A Math Riddle: But the math does not add up

Grandpa had a lot of different books of stamps on his desk. He was counting them.

I was curious. “What are all those stamps Grandpa?”

“I just got back from the Post Office son and bought a bunch of different stamps! All books.”

“Yea? What did you get?” Not sure why I asked.

“Well, let’s see. Two books of International stamps. One book of 75 cent stamps. And I got – 20 cent stamps, 50 cent stamps and also 55 cent stamps. That is it!”

“Lot of books, Grandpa.”

“Yes, son. I spent 50 dollars in total. Let’s see. Wow. I have a total of 13 books of various stamps. So do your math and tell me son, how many books from each stamp category do I have? Here is the pricing. I only bought stamp books; no individual stamps”

I did the math. It did not add up.

“Are you sure you counted the books right, Grandpa?”

“Yes, absolutely!”

“No other stamps than what you told me?”

“Nope. Only full books, 10 stamps in each book.”

Hmmm. Why was he right?

Postage Pricing
1 Book = 10 Stamps

2 cent stamp 20 cents per book
5 cent stamp 50 cents per book
10 cent stamp \$1.00 per book 20 cent stamp \$2.00 per book
50 cent stamp \$5.00 per book 55 cent stamp \$5.50 per book
75 cent stamp \$7.50 per book International stamp \$12.00 per book

To solve this, we need to examine more closely Grandpa's words when he says:

Two books of International stamps. One book of 75 cent stamps. And I got – 20 cent stamps, 50 cent stamps and also 55 cent stamps. That is it!

Knowing that Grandpa spent exactly \$50.00 and bought 13 books, what can we say for sure? We know the exact quantities of international stamps and 75-cent stamps that Grandpa book - these cost \$31.50 for 3 books (2 lots at \$12.00 and 1 lot at \$7.50).

This leaves us with needing to account for the remaining \$18.50 from 10 more books of stamps. If we take his words at face value then we run into the problem our narrator had: If Grandpa definitely bought books of 20-cent stamps (price \$2.00 each), 50-cent stamps (price \$5.00 each) and 55-cent stamps (price \$5.50 each), there is no way that some combination of 10 books of these stamps can add up to only \$18.50 - at the very least, we would be looking at paying \$26.50, for 8 books of the cheapest and 1 each of the others.

However, if our narrator misheard Grandpa, who actually said:

And I got – 20 cent stamps, 50 cent stamps and also 50 5-cent stamps... (instead of '55 cent stamps')

then the maths can work out!

Fifty 5-cent stamps works out as 5 books of 5-cent stamps, costing a total of \$2.50 (50 cents a book). This leaves \$16.00 to be accounted for, which can easily be divided into 3 books of the 20-cent stamps (\$2.00 each) and 2 books of the 50-cent stamps (\$5.00 each).

So Grandpa actually bought:

2x books of International stamps (\$12.00 x2) 1x book of 75-cent stamps (\$7.50)
3x books of 20-cent stamps (\$2.00 x3) 2x books of 50-cent stamps (\$5.00 x2)
5x books of 5-cent stamps (\$0.50 x5) TOTAL: 13 books of stamps, costing \$50.00

• Wow. @Stiv. Are you sure you don't know my Grandpa?
– DrD
Feb 22, 2021 at 13:37
• @DrD Not personally - I was just behind him in the Post Office queue :)
– Stiv
Feb 22, 2021 at 13:47
• LOLOLOL Now we know.
– DrD
Feb 22, 2021 at 13:48