A lot of the Math Kangaroo problems ask you to find a solution that is either the minimum value or the maximum value of all possible solutions. Let us see what kind of strategies and techniques we can use to solve such problems through an example.

Alice has 10 siblings (including herself). Alice is the eldest. All her siblings were born in different years. The sum of the ages of the 10 siblings is 180, what is the smallest possible age of Alice?

Step One The first thing to do in solving such Min Max word problem is to find out whether the question is asking you to solve for the minimum or maximum value. In this problem, it is asking for the minimum value.

Step Two The 2nd thing to do is to list out all the known conditions and restrictions. From the problem description, we know:

·         There are 10 people

·         These 10 people have different ages (no 2 people will have the same age)

·         The sum of the ages of the 10 people is 180.

Step Three If possible, try to convert the known conditions and restrictions into numeric expressions or equations. In the above problem, we can convert the known conditions into the following equation:

S1+S2+S3+S4+S5+S6+S7+S8+S9+A=180, where S1<S2<S3<S4<S5<S6<S7<S8<S9<A

Step Four Now, you can try to derive more refined conditions based on the known conditions and restrictions and the problem requirement (of finding max or min values). In our problem, in order for A to be smallest, we need to make S1 to S9 as close to A as possible. Since S1 to S9 and A must all be different, we essentially have an arithmetic sequence A, A-1, A-2, A-3, & A-9. From the formula for the sum of arithmetic sequence, we have: 10*(A+A-9)/2 = 180, or A=22.5>22.

Step Five Get the min or max value from refined conditions. In our example above, since A or Alice age must be greater than 22, the only choice left in the given choices is 23, and that is our answer to the question.