Problem 236 // Project Euler



Luxury Hampers

Suppliers ‘A’ and ‘B’ provided the following numbers of products for the luxury hamper market:

Product ‘A’ ‘B’
Beluga Caviar 5248 640
Christmas Cake 1312 1888
Gammon Joint 2624 3776
Vintage Port 5760 3776
Champagne Truffles 3936 5664

Although the suppliers try very hard to ship their goods in perfect condition, there is inevitably some spoilage - i.e. products gone bad.

The suppliers compare their performance using two types of statistic:

  • The five per-product spoilage rates for each supplier are equal to the number of products gone bad divided by the number of products supplied, for each of the five products in turn.
  • The overall spoilage rate for each supplier is equal to the total number of products gone bad divided by the total number of products provided by that supplier.

To their surprise, the suppliers found that each of the five per-product spoilage rates was worse (higher) for ‘B’ than for ‘A’ by the same factor (ratio of spoilage rates), m>1; and yet, paradoxically, the overall spoilage rate was worse for ‘A’ than for ‘B’, also by a factor of m.

There are thirty-five m>1 for which this surprising result could have occurred, the smallest of which is 1476/1475.

What’s the largest possible value of m?
Give your answer as a fraction reduced to its lowest terms, in the form u/v.


豪华礼盒

供应商’A’和’B’为豪华礼盒市场提供以下数量的商品:

商品 ‘A’ ‘B’
鲟鱼子酱 5248 640
圣诞蛋糕 1312 1888
猪肉火腿 2624 3776
典藏波特 5760 3776
香槟松露 3936 5664

尽管供应商十分努力保持商品处于良好状态,但总会出现一些不可避免的损耗,也就是说,商品变质了之类的。

供应商用两种不同的统计量来比较他们的表现:

  • 两家供应商的各商品损耗率是每种商品的变质数量除以每种商品的供应量,对于五种商品各自计算。
  • 两家供应商的总损耗率是所有商品的变质数量除以所有商品的总供应量。

令他们惊讶的是,尽管就五种商品各自的损耗率而言,’B’都要比’A’表现得差(损耗率更高),且均为m>1倍,然而就总损耗率而言,’A’却要比’B’表现得差,且同样为m倍。

一共有35个不同的m>1使得上述情况有可能发生,其中最小的是1476/1475。

最大的m是多少?
你的答案应该写成最简分数,格式为u/v。