Ratios, Proportions, and Percents

Master ratio problems and percent calculations on SAT

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Ratios, Proportions, and Percents on the SAT

Why This Topic Matters

Ratios, proportions, and percents make up roughly 15-20% of SAT Math questions. They appear in both Calculator and No-Calculator sections.

Ratios

A ratio compares two quantities. The ratio of aa to bb can be written as: a:boraba:b \quad \text{or} \quad \frac{a}{b}

Ratio Problems Strategy

If the ratio of cats to dogs is 3:53:5 and there are 40 animals total:

  • Total parts = 3+5=83 + 5 = 8
  • Each part = 408=5\frac{40}{8} = 5
  • Cats = 3×5=153 \times 5 = 15
  • Dogs = 5×5=255 \times 5 = 25

Part-to-Part vs. Part-to-Whole

  • Part-to-Part: cats to dogs = 3:53:5
  • Part-to-Whole: cats to total = 3:83:8 or 38\frac{3}{8}

Proportions

A proportion is an equation stating two ratios are equal: ab=cd\frac{a}{b} = \frac{c}{d}

Cross multiplication: ad=bcad = bc

Setting Up Proportions

Keep consistent units on corresponding sides.

Correct: miles1hours1=miles2hours2\frac{\text{miles}_1}{\text{hours}_1} = \frac{\text{miles}_2}{\text{hours}_2}

Wrong: miles1hours1=hours2miles2\frac{\text{miles}_1}{\text{hours}_1} = \frac{\text{hours}_2}{\text{miles}_2}

Unit Conversions

Use dimensional analysis (multiply by conversion factors):

Example: Convert 45 mph to feet per second. 45mileshour×5280 feet1 mile×1 hour3600 seconds=66feetsecond45 \frac{\text{miles}}{\text{hour}} \times \frac{5280 \text{ feet}}{1 \text{ mile}} \times \frac{1 \text{ hour}}{3600 \text{ seconds}} = 66 \frac{\text{feet}}{\text{second}}

Percents

Key Formulas

Percent=PartWhole×100\text{Percent} = \frac{\text{Part}}{\text{Whole}} \times 100

Part=Percent100×Whole\text{Part} = \frac{\text{Percent}}{100} \times \text{Whole}

Percent Change

Percent Change=New−OriginalOriginal×100\text{Percent Change} = \frac{\text{New} - \text{Original}}{\text{Original}} \times 100

  • Percent increase: New > Original → positive result
  • Percent decrease: New < Original → negative result

Multiplier Method (SAT Shortcut!)

Instead of calculating the change, use a multiplier:

| Situation | Multiplier | |---|---| | 20% increase | ×1.20\times 1.20 | | 15% decrease | ×0.85\times 0.85 | | 8% sales tax | ×1.08\times 1.08 | | 30% discount | ×0.70\times 0.70 |

Example: A $80 item with 25% discount and 8% tax: 80×0.75×1.08=$64.8080 \times 0.75 \times 1.08 = \$64.80

Successive Percent Changes

NEVER add percents! Use multipliers instead.

A 20% increase followed by a 10% decrease: ×1.20×0.90=×1.08\times 1.20 \times 0.90 = \times 1.08 That's an 8% net increase (NOT 10%!).

SAT Question Types

Type 1: Direct Proportion

"If 3 widgets cost $7.50, how much do 8 widgets cost?" 37.50=8x  ⟹  x=$20\frac{3}{7.50} = \frac{8}{x} \implies x = \$20

Type 2: Percent of a Number

"What is 35% of 240?" → 0.35×240=840.35 \times 240 = 84

Type 3: Percent Change

"A price increased from $40 to $52. What is the percent increase?" 52−4040×100=30%\frac{52 - 40}{40} \times 100 = 30\%

Type 4: Working Backward

"After a 20% discount, a jacket costs $56. What was the original price?" Original×0.80=56  ⟹  Original=$70\text{Original} \times 0.80 = 56 \implies \text{Original} = \$70

Type 5: Scale and Maps

"On a map where 1 inch = 25 miles, two cities are 3.5 inches apart. What is the actual distance?" 3.5×25=87.5 miles3.5 \times 25 = 87.5 \text{ miles}

