# Hot Dogs

How many hot dogs did this hot dog lover eat on the first day of her binge?

A hot dog lover ate 60 hot dogs over an 8-day period.

Every day, she ate one more hot dog than she had eaten the day before.

How many hot dogs did she eat on the first day?

Every day, she ate one more hot dog than she had eaten the day before.

How many hot dogs did she eat on the first day?

*This problem is adapted from the World Mathematics Championships***Answer**: 4

**Using the average**

60 hot dogs in 8 days is an average of 60$\div$8 = 7.5 hot-dogs per day.

7.5 will be in the middle of the numbers

7.5

4 5 6 7 8 9 10 11

4 hot-dogs on the first day

**Sensible trial**

6 hot-dogs on the first day, total 6+7+8+9+10+11+12+13=76 hot-dogs. Too many

5 hot-dogs on the first day, total 5+6+7+8+9+10+11+12=68 hot-dogs. Too many

4 hot-dogs on the first day, total 4+5+6+7+8+9+10+11=60 hot-dogs.

So she must have eaten 4 hot-dogs on the first day.

**Using algebra**

Suppose she ate $n$ hot-dogs on the first day. Then she ate $n+ 1$ hot-dogs on the second day, and $n+ 2$ hot-dogs on the third day, and so on until $n+ 7$ hot-dogs on the eighth day.

Altogether that is $n+1+n+2+n+3+...+n+7=8n+1+2+3+4+5+6+7=8n+28$ hot-dogs.

So $8n+28=60$, so $8n=32$, so $n=4$.