There are simpler approaches.
Method 1: We place the consonants first.
Since the word EQUATION has eight letters, there are eight positions to fill. Therefore, there are eight ways to place the Q, seven ways to place the T, and six ways to place the N. Since the vowels must appear in the original order, there is only one way to place the five vowels in the remaining five positions. Hence, the number of arrangements of the letters of the word EQUATION in which the vowels appear in the original order is $8 \cdot 7 \cdot 6 = 336$.
Method 2: We use symmetry.
Since the word EQUATION has eight distinct letters, there are $8!$ ways to arrange its letters. Within a given arrangement, the five distinct vowels can be permuted among themselves in $5!$ ways. Of these $5!$ ways of arranging the vowels, only one leaves the vowels in the original order. Hence, the number of admissible arrangements of the letters is
$$\frac{8!}{5!} = \frac{8 \cdot 7 \cdot 6 \cdot 5!}{5!} = 8 \cdot 7 \cdot 6 = 336$$