## 2500.63 – Kumquat House Numbers

A new housing development, Mango Manors, is being built just outside of town. New homeowners are buying house numbers. It costs $0.05 to buy a single digit, but two digits can be purchased for $0.09. If each homeowner on Kumquat Street—a street with only single- and two-digit house numbers—buys the numbers for their own house, the total amount paid is $2.15. However, if one person buys all the numbers, the total amount paid is $2.12. How many single-digit house numbers are there on Kumquat Street?

- 3
- 4
- 5
- 6
- 7

## Solution

Suppose one person buys all the house numbers for $2.12. They buy pairs at $0.09 each, meaning that, if $n$ is the number of pairs bought, then:

$\text{\textdollar}0.09n=\text{\textdollar}2.12$

$n=\frac{\text{\textdollar}2.12}{\text{\textdollar}0.09}=23.555.$

This, however, cannot work, as $n$ would have to be a whole number. This means that there must be one single number that does not belong to a pair. If we subtract this number and the $0.05 being spent on it from the equation, we’re left with $2.07. Again, if the one person buys $n$ pairs, then:

$\text{\textdollar}0.09n=\text{\textdollar}2.07$

$n=\frac{\text{\textdollar}2.07}{\text{\textdollar}0.09}=23$

This tells us that, if twenty-three pairs plus one single number are being bought, there are forty-seven digits in total on Kumquat Street.

However, all homeowners are buying their own numbers for a total cost of $2.15. If we set aside the one single digit from above, we’re left with forty-six numbers being bought for $2.10. Notice, now, that two digits bought as a pair cost $0.09, but, if they’re bought separately, they will cost $0.10—a difference of a penny. If the forty-six remaining numbers are all bought as pairs, the cost of those forty-six would be $2.07, which is $0.03 less than the $2.10 that is actually being spent.

Because of this $0.03 difference, we know that there are three pairs of digits that are not being bought as pairs, but as six individual digits. If we add the other single from before, we see that there are seven digits being bought individually, which also means that there are twenty pairs. To check our answer:

$0.05×7 singles=$0.35

$0.09×20 pairs=$1.80

$0.35+$1.80=$2.15

Once again, there are seven single-digit house numbers, meaning that the correct answer is e.