Solve for Table Size When Per-Person Costs Are Equal

Number Puzzle 9th-10th Grade
Problem

At a restaurant, table A's bill of $72 is split equally among its people. Table B has two more people and a bill of $108, also split equally. Each person at table A pays the same amount as each person at table B. How many people were at table A?

Already Got the Answer?

Setup:72/x = 108/(x + 2), where x is the number of people at table A.

Answer: 4 people at table A (6 at table B, and everyone pays $18).

One-line check:72 ÷ 4 = 18 and 108 ÷ 6 = 18. If you got a different number, jump to "What Trips People Up" below.

What You Will Learn

  • Translating "equal share" into an equation. "Split equally" always means total ÷ number of people, and "pays the same" means setting two such quotients equal.
  • Solving a proportion with a variable in both denominators. Cross-multiplication turns a rational equation into a plain linear one, provided you distribute correctly.
  • Reasoning with differences. When two groups pay the same rate, the extra money must come from the extra people. This often lets you skip the algebra entirely.
  • Using ratios as a shortcut. Equal unit price means the bills and the headcounts are in the same ratio.
  • Letting context check your algebra. A headcount has to be a whole number, which gives you a built-in test of the answer.

Solution: Method 1 — Set the Per-Person Costs Equal

This problem looks like it has two unknowns, the size of table A and the size of table B. But the problem tells us how they are connected (B has exactly two more people), so one variable is enough. The equation comes from the one fact that ties the tables together: everybody pays the same amount.

Step 1 — Define the variable

Let x be the number of people at table A. Table B has two more people, so it has x + 2 people.

Step 2 — Write each per-person cost

Splitting a bill equally means dividing the bill by the number of people.

Table A, each person pays: 72 / x
Table B, each person pays: 108 / (x + 2)

Step 3 — Set the costs equal

The problem says each person at table A pays the same as each person at table B, so those two expressions are equal.

72 / x = 108 / (x + 2)

Step 4 — Cross-multiply

Multiply both sides by x and by (x + 2) to clear the denominators. In effect, each numerator is multiplied by the opposite denominator.

72(x + 2) = 108x

Distribute the 72 across both terms in the parentheses:

72x + 144 = 108x

Step 5 — Solve the linear equation

Collect the x-terms on one side by subtracting 72x from both sides:

144 = 36x
x = 144 / 36 = 4

Step 6 — Interpret

Table A had 4 people. Table B had 4 + 2 = 6 people. Each person paid 72 ÷ 4 = $18.

Solution: Method 2 — Follow the Extra Money

Here is a way to solve this without setting up a single fraction. Table B has two more people, and it also has a bigger bill. Since everyone pays the same amount, the extra dollars on table B's bill must come from the two extra diners.

Step 1 — Find the extra money

108 − 72 = 36 dollars

Step 2 — Divide it among the extra people

Those $36 are paid by exactly 2 additional people, so each person pays:

36 ÷ 2 = 18 dollars per person

Step 3 — Work back to table A

Table A's bill was $72 and each person paid $18:

72 ÷ 18 = 4 people

This is the same computation as Method 1 in disguise. If you subtract the two equations "bill = cost × people," the table A people cancel and you are left with 36 = c · 2. The advantage is that the reasoning is entirely about money and people, with no algebra to mishandle.

Solution: Method 3 — Ratio Scaling

Here is a third view. If the price per person is the same at both tables, then the bill is proportional to the number of people.

Step 1 — Compare the bills

The ratio of the bills is 72 : 108. Dividing both by 36 gives 2 : 3.

Step 2 — Transfer the ratio to the headcounts

Table A has 2 "parts" of people and table B has 3 parts. The difference of one part is the 2 extra people the problem mentions, so:

1 part = 2 people
Table A = 2 parts = 4 people

Algebraically, this is x : (x + 2) = 2 : 3, so 3x = 2x + 4 and x = 4. Three different routes give the same answer, which is a good sign.

