🎯 What You'll Learn in This Lesson
- How a ratio, rate, proportion, and percent are connected — and how to switch between them.
- Cross-multiplication: the universal proportion solver.
- The single percent formula that handles every percent question on the SAT.
- Percent change vs. percent of vs. percent off — the three flavors students confuse.
- Speed shortcuts that turn 90-second questions into 20-second wins.
- 10 trick-based practice questions with shortcut solutions.
1. Ratios, Rates, and Proportions
A ratio compares two quantities of the same type (boys to girls, red to blue marbles). A rate compares two quantities of different types (miles per hour, dollars per pound). A proportion is just an equation that says two ratios are equal.
Three Ways to Write the Same Ratio
Format | Example | When to Use |
Colon | 3 : 5 | Quick comparisons |
Fraction | 3 / 5 | Calculating with the ratio |
Words | 3 to 5 | Word problems |
Cross-multiplication
a / b = c / d ⇒ a · d = b · c
💡 Pro Tip — Always Reduce
Reduce ratios just like fractions. The ratio 12 : 18 is the same as 2 : 3. SAT answer choices are almost always given in lowest terms — if your answer is not, you probably need to simplify it.
2. The Universal Proportion Setup
Most ratio and rate questions can be solved by setting up one proportion. The trick is to keep the same units in the same row.
STEP 1 — Identify the two quantities being related
STEP 2 — Match units across the equal sign
STEP 3 — Plug in three known values; let x be the missing one
STEP 4 — Solve the resulting linear equation
📝 Worked Example
If 8 notebooks cost $20, how much do 14 notebooks cost?
Setup: 8 / 20 = 14 / x
Cross-multiply: 8x = 280, so x = $35.
3. Percentages — One Formula to Rule Them All
(part / whole) × 100 = percent
Every percent question in SAT Math is a rearrangement of this single formula. Identify which of the three pieces (part, whole, percent) is missing and solve for it.
The Three Question Types
Question Style | What's Missing | How to Solve |
What is 30% of 80? | Part | 0.30 × 80 = 24 |
12 is what % of 50? | Percent | 12/50 × 100 = 24% |
15 is 25% of what? | Whole | 15 / 0.25 = 60 |
⚡ Speed Trick — Decimal Conversion
Always convert a percent to its decimal form (move the decimal point two places left) before multiplying.
25% → 0.25, 7.5% → 0.075, 140% → 1.40
This avoids mistakes and lets you compute mentally for round numbers.
4. Percent Change — The Formula Students Mess Up
Percent Change = (New − Old) / Old × 100
If the answer is positive, it's a percent INCREASE. If negative, it's a percent DECREASE. The denominator is always the ORIGINAL (old) value — not the new one.
❓ Is the new value bigger than the old value? | ||
✅ YES ▼ | ❌ NO ▼ | |
Increase — answer is positive | Decrease — answer is negative | |
⚡ Speed Trick — The Multiplier Method
Replace 'increase by p%' and 'decrease by p%' with one multiplication.
- Increase by 20%: multiply by 1.20
- Decrease by 20%: multiply by 0.80
- Multiple changes: multiply the multipliers in sequence
Example: A $50 item is marked up 20%, then discounted 10%. Final price = 50 × 1.20 × 0.90 = $54.
⚠️ Watch Out — Successive Percents Don't Add
A 20% increase followed by a 20% decrease does NOT bring you back to the original. (You end at 96% of the original.)
Always apply percent changes one at a time using multipliers.
5. Speed Tricks Summary
⚡ Use multipliers, not arithmetic
Replace any percent change with a single multiplication. Chains of changes become products of multipliers.
⚡ Match units across a proportion
Keep miles over hours on both sides. If you flip one ratio, you'll get the wrong answer.
⚡ Convert percents to decimals immediately
Mental math gets dramatically faster once you stop seeing the % sign.
