To solve this problem, let's calculate the value of $d$ step-by-step.

Problem Setup:

The cards display the following expressions:

• $3d$
• $d$
• $3d - 2$
• $2d - 1$
• $d + 3$

The mean of the values on these cards is $10$. The formula for the mean is:

$\text{Mean} = \frac{\text{Sum of all terms}}{\text{Number of terms}}$

Step 1: Sum up the terms

The sum of the terms on the cards is:

$3d + d + (3d - 2) + (2d - 1) + (d + 3)$

Simplify the sum: $3d + d + 3d - 2 + 2d - 1 + d + 3 = 10d \, + \, 0$

So, the total sum is $10d$.

Step 2: Use the mean formula

The mean is given to be $10$. Using the mean formula:

$\frac{\text{Sum of all terms}}{\text{Number of terms}} = 10$

Substitute the sum $10d$ and the number of terms $5$:

$\frac{10d}{5} = 10$

Step 3: Solve for $d$

Simplify the equation: $2d = 10$

Solve for $d$: $d = 5$

Final Answer:

$\boxed{d = 5}$

Would you like a further explanation of any step?

Follow-Up Questions:

1. How is the mean formula applied in general situations?
2. What happens if the mean is changed?
3. How do you simplify expressions with multiple terms effectively?
4. Can this problem be solved graphically?
5. What are the properties of equations with a single variable like this one?

Tip:

Always double-check the total number of terms in problems involving averages to avoid missteps.