Question

Emma and her friends played a board game that uses cards and dice to determine how many spaces they move. On her first turn, Emma rolled the dice and then played a card that increased the number of spaces she moved by 8. On her second turn, she rolled 3 times her previous roll, but her friend played a card that decreased the number of spaces she moved by 4. In the end, she moved the same number of spaces on both turns.
Which equation can you use to find r, the number Emma rolled on her first turn?
What number did Emma roll on her first turn?

Answer 1

Let \( r \) be the number Emma rolled on her first turn.

On her first turn, after rolling \( r \), she played a card that increased her movement by 8 spaces. Thus, the total number of spaces she moved on her first turn is:

\[
r + 8
\]

On her second turn, she rolled three times her previous roll, which is \( 3r \). However, her friend played a card that decreased her movement by 4 spaces. Therefore, the total number of spaces she moved on her second turn is:

\[
3r - 4
\]

According to the problem, the total spaces moved on both turns were the same. We can set up the equation:

\[
r + 8 = 3r - 4
\]

Now, let's solve for \( r \).

1. Start by simplifying the equation:

\[
r + 8 = 3r - 4
\]

2. Subtract \( r \) from both sides:

\[
8 = 2r - 4
\]

3. Add 4 to both sides:

\[
12 = 2r
\]

4. Divide both sides by 2:

\[
r = 6
\]

Thus, Emma rolled a **6** on her first turn.

To verify:
- On her first turn: \( 6 + 8 = 14 \)
- On her second turn: \( 3 \times 6 - 4 = 18 - 4 = 14 \)

Since both turns resulted in moving **14 spaces**, the solution is correct. Emma rolled **6** on her first turn.