Lesson 7.11

Word Problems: Value & Cost

Tickets, coins, and stamps. Life is full of things that have both a quantity (how many?) and a value (how much?). Systems let us track both at once.

Introduction

If I have 10 bills in my wallet, do I have $10 or $1000? It depends on the Value of each bill. We need one equation for the piece of paper (quantity) and one equation for the money (value).

Past Knowledge

Lesson 7.10 (Sum Problems). The "Quantity" equation is usually just .

Today's Goal

Set up and solve systems involving ticket sales and coin counting.

Future Success

This is the blueprint for business inventory: tracking "Units Sold" vs "Revenue".

Key Concepts

The Two Equation Structure

Every Value Problem has the same skeleton:

1. Quantity Equation

Counts the physical items.

2. Value Equation

Counts the total worth/money.

Pro Tip: Usually, you multiply the Quantity Equation by

to solve via Elimination.

Worked Examples

Example 1: The School Play

Basic

Adult tickets cost $10. Student tickets cost $5. We sold 100 tickets total and made $800. How many adults went?

Step 1: Set Up

Let = adults, = students.

Quantity: (Total tickets)

Value: (Total money)

Step 2: Solve

Target . Multiply top by -5.

(Add them!)

Answer: 60 Adults

Example 2: Coins (Decimals)

Intermediate

I have 20 coins, all Dimes and Quarters. I have $4.10 total. How many of each?

Step 1: Set Up & Clear Decimals

Quantity:

Value:

HACK: Multiply Value Eq by 100 to kill decimals!

Step 2: Solve

Multiply Quantity by -10.

(Add)

14 Quarters, 6 Dimes

Example 3: Break-Even Point

Advanced

A business spends $500 on equipment and $10 per shirt. They sell shirts for $25. How many shirts to break even?

Step 1: Cost vs Revenue

Cost:

Revenue:

Step 2: Substitution

Set them equal ().

Must sell 34 shirts.

Common Pitfalls

Mixing Up Eq 1 and Eq 2

Don't put prices in the quantity equation! is wrong. The number of tickets (100) has nothing to do with their price ($10).

Decimal Drifting

In coin problems, is a quarter. Some students write but leave the total as . Keep units consistent (either all cents or all dollars).

Real-Life Applications

Business Operations:

Every coffee shop calculates this daily.
"We sold 300 cups (Quantity). We made $1200 (Value). How many were Lattes vs Drip Coffee?"
This data drives what inventory they buy for tomorrow.