The Answer

There were 4 people at table A.

Table B had 6 people, and each person at both tables paid $18.

Verification

Substitute x = 4 back into the original statement of the problem, not just into the equation you built.

Table A:$72 ÷ 4 = $18 per person.

Table B: has 4 + 2 = 6 people, and $108 ÷ 6 = $18 per person.

Equation:72/4 = 18 and 108/6 = 18, so both sides match. ✓

Cross-multiplied form:72(4 + 2) = 432 and 108 · 4 = 432. ✓

Sanity Check: Is a Whole Number Guaranteed?

Not at all. The people at a table can't be fractional, so a well-posed version of this problem has to produce a whole number. Here is what happens when the bills change:

Bills (A, B)EquationxValid?
$72, $10872(x+2) = 108x4Yes
$72, $10072(x+2) = 100x144/28 ≈ 5.14No: not a whole number
$72, $7272(x+2) = 72xno solutionNo: 144 = 0 is impossible

The last row is instructive: two extra people can't leave the bill unchanged if everyone pays the same positive amount. The fact that this problem's answer comes out whole is a sign the numbers were chosen carefully. If your own work ever produces something like 5.14 people, recheck your setup before assuming the problem is odd.

What Trips People Up

✗ 72x + 2 = 108x

This is the classic cross-multiplication slip: the 72 multiplies the whole quantity (x + 2), not just the x. The correct line is 72(x + 2) = 108x, which expands to 72x + 144 = 108x. With the slip you would get 2 = 36x and a nonsensical fraction of a person.

✗ Answering "18" or "6"

Method 2 produces 18 along the way, but that is the dollars per person, not the headcount. Table B's 6 people is also a number you compute en route. The question asks about table A, so reread the final sentence before you box an answer. A quick sanity test: 18 people sharing a $72 bill would pay $4 each, which is nowhere near $18.

✗ Putting the extra 2 in the wrong place: 72/x + 2 = 108/x

"Two more people" adds to the headcount in table B's denominator, giving x + 2. It does not add two dollars to anything, and it does not mean the tables share a denominator. Each table has its own count, and the "+2" belongs only to table B's.

The Pattern Behind This

Behind this problem is a constant unit rate. Call the shared cost per person c. Then the two bills are 72 = c·n and 108 = c·(n + k), where n is table A's headcount and k = 2 is the extra people. Subtracting the equations eliminates n:

c = (B₂ − B₁) / k
n = B₁ / c = k · B₁ / (B₂ − B₁)

With our numbers: n = 2 · 72 / 36 = 4. ✓

Limits of the shortcut: it requires B₂ > B₁ when table B is the larger table, and it only makes sense in context when n comes out as a whole number. It also assumes a single, shared price per person, so no tips or taxes that differ by table. (The fourth What-If below shows how to handle that wrinkle.)

The same structure appears in "two cars travel at the same speed" problems, "two workers have the same hourly pay" problems, and any situation where total = rate × quantity with the same rate on both sides.

How to Spot This Problem Type

  • Phrases like "split equally,""each pays the same," or "the same cost per person."
  • Two groups whose sizes differ by a known amount ("two more," "three fewer") but whose totals are given.
  • A question about the count of something, with totals provided instead of unit rates.

A disguised version: "Two printers produce 72 and 108 pages in the same number of minutes, but the second printer has two more nozzles. If every nozzle prints at the same speed, how many nozzles does the first printer have?" It has the same structure and the same answer.

Try These Variations

These four problems build on each other. Each one has a whole-number answer, so if your result isn't a whole number, recheck your setup. Try each one on paper before you open the solution.

1 Three extra people
Table A's bill of $90 is split equally. Table B has three more people and a bill of $135, also split equally. Each person pays the same amount at both tables. How many people were at table A?
Step 1 — Define the variable

Let x be the number of people at table A. Table B has x + 3 people.

Step 2 — Set up the equation

Equal per-person costs give 90/x = 135/(x + 3).

