Solve: How Many Children vs. Adults at the Pool?

What This Problem Teaches

• Setting up systems of equations from word problems with two unknowns
• Recognizing when you need both a "count equation" and a "value equation"
• Using elimination method to solve systems efficiently
• Interpreting solutions in the context of real-world constraints
• Verifying answers by substituting back into both original equations

Let's Draw It

Solution: Method 1 — The Elimination Approach

Step 1 — Define variables

Let c = number of children and a = number of adults.

Step 2 — Write the system of equations

We have two pieces of information that give us two equations:

Step 3 — Use elimination to solve

To eliminate c, multiply the first equation by -1.25:

Step 4 — Add the equations

Adding the left sides and right sides:

Step 5 — Solve for adults

Divide both sides by 0.75:

Step 6 — Find the number of children

Substitute a = 350 into the first equation:

Solution: Method 2 — The Value-Shift Strategy

Step 1 — Consider all people as children first

If all 661 people were children paying $1.25 each:

Step 2 — Find the revenue shortfall

The actual revenue is higher than the "all children" scenario:

Step 3 — Calculate the price difference per adult

Each adult pays $2.00 - $1.25 = $0.75 more than a child would pay.

Step 4 — Find the number of adults

The number of adults equals the shortfall divided by the extra amount each adult pays:

Step 5 — Find the number of children

Subtract to find children:

Verification

Let's check our answer in both original equations:

Both equations are satisfied, confirming our solution is correct.

Watch Out For These

Does This Seem Reasonable?

Let's do a quick sanity check. We found 311 children and 350 adults, so slightly more adults than children. Since adults pay $2.00 versus $1.25 for children, having more of the higher-paying customers should push the average price above the midpoint of $1.625.

This is indeed above $1.625, which makes sense given our slightly adult-heavy mix. The numbers check out!

The Pattern Behind This

This is a classic two-category mixture problem. The general pattern is:

• Let x and y represent quantities of each category
• Write one equation for total quantity: x + y = total
• Write one equation for total value: price₁ × x + price₂ × y = total_value
• Solve the system using elimination or substitution

This same structure applies to coin problems (quarters and dimes), ticket sales (adult and student), inventory problems (two product types), and many other scenarios where you have two categories with different unit values.

Why This Matters

Systems of equations appear constantly in business and economics. Retailers use them to analyze sales mix, manufacturers use them for production planning, and event organizers use them for pricing strategies. The pool admission problem is actually a simplified version of revenue management—the same mathematical thinking that airlines use to optimize ticket pricing across different passenger categories.

What If?

2026-10-29