Solve for Table Size When Per-Person Costs Are Equal

What This Problem Teaches

• Setting up equations from equal ratios in real-world contexts
• Working with variables in both numerator and denominator positions
• Cross multiplication as a fraction-solving technique
• Translating "same amount per person" constraints into mathematical equations
• Verifying solutions by checking that per-person costs actually match

Solution: Method 1 — The Equal Payment Equation

The key insight is that "each person pays the same amount" means the per-person cost at both tables is identical. This creates an equation we can solve.

Step 1 — Define the variable

Let n = the number of people at table A. Since table B has two more people, table B has n + 2 people.

Step 2 — Set up the equal payment equation

Each person at table A pays: $72 ÷ n = 72/n

Each person at table B pays: $108 ÷ (n + 2) = 108/(n + 2)

Since these amounts are equal:

Step 3 — Cross multiply to eliminate fractions

Cross multiplying gives us:

Step 4 — Expand and solve for n

Distribute the 72:

Subtract 72n from both sides:

Solution: Method 2 — Working Backwards from Payment Amount

Instead of setting up the equation first, let's think about what the actual per-person payment must be, then work backwards to find the group sizes.

Step 1 — Find a relationship between the bills

Notice that $108 ÷ $72 = 1.5. So table B's bill is 1.5 times table A's bill.

Step 2 — Use the people difference constraint

If table B has exactly 2 more people than table A, but pays 1.5 times as much, what does this tell us about the per-person amount?

Let's say table A has n people and table B has n + 2 people. For the per-person amounts to be equal:

Step 3 — Rearrange using the bill ratio

Since 108 = 1.5 × 72, we can substitute:

Step 4 — Cross multiply and solve

Cross multiplying:

Verification

Let's check that our answer produces equal per-person payments:

Table A: 4 people, $72 bill → $72 ÷ 4 = $18 per person

Table B: 6 people, $108 bill → $108 ÷ 6 = $18 per person

Common Pitfalls

The Pattern Behind This

This problem demonstrates the equal ratio principle. Whenever you see "each person pays the same amount" or "same cost per unit," you're dealing with proportional relationships.

The power of this approach is that it works regardless of the specific numbers. Whether the difference is 2 people, 3 people, or any other amount, the same cross-multiplication technique applies.

How to Spot This Problem Type

Watch for these key phrases that signal an equal ratio setup:

• "Each person pays the same amount"
• "Split equally" combined with "same cost per person"
• "Equal per-unit cost" or "same rate"
• One group having "more people" but both having "equal individual shares"

The structure is always: two groups with different totals and different sizes, but the same per-person (or per-unit) amount. This creates the proportion that you can solve with cross multiplication.