Common SAT Mistakes

  1. Adding successive percents instead of using multipliers
  2. Using the wrong base for percent change (always use the ORIGINAL value)
  3. Inconsistent units in proportions
  4. Confusing "percent of" with "percent more than" — 30% more than 100 is 130, not 30
  5. Forgetting to convert between decimals, fractions, and percents

Conversion Quick Reference

| Fraction | Decimal | Percent | |---|---|---| | 12\frac{1}{2} | 0.50 | 50% | | 13\frac{1}{3} | 0.333... | 33.3% | | 14\frac{1}{4} | 0.25 | 25% | | 15\frac{1}{5} | 0.20 | 20% | | 18\frac{1}{8} | 0.125 | 12.5% | | 34\frac{3}{4} | 0.75 | 75% | | 23\frac{2}{3} | 0.667... | 66.7% |

📚 Practice Problems

1Problem 1easy

❓ Question:

What is 15% of 200?

💡 Show Solution

Solution:

Convert to decimal and multiply: 0.15×200=300.15 \times 200 = 30

Answer: 3030

SAT Tip: 15% = 0.15 (move decimal two places left)

2Problem 2medium

❓ Question:

A price increased from \80toto$100$. What is the percent increase?

💡 Show Solution

Solution:

Use percent change formula: Percent Change=100−8080×100%\text{Percent Change} = \frac{100 - 80}{80} \times 100\%

=2080×100%= \frac{20}{80} \times 100\%

=0.25×100%=25%= 0.25 \times 100\% = 25\%

Answer: 25%25\% increase

3Problem 3hard

❓ Question:

In a class, the ratio of boys to girls is 3:53:5. If there are 40 students total, how many are girls?

💡 Show Solution

Solution:

Total ratio parts: 3+5=83 + 5 = 8

Girls represent 58\frac{5}{8} of total: 58×40=25\frac{5}{8} \times 40 = 25

Answer: 25 girls

Check: Boys = 38(40)=15\frac{3}{8}(40) = 15; Total = 15+25=4015 + 25 = 40 ✓

4Problem 4easy

❓ Question:

The ratio of boys to girls in a class is 3:53:5. If there are 24 students in the class, how many girls are there?

💡 Show Solution

Step 1: Total parts = 3+5=83 + 5 = 8

Step 2: Each part = 248=3\frac{24}{8} = 3 students

Step 3: Girls = 5×3=155 \times 3 = 15

Check: Boys = 3×3=93 \times 3 = 9. Total = 9+15=249 + 15 = 24 ✓

Answer: 15 girls

5Problem 5easy

❓ Question:

The ratio of boys to girls in a class is 3:53:5. If there are 24 students in the class, how many girls are there?

💡 Show Solution

Step 1: Total parts = 3+5=83 + 5 = 8

Step 2: Each part = 248=3\frac{24}{8} = 3 students

Step 3: Girls = 5×3=155 \times 3 = 15

Check: Boys = 3×3=93 \times 3 = 9. Total = 9+15=249 + 15 = 24 ✓

Answer: 15 girls

6Problem 6medium

❓ Question:

A shirt originally priced at $45 is on sale for 20% off. Sales tax of 8% is applied after the discount. What is the total cost?

💡 Show Solution

Step 1: Apply the 20% discount using a multiplier: $45×0.80=$36\$45 \times 0.80 = \$36

Step 2: Apply 8% sales tax: $36×1.08=$38.88\$36 \times 1.08 = \$38.88

One-step method: \45 \times 0.80 \times 1.08 = $38.88$

Answer: $38.88

SAT Tip: Use multipliers for efficiency. Discount of 20% → multiply by 0.80. Tax of 8% → multiply by 1.08.

7Problem 7medium

❓ Question:

A shirt originally priced at $45 is on sale for 20% off. Sales tax of 8% is applied after the discount. What is the total cost?

💡 Show Solution

Step 1: Apply the 20% discount using a multiplier: $45×0.80=$36\$45 \times 0.80 = \$36

Step 2: Apply 8% sales tax: $36×1.08=$38.88\$36 \times 1.08 = \$38.88

One-step method: \45 \times 0.80 \times 1.08 = $38.88$

Answer: $38.88

SAT Tip: Use multipliers for efficiency. Discount of 20% → multiply by 0.80. Tax of 8% → multiply by 1.08.

8Problem 8medium

❓ Question:

A population increased from 12,000 to 15,600 over 5 years. What was the percent increase?