Step 3 — Cross-multiply

90(x + 3) = 135x, which becomes 90x + 270 = 135x.

Step 4 — Solve

270 = 45x, so x = 6.

Step 5 — Verify

Table A: 90 ÷ 6 = 15. Table B has 9 people: 135 ÷ 9 = 15. ✓ Table A had 6 people.

2 Reverse the unknown
Table A's bill of $84 is split equally, and each person at table A pays $12. Table B has two more people than table A, and each person at table B also pays $12. What is table B's total bill?
Step 1 — Find table A's headcount

Bill ÷ cost per person: 84 ÷ 12 = 7 people.

Step 2 — Find table B's headcount

Two more people: 7 + 2 = 9.

Step 3 — Compute table B's bill

Cost per person × people: 12 × 9 = 108.

Step 4 — Verify

The extra two people add 2 × 12 = 24, and 84 + 24 = 108. ✓ Table B's bill is $108.

3 A third table joins
Table A's bill is $80. Table B has two more people than table A and a bill of $112. Table C has three more people than table A and a bill of $128. Everyone at all three tables pays the same amount. How many people were at table A?
Step 1 — Use two tables to find x

Let x be table A's headcount. Tables A and B give 80/x = 112/(x + 2).

Step 2 — Cross-multiply

80(x + 2) = 112x, so 80x + 160 = 112x.

Step 3 — Solve

160 = 32x, so x = 5. Everyone pays 80 ÷ 5 = $16.

Step 4 — Use table C as a consistency check

Table C has 5 + 3 = 8 people, and 128 ÷ 8 = 16. This matches, so the third table adds no new information but confirms the data are consistent.

Step 5 — Verify

Table B has 7 people and 112 ÷ 7 = 16. ✓ Table A had 5 people.

4 Tips change the totals
Table A's food bill is $75, and a 20% tip is added before the total is split equally. Table B has two more people and a food bill of $120, with a 5% tip added before its total is split equally. Each person at table A pays the same amount as each person at table B. How many people were at table A?
Step 1 — Find each table's total with tip

Table A: 75 × 1.20 = 90. Table B: 120 × 1.05 = 126. The amount split is the total including tip, not the food bill.

Step 2 — Set up the equation

Let x be table A's headcount: 90/x = 126/(x + 2).

Step 3 — Cross-multiply

90(x + 2) = 126x, so 90x + 180 = 126x.

Step 4 — Solve

180 = 36x, so x = 5.

Step 5 — Verify

Table A: 90 ÷ 5 = 18. Table B has 7 people: 126 ÷ 7 = 18. ✓ The shortcut formula agrees: 2 · 90 / (126 − 90) = 5. Table A had 5 people.

Frequently Asked Questions

Let x be the number of people in the first group, write each group's per-person cost as bill ÷ headcount, and set the two expressions equal. In this example, one table has a $72 bill and x people, and the other has a $108 bill and x + 2 people, so 72/x = 108/(x + 2). Cross-multiplying gives 72x + 144 = 108x, so 36x = 144 and x = 4 people.
Because everyone pays the same amount, the extra money in the larger bill comes entirely from the extra people. Divide the difference in bills by the difference in headcount to get the cost per person, then divide the smaller bill by that cost. In this example, (108 − 72) ÷ 2 = $18 per person, and 72 ÷ 18 = 4 people at the smaller table.
Divide each bill by its headcount and confirm both give the same per-person amount, and that the headcount is a whole number. In this example, 72 ÷ 4 = $18 and 108 ÷ 6 = $18, so the answer of 4 people at the first table is consistent.
NJ
Neven Jurkovic, PhD

Professor of Computer Science, Palo Alto College, Alamo Colleges District, San Antonio, TX

Developer of Algebrator

Contact

This solution was prepared with AI assistance and reviewed by Dr. Jurkovic for mathematical accuracy and pedagogical clarity.

2026-09-19

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