⚡ For 'what % is x of y', use x/y directly
Don't translate to algebra — divide and move the decimal.
⚡ Memorize 1/8 = 12.5%, 1/6 ≈ 16.67%, 1/3 ≈ 33.3%
These tiny conversions appear in dozens of SAT questions.
6. Summary
📌 The Whole Topic in 30 Seconds
- Ratio: compares same-type quantities
- Rate: compares different-type quantities (e.g., $/lb)
- Proportion: two equal ratios — solve by cross-multiplying
- Percent: (part / whole) × 100
- Percent change: (new − old) / old × 100
- Multiplier rule: +p% → ×(1 + p/100); −p% → ×(1 − p/100)
7. Practice — 10 Trick-Based Questions
⏱️ Instructions
Try each in under 60 seconds. Solutions follow.
Q1. The ratio of cats to dogs in a shelter is 3:4. If there are 21 cats, how many dogs?
A) 24 B) 28 C) 32 D) 36
Hint: Set up 3/4 = 21/x and cross-multiply.
Q2. A car travels 150 miles in 3 hours. At the same rate, how far in 5 hours?
A) 200 B) 225 C) 250 D) 275
Hint: Find rate (50 mph) then multiply, OR set up the proportion.
Q3. What is 35% of 200?
A) 35 B) 50 C) 70 D) 80
Hint: Just multiply: 0.35 × 200.
Q4. 18 is what percent of 60?
A) 18% B) 24% C) 30% D) 36%
Hint: Divide: 18/60 = 0.30 → 30%.
Q5. A jacket originally priced at $80 is on sale for $60. What is the percent decrease?
A) 20% B) 25% C) 30% D) 33%
Hint: (80 − 60)/80 = 20/80 = 25%.
Q6. A salary of $50,000 increases by 10%, then by another 10%. What is the new salary?
A) $55,000 B) $60,000 C) $60,500 D) $61,000
Hint: 50,000 × 1.10 × 1.10 = 50,000 × 1.21.
Q7. If 4 workers can finish a job in 9 hours, how long would 6 workers take at the same rate?
A) 4 B) 6 C) 7 D) 8 hours
Hint: Inverse: more workers = less time. Total work = 4 × 9 = 36 worker-hours; 36/6 = 6.
Q8. A shirt is marked up 25%, then discounted 20%. What percent of the original is the final price?
A) 95% B) 100% C) 105% D) 110%
Hint: Multipliers: 1.25 × 0.80 = 1.00 → 100%.
Q9. If 3 pounds of cherries cost $12, how much do 5 pounds cost?
A) $15 B) $18 C) $20 D) $25
Hint: Unit price: $4/lb. Then $4 × 5 = $20.
Q10. A number is decreased by 40% and then increased by 50%. What percent of the original is the result?
A) 80% B) 90% C) 100% D) 110%
Hint: 0.60 × 1.50 = 0.90 → 90%.
Answer Key & Shortcut Solutions
# | Answer | Shortcut Reason |
Q1. | B) 28 | 3/4 = 21/x → x = 28. |
Q2. | C) 250 | Rate is 50 mph; 50 × 5 = 250. |
Q3. | C) 70 | 0.35 × 200 = 70. |
Q4. | C) 30% | 18/60 = 0.30. |
Q5. | B) 25% | 20/80 = 0.25. |
Q6. | C) $60,500 | 50,000 × 1.21 = 60,500. |
Q7. | B) 6 | Total work = 36 worker-hours; 36/6 = 6. |
Q8. | B) 100% | 1.25 × 0.80 = 1.00. |
Q9. | C) $20 | Unit rate $4/lb; 4 × 5 = 20. |
Q10. | B) 90% | 0.60 × 1.50 = 0.90. |
🚀 You've Got This! 💪
Ratios and percents account for several questions per SAT — and the multiplier trick alone saves minutes. Next: unit conversions, where the same proportion logic shines.