Maths Olympiad Prep

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Number theory Difficulty 3.2 AMC 10/12 Find the answer

The Amaco Middle School bookstore sells pencils costing a whole number of cents. Some seventh graders each bought a pencil, paying a total of 1.431.43 dollars. Some of the 3030 sixth graders each bought a pencil, and they paid a total of 1.951.95 dollars. How many more sixth graders than seventh graders bought a pencil?

Pick one

Solution

Because the pencil costs a whole number of cents, the cost must be a factor of both 143143 and 195195. They can be factored into 111311\cdot13 and 35133\cdot5\cdot13. The common factor cannot be 11 or there would have to be more than 3030 sixth graders, so the pencil costs 1313 cents. The difference in costs that the sixth and seventh graders paid is 195143=52195-143=52 cents, which is equal to 52/13=(D) 452/13 = \boxed{\textbf{(D)}\ 4} sixth graders.

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.