# Kumquat House Numbers

A new housing development, Mango Manors, is being built outside of town. It costs five cents to buy a single digit house number, but two digits can be bought for nine cents. If each home owner on Kumquat Street, a street with only single digit and 2-digit house numbers, buys the numbers for his own house, the total amount paid is \$2.15. If one person buys them all, then the total amount paid is \$2.12. How many single digit house numbers are on Kumquat Street?

1. 3
2. 4
3. 5
4. 6
5. 7

If one person buys all the digits for \$2.12, suppose s/he buys n pairs at 9¢ each. Then 9n = 212 → n = 212/9 = 23.555…. This is no good, because n has to be a whole number. So there must be one single digit at 5¢ out of the \$2.12. So now s/he buys n pairs for \$2.07. Then 9n = 207 → n = 23 pairs, or 46 digits, plus the one additional single digit.

But if all homeowners buy their own numbers, the total cost is \$2.15. If we set aside the one single digit from above, we have all the rest (46) costing \$2.10. Now, we note that two dgits singly cost 10¢, whereas bought as a pair they cost 9¢, 1¢ cheaper. The \$2.07 paid for 23 pairs of digits is 3¢ less than \$2.10, so we see that there must be 3 pairs of single digits, or six singles bought as pairs. So, these six plus that other one make 7 single digits.

Check: There are 47 digits total: 7 singles and 20 pairs.

7 x 5 + 20 x 9 = 215 and 1 x 5 + 23 x 9 = 212.