Solve problems involving percent increase or decrease. |...

SECTION 1

Historical Context & Motivation

Percentages emerged in ancient civilizations to simplify trade and taxation calculations. Merchants in Babylon around 2000 BCE used ratios out of 60, laying groundwork for modern percents. By the Renaissance, Italian bankers refined these into the % symbol for precise profit tracking. This addressed the need to compare relative changes across varying quantities in commerce.

2000 BCE

Babylonian Ratios

Sexagesimal system introduces proportional thinking for trade.

15th Century

Italian Banking

Luca Pacioli popularizes double-entry bookkeeping with percentages.

1786

% Symbol Standardized

French mathematicians formalize the per cent sign for widespread use.

Modern Era

Statistics Boom

Percent changes become essential in economics and data analysis.

These developments solved the challenge of expressing proportional growth or shrinkage without fixed units. On the SSAT, you'll use this to interpret data trends confidently.

SECTION 2

Core Principles & Definitions

A percent increase occurs when a value grows relative to its original amount, calculated as the change divided by the original times 100. Conversely, a percent decrease measures shrinkage using the same base. These metrics shine in economics, like tracking stock rises or sales drops.

Percent Change Formula

((New − Original) / Original) × 100%

Multiplier Method

Increase: × (1 + r/100); Decrease: × (1 − r/100)

Original Recovery

After decrease of x%, recovery needs x/(100−x)% increase.

Successive Changes

Combine as (1 + r₁/100)(1 + r₂/100) for net effect.

🚀 Real-World Analogy

Think of percent increase like fueling a car: a 20% boost adds more miles if you start with a full tank than a half-empty one. The original value sets the scale for change.

SECTION 3

Visualizing Percent Increase

The bar grows from $100 to $120, a 20% increase. Note the added height matches 20% of the original bar.

This diagram illustrates how percent increase scales with the original value. The new bar's extra length equals 20% of the first bar, emphasizing relative growth. Visualizing changes this way builds intuition for SSAT problems.

SECTION 4

Mathematical Framework

The core formula for percent change uses the original as the base. For increases, add the percentage to 1 and multiply; for decreases, subtract. This framework handles multi-step changes elegantly.

PERCENT CHANGE

Percent Change = ((New − Original) / Original) × 100%

Positive for increase, negative for decrease. Original ≠ 0.

MULTIPLIER

New = Original × (1 ± (r / 100))

+ for increase, − for decrease; r = percent rate.

SUCCESSIVE

Final = Original × (1 + r₁/100) × (1 + r₂/100)

Net percent ≠ simple sum of individual percents.

SECTION 5

Detailed Breakdown: Increase vs. Decrease

Dropping 20% to 80 requires a 25% rise from 80 to recover, not 20%.

The diagram reveals asymmetry: a 20% drop demands over 20% growth to recover because percentages apply to different bases. This common trap tests SSAT reasoning. Always identify the original value clearly.

Successive changes compound multiplicatively, not additively. A 10% increase followed by 10% decrease yields a net 1% loss on 100: 110 then 99.

SECTION 6

Worked Example: Population Growth

A town had 5000 residents. It grew by 15% then shrank by 10%. What is the final population and net percent change?

Step-by-Step Solution

Step 1: Apply 15% IncreaseNew = 5000 × (1 + 15/100) = 5000 × 1.15

5750

Step 2: Apply 10% DecreaseFinal = 5750 × (1 − 10/100) = 5750 × 0.90

5175

Step 3: Net Percent Change((5175 − 5000) / 5000) × 100% = (175 / 5000) × 100%

3.5%

SECTION 7

Strengths & Limitations

Percent change excels in relative analysis but pair with absolutes for full picture.

Aspect	Strengths	Limitations
Comparability	Scales across different sizes (e.g., compare 10% of 100 vs. 1000).	Ignores absolute impact (small % on large base = big change).
Successive Use	Multiplier chains for compound effects.	Not additive; 10% + 10% ≠ 20% net.

✦ KEY TAKEAWAY

Like a rubber band: stretching 10% then relaxing 10% doesn't snap back exactly due to changing length.

SECTION 8

Connection to Advanced Topics

Percent changes extend to compound interest and exponential growth, where rates apply repeatedly over time. SSAT previews this with successive percents, building toward rates in higher math.

Basic	Advanced
Single or two-step % change.	A = P(1 + r/n)^(nt) for continuous compounding.
Net % from formula.	Effective annual rate (EAR) = (1 + r/n)^n − 1.

Mastering basics here equips you for exponential models in finance and science.

SECTION 9

Practice Problems

PROBLEM 1 — CONCEPTUAL

If a value decreases by 25%, what percent increase is needed to return to original? A. 20% B. 25% C. 33 1/3% D. 50% E. 75%

PROBLEM 2 — BASIC CALCULATION

A stock rises from 80 to 100. Percent increase? A. 20% B. 25% C. 25% D. 20% E. 30%

PROBLEM 3 — INTERMEDIATE

After 20% off sale price of 80, original price? A. 90 B. 96 C. 100 D. 104 E. 120

PROBLEM 4 — APPLIED

Team score: +50% to 150, then −20%. Net change from original? A. +10% B. +20% C. 30% D. +30% E. −10%

PROBLEM 5 — CRITICAL THINKING

Price 200, down 10% twice. Equivalent single %? A. 19% B. 20% C. 19% D. 21% E. 18%

SUMMARY

Key Concepts Review

Master percent change with ((new−orig)/orig)×100%, using multipliers for efficiency. Visualize bars to grasp relative scales.

Successive percents compound; recovery after decrease exceeds the drop. Practice reveals traps like base confusion. You're ready to ace SSAT Quantitative!