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SSAT Middle Level: Quantitative • Fractions, Decimals, and Percents

Solve problems involving percent increase or decrease.

Master how values grow or shrink by percentages, like game scores or store sales.

Explore the lesson 4 min read
SSAT Middle Level illustration with a test notebook, quantitative function, and reading book
Inside this lesson 10 sections 5 questions 6 figures

SECTION 1

Historical Context & Motivation

Long ago, people needed a way to talk about changes in amounts, like more crops or fewer coins. Percents come from the Latin words "per centum," meaning "by the hundred." They help us see how much something changes compared to before. Think of it like tracking points in a soccer game.

15th Century • Birth of Percents

Italian merchants used percents for trade deals and profits.

1700s • Taxes and Growth

Governments tracked tax increases with percents.

1900s • Everyday Use

Schools taught percent change for sales and sports stats.

Today • Test Prep

SSAT uses it to build your math superpowers!

Before percents, changes were hard to compare. Now, you can quickly see if your video game high score jumped by 20%. This solves real-world puzzles like price tags or team wins.

SECTION 2

Core Principles & Definitions

A percent increase happens when a value grows bigger. Percent decrease is when it gets smaller. Both compare the change to the original amount.

1

Original Value

The starting amount, like 100 points in a game.
2

Change Amount

New minus original, like +20 points.
3

Percent Change

(Change / original) × 100%. Always out of 100!
4

Increase vs Decrease

Positive for growth, negative for shrink.
⚽

Quick Tip

Imagine your soccer team's score goes from 5 to 7 goals. That's a 40% increase! Like leveling up in a game—small changes add up big.

SECTION 3

Visual Explanation

Percent Increase: From $10 to $12 Understanding how a price change relates to percent increase $0 $2 $4 $6 $8 $10 Price ($) $10 Original Price $12 New Price +$2 increase How to Calculate 1 Find the change: $12 − $10 = $2 2 Divide by original: $2 $10 = 0.20 3 Convert to percent: 0.20 × 100 = 20% A price going from $10 to $12 is a 20% increase!
The blue bar shows the original $10 price. The cyan bar is taller for the new $12 price. The difference is 20% of original.

See how the new bar is taller? That's growth! The diagram makes it easy to spot percent increase visually, like comparing candy bar sizes before and after a sale.

SECTION 4

Mathematical Framework

The math is simple. Find the change, divide by original, then multiply by 100 for percent. Always use the original as the bottom number!

PERCENT CHANGE

((new−original)/original)×100((new − original) / original) × 100%

If new > original, it's increase. If new < original, decrease. original ≠ 0.

MULTIPLIER

new=original×(1±(percent/100))new = original × (1 ± (percent / 100))

+ for increase, − for decrease. Great for checking work!

Plug in numbers step by step. You'll feel like a math wizard on the SSAT!

SECTION 5

Detailed Breakdown

Increase and decrease work the same way but flip the sign. Watch for words like "rose by" or "dropped."

Percent Increase & Decrease Starting from an original value of 100 60 80 100 120 140 Value 100 Original 120 +20% ▲ Increase 100 × 1.20 = 120 80 −20% ▼ Decrease 100 × 0.80 = 80 💡 Key Formulas Increase: value × (1 + r) Decrease: value × (1 − r) where r = percent ÷ 100 A 20% change means multiplying by 20 ÷ 100 = 0.20 of the original value +20 −20
Circle starts at 100. Green arrow up to 120 (+20%). Red arrow down to 80 (−20%). Same percent, different results.

The diagram shows both paths from the same start. Up for more, down for less—like jumping higher or sinking in a game.

SECTION 6

Worked Example

Candy Price Jump

  1. 1

    Step 1 — Find Change

    Original: 50¢, New: 60¢. Change = 60 − 50 = 10¢
    10¢
  2. 2

    Step 2 — Divide by Original

    10 / 50 = 0.2
    0.2
  3. 3

    Step 3 — To Percent

    0.2 × 100 = 20% increase!
    20%

See? Easy steps. Practice this, and SSAT problems will seem like fun challenges.

SECTION 7

Strengths & Common Mistakes

Do ThisAvoid This
Use original as denominator.Don't use new value.
Check if increase or decrease.Forget the sign.
Multiply by 100 last.Skip ×100%.

Spot traps to boost your score!

🛑

Key takeaway

Percents shine for quick comparisons, like sale prices. But always double-check the base—like aiming before kicking the ball.

SECTION 8

Connections to Advanced Ideas

Basic Percent ChangeAdvanced (Compound)
One-time change.Change over time, like interest.
Simple formula.(1+r/100)n(1 + r/100)^n.

Master basics now, and compound growth is next—like daily training making you a sports star. SSAT preps you perfectly!

SECTION 9

Practice Problems

Question 1: CONCEPTUAL

What does a 0% increase mean? A) Value halves B) Value doubles C) No change D) Value triples E) Value disappears
Show answer
C) No change. 0% means (0/original)×100=0, so new=original. A halves (50% decrease), B doubles (100% increase), D triples (200%), E is silly.

Question 2: BASIC CALCULATION

A shirt costs $20, now $25. Percent increase? A) 10% B) 20% C) 25% D) 30% E) 50%
Show answer
C) 25%. Change=5, 5/20=0.25×100=25%. A uses 2/20 wrong change, B halves change, D adds extra, E wrong base.

Question 3: INTERMEDIATE

Team score: 40 to 32. Percent decrease? A) 10% B) 20% C) 25% D) 30% E) 40%
Show answer
B) 20%. Change=−8, |−8/40|=0.2×100=20%. C uses 8/32 wrong base, A too small, D/E too big.

Question 4: APPLIED

Game level 50 gems to 65. Increase? A) 15% B) 25% C) 30% D) 50% E) 65%
Show answer
C) 30%. 15/50=0.3×100=30%. A is change only, B wrong math, D doubles, E is new value.

Question 5: CRITICAL THINKING

Price drops from 80¢ to 64¢, then up 25%. Final percent from original? A) −10% B) 0% C) +10% D) −20% E) +20%
Show answer
B) 0%. First −20% to 64¢, +25% of 64=16, 64+16=80¢ original. Not cumulative on original. A/D wrong direction, C/E ignore steps.

SUMMARY

Lesson Summary

Summary

You can now crush percent increase/decrease! Use ((new−original)/original)×100%. Check original base.

Practice builds speed for SSAT. You're ready—go get those top scores! 🚀

Varsity Tutors • SSAT Middle Level • Solve problems involving percent increase or decrease.