💡 Show Solution

Use the percent change formula: Percent Change=New−OriginalOriginal×100\text{Percent Change} = \frac{\text{New} - \text{Original}}{\text{Original}} \times 100

=15,600−12,00012,000×100= \frac{15{,}600 - 12{,}000}{12{,}000} \times 100

=3,60012,000×100= \frac{3{,}600}{12{,}000} \times 100

=0.30×100=30%= 0.30 \times 100 = 30\%

Answer: 30% increase

Key: Always divide by the ORIGINAL value, not the new value.

9Problem 9medium

❓ Question:

A population increased from 12,000 to 15,600 over 5 years. What was the percent increase?

💡 Show Solution

Use the percent change formula: Percent Change=New−OriginalOriginal×100\text{Percent Change} = \frac{\text{New} - \text{Original}}{\text{Original}} \times 100

=15,600−12,00012,000×100= \frac{15{,}600 - 12{,}000}{12{,}000} \times 100

=3,60012,000×100= \frac{3{,}600}{12{,}000} \times 100

=0.30×100=30%= 0.30 \times 100 = 30\%

Answer: 30% increase

Key: Always divide by the ORIGINAL value, not the new value.

10Problem 10hard

❓ Question:

After a 15% discount, a laptop costs $680. What was the original price?

💡 Show Solution

Step 1: A 15% discount means the customer pays 85% of the original price. Original×0.85=680\text{Original} \times 0.85 = 680

Step 2: Solve for the original price: Original=6800.85=$800\text{Original} = \frac{680}{0.85} = \$800

Check: 800×0.85=680800 \times 0.85 = 680 ✓

Answer: $800

Common mistake: Don't calculate 15% of 680 and add it. That gives 680+102=782680 + 102 = 782, which is WRONG because the 15% should be based on the original price, not the sale price.

11Problem 11hard

❓ Question:

After a 15% discount, a laptop costs $680. What was the original price?

💡 Show Solution

Step 1: A 15% discount means the customer pays 85% of the original price. Original×0.85=680\text{Original} \times 0.85 = 680

Step 2: Solve for the original price: Original=6800.85=$800\text{Original} = \frac{680}{0.85} = \$800

Check: 800×0.85=680800 \times 0.85 = 680 ✓

Answer: $800

Common mistake: Don't calculate 15% of 680 and add it. That gives 680+102=782680 + 102 = 782, which is WRONG because the 15% should be based on the original price, not the sale price.

12Problem 12expert

❓ Question:

The value of an investment increases by 10% in the first year and decreases by 10% in the second year. If the initial investment was $1,000, what is the value after two years, and what is the net percent change?

💡 Show Solution

Step 1: After year 1 (10% increase): $1,000×1.10=$1,100\$1{,}000 \times 1.10 = \$1{,}100

Step 2: After year 2 (10% decrease): $1,100×0.90=$990\$1{,}100 \times 0.90 = \$990

Step 3: Net percent change: 990−10001000×100=−1%\frac{990 - 1000}{1000} \times 100 = -1\%

Answer: $990 after two years; net change is a 1% DECREASE.

Key Insight: A 10% increase followed by a 10% decrease does NOT return to the original value! The multiplier is 1.10×0.90=0.991.10 \times 0.90 = 0.99, which is a 1% net decrease. This is because the 10% decrease is applied to a LARGER number.

13Problem 13expert

❓ Question:

The value of an investment increases by 10% in the first year and decreases by 10% in the second year. If the initial investment was $1,000, what is the value after two years, and what is the net percent change?

💡 Show Solution

Step 1: After year 1 (10% increase): $1,000×1.10=$1,100\$1{,}000 \times 1.10 = \$1{,}100

Step 2: After year 2 (10% decrease): $1,100×0.90=$990\$1{,}100 \times 0.90 = \$990

Step 3: Net percent change: 990−10001000×100=−1%\frac{990 - 1000}{1000} \times 100 = -1\%

Answer: $990 after two years; net change is a 1% DECREASE.

Key Insight: A 10% increase followed by a 10% decrease does NOT return to the original value! The multiplier is 1.10×0.90=0.991.10 \times 0.90 = 0.99, which is a 1% net decrease. This is because the 10% decrease is applied to a LARGER number.

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