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 $n=4$.