Percent Change, Error, and Multistep Problems

By the end of this lesson you will be able to:

• Calculate percent increase and percent decrease
• Compute percent error and interpret it in context
• Solve multistep problems by chaining percent relationships

What You Will Learn Today

By the end of this lesson, you will be able to:

4. Compute percent increase and percent decrease using
5. Calculate percent error as and interpret it
6. Solve multistep problems by chaining proportional relationships and checking reasonableness

Does Equal Up and Down Cancel Out?

An item costs $200.

• After a 20% increase:
• After a 20% decrease:

Final price: $192 — not $200!

The second percent applies to a different base than the first.

The Percent Change Formula Explained

• Positive result → percent increase
• Negative result → percent decrease
• The denominator is always the original (before) value

"Percent of WHAT?" → percent of the ORIGINAL

Worked Example: Computing Percent Increase

A store's weekly sales rose from $4,000 to $4,600. What is the percent increase?

Base = $4,000 (the original sales) — not $4,600

Worked Example: Computing Percent Decrease

A $250 coat is marked down to $190. What is the percent decrease?

The sale price is 24% less than the original — base is $250

Check: Percent Increase From 80 to 92

A student's score improved from 80 to 92. What is the percent increase?

What is the correct base for this calculation?

Compute the percent increase.

Answer: The Score Increased by Fifteen Percent

Using 92 as the base gives — incorrect

Worked Example: Finding the New Value

A town's population of 3,200 increases by 12.5%. What is the new population?

Chain of Changes: The Trap

$200 item: +10% then −10%

The Percent Error Formula Explained

• Numerator: absolute value of the difference (always positive)
• Denominator: the actual (true) value — not the measured value
• Use when comparing a measurement or estimate to a known true value

Worked Example: Percent Error (Length)

A student measures a room as 28 ft. The actual length is 30 ft.

Denominator = 30 (actual) — if you use 28 (measured), the formula is wrong

Worked Example: Percent Error (Density)

A lab measures a liquid's density as 0.95 g/mL. Actual density is 1.00 g/mL.

5% error is moderate — in many science contexts, under 5% is "acceptable"

Check: Percent Change or Percent Error?

Classify each — which formula applies?

1. Population: 5,000 → 5,750 over a decade
2. Thermometer: reads 99.2°F; actual temperature is 98.6°F
3. Stock price: $80 → $64 this year
4. Building estimate: 45 m; actual height is 48 m

Answer: Classifying Percent Change vs. Error

Change: quantity shifted over time. Error: measurement vs. truth.

Transition: All the Pieces Together

You now have the complete toolkit:

• Tax, tip, markup, markdown, commission (Deck 1)
• Simple interest: (Deck 1)
• Percent increase and decrease
• Chain of percent changes
• Percent error

Now we use all of them in the same problem.

A Framework for Solving Multistep Problems

Before computing: Estimate — "About how much should the answer be?"

During: Identify and compute each percent step in sequence

After: Verify — "Does this answer make sense given my estimate?"

Worked Example: Restaurant Multistep Problem

Wholesale $4.00 · Markup 300% · Tax 8% · Tip 15%

• Menu:
• Post-tax:
• Tip:
• Total: (estimate ~$20 ✓)

Worked Example: Salary and Interest

Salary $48,000 · Raise 6% · Deposit raise amount · Interest 3% for 4 years

Step 1: Raise /year

Step 2:

Step 1 base = salary | Step 2 base = raise amount

Check: Find the Original Price

After a 35% discount, a jacket costs $91. What was the original price?

Identify: What does $91 represent in terms of the original price?

Set up and solve.

Answer: Original Price Was $140

Check: ✓

Guided Practice: Markup Then Discount

A store marks up its wholesale cost by 80%, then puts the item on sale for 30% off the marked-up price.

What is the actual percent change over the wholesale cost?

Hint:

Summary: Percent Change, Error, and Multistep

• Change = — base is original
• New value: original | chain uses a different base each step
• Error = — base is actual

Wrong base | chained ≠ additive | measured ≠ denominator

Connections Across the Proportional Reasoning Unit

• 7.RP.A.1 — unit rates: the rate in every percent formula is a unit rate
• 7.RP.A.2 — proportional relationships: part whole, = percent rate
• 7.RP.A.3 — apply proportional reasoning to all real-world percent contexts

Coming next: ratios in scale drawings and geometric applications