Let's solve the problem step by step.

Let:

• $r$ be the number of red marbles,
• $b$ be the number of black marbles.

According to the problem, the ratio of red marbles to black marbles is $19:1$, meaning there are 19 red marbles for every black marble. This gives the relation:

$r = 19b$

Also, the total number of marbles is 120:

$r + b = 120$

Now, substitute $r = 19b$ into the total equation:

$19b + b = 120$

Simplify:

$20b = 120$

Solve for $b$:

$b = \frac{120}{20} = 6$

Now, substitute $b = 6$ back into the equation $r = 19b$:

$r = 19 \times 6 = 114$

So, there are 114 red marbles in the bag.

Would you like more details, or do you have any questions?

Here are 5 related questions for further exploration:

1. How many black marbles are in the bag?
2. What would happen if there were 150 marbles in total instead of 120?
3. What if the ratio was 5:1 instead of 19:1?
4. How would the number of marbles change if the total was increased by 50%?
5. What if 20% of the black marbles were removed from the bag?

Tip: Always check your equations by plugging the numbers back into the original conditions to ensure accuracy.