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Part 1: Ratios & Rates
Ratios, Proportions & Percentages
Part 1 of 7 โ Ratios and Rates
Ratios
A ratio compares two quantities: a:b or baโ.
If the ratio of boys to girls is 3:5 and there are 40 students total:
- Total parts = 3+5=8
- Boys = (3/8)(40)=15
- Girls = (5/8
Rates
A rate is a ratio with units: miles/hour, dollars/item, people/year.
Unit Rate = rate per one unit. "$7.50 for 3 pounds" โ $2.50 per pound.
Proportions
baโ=dcโ
Example: If 3 widgets cost $14, how much do 7 widgets cost?
143โ=x
Key Takeaways โ Part 1
- Ratio a:b means the total has a+b parts
- Unit rate: divide to find the "per one" rate
- Cross multiplication: a/b=c/d โ
Part 2: Proportional Reasoning
Ratios, Proportions & Percentages
Part 2 of 7 โ Percentages
Percent Basics
p% means 100pโ.
- "What is 15% of 80?" โ
Part 3: Percentages
Ratios, Proportions & Percentages
Part 3 of 7 โ Direct and Inverse Variation
Direct Variation: y=kx
"y is directly proportional to x" means as x doubles, doubles.
Part 4: Unit Conversion
Ratios, Proportions & Percentages
Part 4 of 7 โ Unit Conversions
Dimensional Analysis
Convert units by multiplying by fractions equal to 1:
"Convert 30 mph to feet per second"
30hourmilesโร
Part 5: Scale & Modeling
Ratios, Proportions & Percentages
Part 5 of 7 โ Scale Factors and Similar Figures
Scale Factor
If two figures are similar with a scale factor of k:
- Lengths scale by k
- Areas scale by k2
- Volumes scale by
Part 6: Problem-Solving Workshop
Ratios, Proportions & Percentages
Part 6 of 7 โ Mixture and Work Problems
Mixture Problems
"How many liters of 60% acid solution must be mixed with 10 liters of 20% acid to get a 40% solution?"
Let x = liters of 60% solution:
0.60x+0.20(10)=0.40(x+10)
Part 7: Review & Applications
Ratios, Proportions & Percentages
Part 7 of 7 โ Review & SAT Mixed Practice
Quick Reference
| Topic | Key Formula |
|---|
| Ratio a:b, total T | Part = |
)
(
40
)
=
25
โน
ad=
bc(crossย multiply)
7
โ
โน
3x=
98โน
x=
398โโ
$32.67
ad=bc
Set up proportions with matching units: miles/hours = miles/hours0.15
ร
80=
12
"12 is what percent of 80?" โ 8012โ=0.15=15% "12 is 15% of what?" โ 12=0.15x โ x=80 Percent Change
Percentย change=OldNewโOldโร100%
Example: Price goes from $40 to $52:
4052โ40โร100=30% increase
Multiplier Method (Faster!)
- Increase of p%: multiply by 1+p/100
- Decrease of p%: multiply by 1โp/100
20% increase on $80: 80 \times 1.20 = \96$
15% discount on $200: 200 \times 0.85 = \170$
Successive Percent Changes โ ๏ธ
A 20% increase followed by a 20% decrease is NOT back to the original!
100ร1.20=120, then 120ร0.80=96 โ that's a 4% net decrease.
Key Takeaways โ Part 2
- Percent change: divide the change by the original (not the new) value
- Multiplier method is fastest: increase โ multiply by (1+r), decrease โ (1โr)
- Successive percent changes: multiply the multipliers together
- A% increase then A% decrease โ original (common SAT trap)
y
Constant of proportionality: k=y/x
Example: If y=12 when x=4, then k=3 and y=3x.
Inverse Variation: y=k/x
"y is inversely proportional to x" means as x doubles, y halves.
Product is constant: xy=k
Example: If y=6 when x=8, then k=48 and y=48/x.
