Geometric prices

Number Theory Level 4

Bob will buy a book if and only if:

The price of the book is an integer number of cents, and ranges from $\$1.00$ to $\$9.99$ , inclusive.
The three digits in the price of the book in order form a geometric progression with a positive common ratio.

If Bob goes to a store with a single book of every possible monetary value, how many books will he buy?

Details and assumptions

For example, Bob will buy a book with price $\$1.24$ , but not one with price $\$3.00$ .

The answer is 17.

3 solutions

Finn Hulse
Jan 28, 2014

This question is simply asking "how many geometric progressions do not exceed ten?". This is simply a counting problem. There are 9 obvious ones: 1.11, 2.22, 3.33, etc.. The only slightly tricky part is when you start counting the others. You kind of just have to count them, one by one. They are:

124 241 139 931 248 842 469 964

And we're done! Now for the EXTREMELY difficult LEVEL 5 part. We are going to attempt the impossible... We are going to add 8 to 9. This is going to take some careful calculation. You may want to cover your eyes... Now, after hours of laborious calculations we have found the answer! It's... 15? No... 27? No... The answer, ladies and gentleman, is... 17!!!!!!!!!!!!!

Sure wished you would have left out the word order. It made me think you wanted the digits sorted. Then there are more than 17.

bobbym none - 7 years, 3 months ago

Vishal Jindal
Dec 16, 2013

values are 111,222,...999 ( nine values), 124,139,248,469,421,931,842,964 (eight values), total seventeen values.

Yes, why is this problem 2339...

Bob Bob - 7 years, 5 months ago

Ashish Pathak
Dec 16, 2013

ofcourse all the numbers like 111 ,222, 333,...... 999 contributes in buying book. other then this which is quite easy to notice are 124 ,421 ,139,931,248,842.. now its given that ratio is positive ,that is ratio can be positive fraction as well.. let the 3 digit number be xyz. then to satisfy geometric condition y^2=xz and for 6^2=36=9*4 thus 469 and 964 also satisfies the condition
hence the total number books which can be bought are 9+6+2=17

It needs to be clarified that the digits form a geometric sequence $\textit{in order}$ so prices like $6.94 or $4.28 would not be bought.

Trevor B. - 7 years, 5 months ago

i forgot to mention that.. as i felt that 3 digits being a part of geometric progression its quite obvious that sequence will be either increasing or decreasing.

Ashish Pathak - 7 years, 5 months ago

With the new editing feature I added that clarification!

Daniel Chiu - 7 years, 5 months ago

Geometric prices

The answer is 17.

3 solutions

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