jmc

Question

A street has 20 houses on each side, for a total of 40 houses. The addresses on the south side of the street form an arithmetic sequence, as do the addresses on the north side of the street. On the south side, the addresses are 4, 10, 16, etc., and on the north side they are 3, 9, 15, etc. A sign painter paints house numbers on a house for 1$ per digit. If he paints the appropriate house number once on each of these 40 houses, how many dollars does he collect?

Accepted Answer

Using the formulas for arithmetic sequences, we see that the

2 0^{th}

number for the north side is

3 + 6 (20 - 1) = 117

and the

2 0^{th}

number for the south side is

4 + 6 (20 - 1) = 118

. Also, we see that a north side house number is always 3 more than a multiple of 6, and a south side house number is always 4 more than a multiple of 6. We can then distribute the house numbers for the north and south sides into 3 groups each by the number of digits:

North side: {3, 9}, {15, \dots, 99}, {105, 111, 117}

South side: {4}, {10, \dots, 94}, {100, \dots, 118}

The north side has 2 houses with one digit house numbers, and 3 houses with three digit house numbers, so it must have

20 - 2 - 3 = 15

houses with two digit house numbers.

The south side has 1 house with one digit house numbers, and 4 houses with three digits house numbers, so it must have

20 - 1 - 4 = 15

houses with two digit address. Thus, the total cost is

(1 \times 2 + 2 \times 15 + 3 \times 3) + (1 \times 1 + 2 \times 15 + 3 \times 4) = 84

dollars.