When x=12: y=48/12=4
Joint Variation
"z varies directly with x and inversely with y": z=kx/y
SAT Application
Speed and time for a fixed distance: d=rt, so t=d/r โ time is inversely proportional to rate.
If you double your speed, the trip takes half the time.
Key Takeaways โ Part 3
- Direct: y=kx (ratio is constant: y/x=k)
- Inverse: y=k/x (product is constant: xy=k)
- "Proportional" on SAT usually means direct variation (y=kx)
- Check with a known pair to find k, then use k for the new scenario
1ย mile5,280ย feetโร
3,600ย seconds1ย hourโ=
44ย ft/s
Common Conversions (SAT-relevant)
| Given | Conversion |
|---|
| 1 mile | 5,280 feet |
| 1 kilometer | 1,000 meters |
| 1 hour | 60 minutes = 3,600 seconds |
| 1 gallon | 4 quarts |
| 1 pound | 16 ounces |
SAT Unit Conversion Strategy
- Write the starting quantity as a fraction
- Multiply by conversion factors so unwanted units cancel
- Compute the result
The SAT provides conversion factors in the problem โ you don't need to memorize them. Focus on the METHOD of canceling units.
Key Takeaways โ Part 4
- Dimensional analysis: multiply by fractions equal to 1, cancel units
- Set up so unwanted units appear in numerator AND denominator (they cancel)
- The SAT will give you all needed conversion factors โ focus on the method
- Write units at every step to avoid errors
k3
Similar Triangles
Two triangles are similar if they have the same angles (AA similarity).
If triangle A has sides 3, 4, 5 and triangle B has a side of 6 corresponding to 3:
- Scale factor k=6/3=2
- Other sides of B: 4ร2=8 and 5ร2=10
- Area of B = Area of Aรk2=Aareaโร4
Map/Model Problems
"On a map, 1 inch = 25 miles. Two cities are 3.5 inches apart."
Distance =3.5ร25=87.5 miles.
SAT Application
- Similar triangle problems
- Map and blueprint questions
- Geometry problems with dilations
Key Takeaways โ Part 5
- Lengths โ k, Areas โ k2, Volumes โ k3
- Find the scale factor by comparing corresponding lengths
- Map/model problems: set up a proportion with the scale
- This length/area/volume relationship is a frequent SAT topic
0.60x+2=0.40x+4
x=10ย liters Work/Rate Problems
"Pipe A fills a tank in 6 hours, Pipe B in 4 hours. Together?"
- Rate A: 1/6 tank/hour
- Rate B: 1/4 tank/hour
- Combined: 1/6+1/4=2/12+3/12=5/12 tank/hour
- Time: 12/5=2.4 hours
SAT Strategy for Rate Problems
RateรTime=Work
Add rates when working together. The combined rate is always faster than either individual rate.
Key Takeaways โ Part 6
- Mixtures: set up equation where (amount of substance) is equal on both sides
- Work problems: add the rates (not the times), then find time from combined rate
- Rate ร Time = Work (or Amount)
- For work problems: find individual rates, add them, then use time=work/rate
a+ba
โ
ร
T
| Proportion | baโ=dcโ โ cross multiply |
| Percent of | =wholepartโร100 |
| Percent change | =oldnewโoldโร100 |
| Direct variation | y=kx |
| Inverse variation | xy=k |
| Scale: length | รk |
| Scale: area | รk2 |
| Scale: volume | รk3 |
Common SAT Mistakes
- Dividing percent change by the new value instead of the old
- Assuming successive percent increases/decreases cancel out
- Adding times instead of rates in work problems
- Forgetting that the ratio a:b has a+b total parts, not aรb
Key Takeaways โ Part 7
- Ratios, proportions, and percentages are core SAT topics โ expect 4-6 questions per test
- Percent change: always divide by the original
- "Original price" problems: work backwards using the multiplier
- Workers ร time = constant (inverse variation) is a classic SAT setup