Let's solve this step-by-step.

Let:

• $M$ represent the number of marbles Mark has.
• $S$ represent the number of marbles Sarah has.

Step 1: Set up equations based on the information given

1. Mark has 30 fewer marbles than Sarah: $M = S - 30$

2. 50% of Mark’s marbles is equal to 1/3 of Sarah’s marbles: $0.5 \times M = \frac{1}{3} \times S$ This can be rewritten as: $\frac{M}{2} = \frac{S}{3}$

Step 2: Solve for $M$ in terms of $S$

From the first equation: $M = S - 30$

Substitute $M = S - 30$ into the second equation: $\frac{S - 30}{2} = \frac{S}{3}$

Step 3: Eliminate the fractions by multiplying both sides by 6

$6 \times \frac{S - 30}{2} = 6 \times \frac{S}{3}$ This simplifies to: $3(S - 30) = 2S$

Expanding the left side: $3S - 90 = 2S$

Step 4: Solve for $S$

Subtract $2S$ from both sides: $S = 90$

Step 5: Find $M$ using $S = 90$

Since $M = S - 30$: $M = 90 - 30 = 60$

Step 6: Calculate the total number of marbles

The total number of marbles Mark and Sarah have together is: $M + S = 60 + 90 = 150$

Final Answer:

They have 150 marbles altogether.

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Would you like more details or have any questions? Here are some related questions:

1. How can we verify the solution by substituting values back into the original conditions?
2. What if the difference in marbles was more than 30? How would that change the solution?
3. Can we solve similar problems using a different method, like substitution or elimination?
4. How would the answer change if Mark’s marbles were instead 40% of Sarah's?
5. How can we represent and solve this problem using a graph?

Tip: To check your solution, substitute the values back into each original condition to confirm they hold true.