A Boring Math Lecture Just Got Awesome!

Logic Level 3

Andrew and Nihar were sitting in a math lecture, where the teacher seemed to be boring. She told every student in the lecture to make 5 two-digit numbers using all of the digits from $0,1,2,3,4,5,6,7,8,9$ and then sum them. As an explicit example, if a student makes these numbers: $10,23,45,67,89$ then he sums them to get 234. On taking a survey in the lecture, the teacher declared that Andrew had found the maximum possible sum and Nihar had found the minimum possible sum.

Now your job is to find, what is the ratio of the sums acquired by Andrew and Nihar? Give your answer to 2 decimal places.

The answer is 2.00.

1 solution

Nihar Mahajan
Apr 4, 2016

Let the two-digit numbers made be $\overline{AB},\overline{CD},\overline{EF},\overline{GH},\overline{IJ}$ so our job is to minimize and maximize the following expression:

$10(A+C+E+G+I)+(B+D+F+H+I)$

We should give more weightage to $(A+C+E+G+I)$ since it is being multiplied by $10$ so as to yield maximum possible number. Hence, clearly $(A,C,E,G,I)=(9,8,7,6,5)$ and the remaining digits would be $(B,D,F,H,I)=4,3,2,1,0$ . Thus Andrew's sum $=10(9+8+7+6+5)+4+3+2+1+0=\boxed{360}$ .

Let's find Nihar's sum now. We should give less weightage to $(A+C+E+G+I)$ since it is being multiplied by $10$ so as to yield minimum possible number. Hence, we have $(A,C,E,G,I)=(5,4,3,2,1)$ (since first digit cannot be 0) and the remaining digits would be $(B,D,F,H,I)=9,8,7,6,0$ . Thus Nihar's sum $=10(4+3+2+1+5)+9+8+7+6+0=\boxed{180}$ .

Thus, ratio of their sums is $\dfrac{360}{180}=\boxed{\boxed{2}}$ .

Why can't the number have the first digit of zero? I don't see it explicitly written into the problem.

Tom Barber - 5 years, 1 month ago

If the first digit becomes 0 it won't no longer remain a "two-digit number".

Nihar Mahajan - 5 years, 1 month ago

The question should have explained if the answer should be in the form of Andrew to Nihar or Nihar to Andrew. Or else the answer can be 0.5

Phi Li - 5 years ago

Yes, I answered 0.50 first.. I was first wondering if 05 would be a valid two-digit number when I got it 'wrong'

Sebastiaan Joosten - 1 year, 8 months ago

A Boring Math Lecture Just Got Awesome!

The answer is 2.00.

1 solution

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