Let's represent the total number of pages in the book as $P$.

On the first day:

Jessica read $\frac{1}{5}P + 12$ pages.

On the second day:

Jessica read $\frac{1}{4}P + 15$ pages.

On the third day:

Jessica read $\frac{1}{3}P + 18$ pages.

After reading these pages, there were only 62 pages left, which she read the next day.

The total number of pages read on the first three days and the remaining pages is equal to the total pages in the book. Thus, we can set up the equation:

$\left( \frac{1}{5}P + 12 \right) + \left( \frac{1}{4}P + 15 \right) + \left( \frac{1}{3}P + 18 \right) + 62 = P$

Now let's simplify and solve this equation:

$\frac{1}{5}P + \frac{1}{4}P + \frac{1}{3}P + 12 + 15 + 18 + 62 = P$

First, combine the constants:

$12 + 15 + 18 + 62 = 107$

So the equation becomes:

$\frac{1}{5}P + \frac{1}{4}P + \frac{1}{3}P + 107 = P$

Now, to combine the fractions, let's find a common denominator. The least common denominator of 5, 4, and 3 is 60.

$\frac{1}{5}P = \frac{12}{60}P, \quad \frac{1}{4}P = \frac{15}{60}P, \quad \frac{1}{3}P = \frac{20}{60}P$

Now substitute these into the equation:

$\frac{12}{60}P + \frac{15}{60}P + \frac{20}{60}P + 107 = P$

Combine the fractions:

$\frac{12 + 15 + 20}{60}P + 107 = P$

$\frac{47}{60}P + 107 = P$

Now, subtract $\frac{47}{60}P$ from both sides:

$107 = P - \frac{47}{60}P$

$107 = \frac{60}{60}P - \frac{47}{60}P$

$107 = \frac{13}{60}P$

Now, multiply both sides by 60 to eliminate the denominator:

$107 \times 60 = 13P$

$6420 = 13P$

Now, solve for $P$:

$P = \frac{6420}{13} = 495$

So, the total number of pages in the book is 495 pages.

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Would you like a more detailed breakdown of any part of the solution? Here are some follow-up questions you might find interesting:

1. How would the total number of pages change if Jessica had read different fractions of the book each day?
2. What happens if Jessica reads fewer pages on the first day? How would this affect the total?
3. How would the problem change if the number of pages left on the last day was different?
4. Can you solve a similar problem where Jessica reads different fractions of the book on each day but has more days to read?
5. How would you check if your answer is correct after solving this equation?

Tip: Always double-check the calculations when solving for an unknown in a fraction-based equation, especially when dealing with multiple terms.