=86. If the first statement is true then ¼(x)>y. Since y=86-x, then ¼(x)>86-x. Now find out the greatest possible number that will satisfy the equation ¼(x)>86-x. The largest number y could be is 17. The smallest number x could be is 69.If c) is true, then one of the groups must be even since "couples" are the multiples of 2 (1 couple consists of 2 people, 2 couples consists of 4 people). Then, you figure out that 1 group affects the other group. If there is 1 more person in group x, then there is one less person in group y. So, that means that both group must be even or else both groups are odd and no group will satisfy c). So x must consist of 70, 72, 74, 76, 78, 80, 82, 84, or 86 people and y must consist of 16, 14, 12, 10, 8, 6, 4, 2, or 0 people. Now try to make the groups satisfy e). Substitute c for young people and o for old people. The equation is 8o=c. So, o+c equals the number of people in one of the groups. If c=8o, then that will mean o+8o equals the number of people in one of the groups. Then, simplified the equation.
o+8o=9o.
So 9o equal the number of people in one of the groups. So one of the groups has to be a multiple of 9. Since y consists of a group that has 16 people or under, then the only multiple of 9 is 9. But the groups have to consist of an even number of people, so group x must satisfy e). The only multiples of 9 above 70 and under 86 are 72 and 81. But, the number has to be even. So, group x consists of 72 people and y consists of 14 people.
So, the answer is that there are 72 people that went to see the water mill and 14 people went to the beach.
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