Master ratio problems and percent calculations on SAT
How can I study Ratios, Proportions, and Percents effectively?▾
Start by reading the study notes and working through the examples on this page. Then use the flashcards to test your recall. Practice with the 13 problems provided, checking solutions as you go. Regular review and active practice are key to retention.
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Ratios, Proportions, and Percents is part of the SAT Prep course on Study Mondo, specifically in the Problem Solving and Data Analysis section. You can explore the full course for more related topics and practice resources.
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Ratio Problems Strategy
If the ratio of cats to dogs is 3:5 and there are 40 animals total:
Total parts = 3+5=8
Each part = 840=5
Cats = 3×5=15
Dogs = 5×5=25
Part-to-Part vs. Part-to-Whole
Part-to-Part: cats to dogs = 3:5
Part-to-Whole: cats to total = 3:8 or 83
Proportions
A proportion is an equation stating two ratios are equal:
ba=dc
Cross multiplication:ad=bc
Setting Up Proportions
Keep consistent units on corresponding sides.
Correct:hours1miles1=hours2miles2
Wrong:hours1miles1=miles2hours2
Unit Conversions
Use dimensional analysis (multiply by conversion factors):
Example: Convert 45 mph to feet per second.
45hourmiles×1 mile5280 feet×3600 seconds1 hour=66secondfeet
Percents
Key Formulas
Percent=WholePart×100
Part=100Percent×Whole
Percent Change
Percent Change=OriginalNew−Original×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
15% decrease
×0.85
8% sales tax
×1.08
30% discount
×0.70
Example: A $80 item with 25% discount and 8% tax:
80×0.75×1.08=$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
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?"
7.503=x8⟹x=$20
Type 2: Percent of a Number
"What is 35% of 240?" → 0.35×240=84
Type 3: Percent Change
"A price increased from $40 to $52. What is the percent increase?"
4052−40×100=30%
Type 4: Working Backward
"After a 20% discount, a jacket costs $56. What was the original price?"
Original×0.80=56⟹Original=$70
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 miles
Common SAT Mistakes
Adding successive percents instead of using multipliers
Using the wrong base for percent change (always use the ORIGINAL value)
Inconsistent units in proportions
Confusing "percent of" with "percent more than" — 30% more than 100 is 130, not 30
Forgetting to convert between decimals, fractions, and percents
Conversion Quick Reference
Fraction
Decimal
Percent
21
0.50
50%
31
0.333...
33.3%
41
0.25
25%
51
0.20
20%
81
0.125
12.5%
43
0.75
75%
32
0.667...
66.7%
Answer:
30
SAT Tip: 15% = 0.15 (move decimal two places left)
2Problem 2medium
❓ Question:
A price increased from \80to$100$. What is the percent increase?
💡 Show Solution
Solution:
Use percent change formula:
Percent Change=80100−80×100%
=8020×100%
=0.25×100%=25%
Answer:25% increase
3Problem 3hard
❓ Question:
In a class, the ratio of boys to girls is 3:5. If there are 40 students total, how many are girls?
💡 Show Solution
Solution:
Total ratio parts: 3+5=8
Girls represent 85 of total:
85
Answer: 25 girls
Check: Boys = 83(40)=15; Total = 15+25= ✓
4Problem 4easy
❓ Question:
The ratio of boys to girls in a class is 3:5. If there are 24 students in the class, how many girls are there?
💡 Show Solution
Step 1: Total parts = 3+5=8
Step 2: Each part = 824=3 students
Step 3: Girls = 5×3=15
Check: Boys = 3×3=9. Total = 9+15=24 ✓
Answer: 15 girls
5Problem 5easy
❓ Question:
The ratio of boys to girls in a class is 3:5. If there are 24 students in the class, how many girls are there?
💡 Show Solution
Step 1: Total parts = 3+5=8
Step 2: Each part = 824=3 students
Step 3: Girls = 5×3=15
Check: Boys = 3×3=9. Total = 9+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
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=OriginalNew−Original×100
=12,00015,600−12,000×100
=12,0003,600×100
=0.30×100=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=OriginalNew−Original×100
=12,00015,600−12,000×100
=12,0003,600×100
=0.30×100=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
Step 2: Solve for the original price:
Original=0.85680=$800
Check:800×0.85=680 ✓
Answer: $800
Common mistake: Don't calculate 15% of 680 and add it. That gives 680+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
Step 2: Solve for the original price:
Original=0.85680=$800
Check:800×0.85=680 ✓
Answer: $800
Common mistake: Don't calculate 15% of 680 and add it. That gives 680+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
Step 2: After year 2 (10% decrease):
$1,100×0.90=$990
Step 3: Net percent change:
1000990−1000×100=−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=0.99, 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
Step 2: After year 2 (10% decrease):
$1,100×0.90=$990
Step 3: Net percent change:
1000990−1000×100=−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=0.99, which is a 1% net decrease. This is because the 10% decrease is applied to a LARGER number.
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Yes, this page includes 13 practice problems with detailed solutions. Each problem includes a step-by-step explanation to help you understand the approach.