Question one
Many shops have many prices set just under a round figure, e.g. £9.99 instead of £10.00 or £99.95 instead of £100.00. It is assumed that this is done because the price seems lower to the consumer. But this is not the reason the practice started. What was the original reason for this pricing method?
Question two
How many integers (whole numbers) greater than or equal to zero and less than 1000 are not divisible by 2 or 5? What is the average value of these integers?
Question three
How many integers greater than or equal to zero and less than 9261 are not divisible by 3 or 7? What is the average value of these integers?
The practice originated to ensure that the clerk had to open the till and give change for each transaction, thus recording the sale and preventing him from pocketing the bank notes.
Question two
Only integers ending in 1, 3, 7, or 9 are not divisible by 2 or 5. This is 4/10 of the possible integers, so the total number of such integers is 4/10 of 1,000, i.e. 400.
The integers can be added in pairs:
sum = (1 + 999) + (3 + 997) + · · · + (499 + 501) .
There are 200 such pairs, so the sum is 1,000 × 200 and the average is 500. A simpler argument would be to say that this is obvious by symmetry: there is nothing in the problem that favours an answer greater (or smaller) than 500.
Alternatively, we can find the number of integers divisible by both 2 and 5 by adding the number divisible by 2 (i.e. 500) to the number divisible by 5 (i.e. 200), and subtracting the number divisible by both 2 and 5 (i.e. 100) since these have been counted twice. To find the sum, we can sum those divisible by 2 (using the formula for a geometric progression), add the sum of those divisible by 5 and subtract the sum of those divisible by 10.
Question three
For the second part, either of these methods will work. In the first method you consider integers in blocks of 21 (essentially arithmetic to base 21): there are 12 integers in each such block that are not divisible by 3 or 7 (namely 1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 20) so the total number is 9261×12/21 = 5292 . The average is 9261/2 as can be seen using the pairing argument (1 + 9260) + (2 + 9259) +· · · or the symmetry argument.
Did you find the solution difficult or unusual?
This kind of result is quite normal for practical problem solvers who use maths to help businesses. They do not always have the problem statement clearly defined. To start to develop a solution to a real-world business problem they have to make certain assumptions. These assumptions are then updated as the results are analysed.
With thanks to the Tesco commercial science team.
