SAT Ratios, Rates, Proportional Relationships & Units
SAT Ratios, Rates, Proportional Relationships & Units
Quantitative relationships, unit reasoning, conversions, scale factors, and real-world applications
What this lesson covers
The SAT uses ratios, rates, proportional relationships, and units in many different contexts. The same underlying idea can appear as a scale drawing, a scientific measurement, a population density, an exchange rate, or a multistep unit conversion.
The goal is not to memorize one procedure. The goal is to recognize the relationship between the quantities and keep the units consistent.
The SAT mindset
Ask what one unit means.
If a quantity is given as “12 dollars per hour,” then the number 12 tells you how many dollars correspond to 1 hour.
One idea connects the whole lesson
Ratio
Compares two quantities.
Rate
Compares quantities with different units.
Proportion
States that two ratios are equivalent.
Derived unit
Combines units through multiplication or division.
1. Ratios: identify what is being compared
A ratio describes a relationship between two quantities. Before calculating, identify exactly what each number represents.
Part-to-part
If there are 3 red markers for every 5 blue markers, the ratio of red to blue is \(3:5\).
Part-to-whole
If there are 3 red and 5 blue markers, the ratio of red markers to all markers is \(3:8\).
2. Proportional relationships
Two quantities are proportional when their ratio stays constant.
Here, \(k\) is the constant of proportionality. In context, \(k\) often has units.
For example, if a printer produces 18 pages per minute, then after \(t\) minutes:
The constant \(18\) means 18 pages for every 1 minute.
3. Scale factors
If two quantities are proportional and one changes by a scale factor, the other changes by the same factor.
Example: A model is enlarged from 20 cm to 30 cm in length.
If another proportional length was 16 cm, its new length is \(1.5*16=24\) cm.
4. Scale drawings
A scale drawing gives a proportional relationship between a measurement on the drawing and the corresponding real-world measurement.
If 1 inch represents 4 feet, then 7.5 inches represents:
5. Rates and unit rates
A rate compares quantities with different units. A unit rate tells how much corresponds to exactly 1 unit of the second quantity.
| Rate | Meaning | Unit rate |
|---|---|---|
| 180 miles in 3 hours | 180 miles for every 3 hours | 60 miles/hour |
| $42 for 6 kilograms | $42 for every 6 kilograms | $7/kg |
6. Units are part of the mathematics
Units are not decoration. They help you decide which operation makes sense.
Watch the units cancel
The kilometers and hours cancel, leaving meters per second.
7. One-step unit conversions
When the conversion factor is given, multiply by a fraction equal to 1 so that the unwanted unit cancels.
8. Multistep and multidimensional conversions
Some SAT questions require more than one conversion. Treat each conversion as a separate step.
The same idea works when the units involve area, volume, or compound rates. Do not skip a conversion just because the numbers look familiar.
9. Derived units
A derived unit is created by combining other units through multiplication or division.
Quotient
Population density
Product
Energy use over time
10. Combining rates correctly
When two regions, groups, or sources are combined, first find their totals. Then calculate the rate for the combined group.
Suppose Region A has area \(40\text{ km}^2\) and density \(300\) people/km².
Its population is:
If Region B has area \(60\text{ km}^2\) and density \(500\) people/km²:
Combined density:
11. Natural-science and social-science applications
College Board can place proportional reasoning inside a scientific or social-science context. The context may look unfamiliar, but the mathematics is often a familiar ratio, rate, or unit relationship.
Science
mass per volume, distance per time, energy per time, concentration, density
Social science
population density, survey rates, economic rates, exchange rates, production rates
12. Exchange rates are unit rates
An exchange rate is simply a conversion factor between currencies.
Then 4,800 units correspond to:
13. A reliable SAT strategy
When the problem looks complicated
Advanced Practice
These original SATMath800 questions are modeled on the kinds of quantitative relationships, unit conversions, scale factors, derived units, and real-world contexts represented in College Board’s official SAT materials.
A school club has red and blue folders in a ratio of 3 to 5. If the club has 64 folders in total, how many of the folders are blue?
- A) 24
- B) 40
- C) 32
- D) 36
A recipe uses 6 ounces of cheese for every 8 servings. At this rate, how many ounces of cheese are needed for 20 servings?
- A) 15
- B) 12
- C) 13
- D) 14
On a scale drawing, 1 inch represents 4 feet. A wall measures 9.5 inches on the drawing. What is the actual length of the wall, in feet?
- A) 19
- B) 28
- C) 38
- D) 34
A runner’s average pace is 5.6 minutes per mile. At this pace, how many minutes will it take the runner to complete 8.5 miles?
- A) 42.4
- B) 45.6
- C) 46.8
- D) 47.6
One gallon is equal to 128 fluid ounces. How many fluid ounces are in 3.5 gallons?
- A) 384
- B) 448
- C) 416
- D) 256
A cyclist travels at a constant speed of 72 kilometers per hour. What is this speed, in meters per second?
- A) 20
- B) 24
- C) 30
- D) 36
A granite sample has a mass of 18 kilograms and a volume of 6 cubic meters. What is the density of the granite, in kilograms per cubic meter?
- A) 0.33
- B) 12
- C) 3
- D) 108
Region A has an area of 40 square kilometers and a population density of 300 people per square kilometer. Region B has an area of 60 square kilometers and a population density of 500 people per square kilometer. What is the population density of the two regions combined?
- A) 320
- B) 380
- C) 400
- D) 420
A water pump moves water at a constant rate of 2.4 liters per minute. How many liters of water will the pump move in 35 minutes?
- A) 72
- B) 84
- C) 78
- D) 92
The length of a model is increased from 20 centimeters to 30 centimeters. If the original mass of the model is 240 grams and mass is directly proportional to the length for this model, what is the new mass?
- A) 280
- B) 300
- C) 360
- D) 320
A laboratory solution contains 2.5 milligrams of a substance per liter of solution. How many milligrams of the substance are contained in 18 liters of the solution?
- A) 45
- B) 40
- C) 36
- D) 20
An exchange rate is 150 foreign currency units for 1 US dollar. An item costs 4,800 foreign currency units. What is the cost of the item in US dollars?
- A) 24
- B) 32
- C) 36
- D) 40
An exchange rate is 150 foreign currency units for 1 US dollar. A traveler buys an item for 7,200 foreign currency units. A service fee equal to 8% of the item’s price in US dollars is added to the purchase. What is the total cost, in US dollars?
- A) 49.92
- B) 50.40
- C) 51.20
- D) 51.84
A car uses fuel at a rate of 6.4 liters per 100 kilometers. The car travels 375 kilometers. If 1 gallon is equivalent to 3.8 liters, approximately how many gallons of fuel does the car use?
- A) 5.26
- B) 5.84
- C) 6.32
- D) 6.08
A travel card charges a 3% fee when money is loaded onto the card. After the fee is deducted, each US dollar on the card can be exchanged for 0.92 euros. What is the least number of US dollars that must be loaded onto the card so that at least 180 euros can be spent?
- A) 196
- B) 198
- C) 200
- D) 202
Final check: what should you notice?
Ratio
What is being compared?
Rate
What does one unit correspond to?
Proportion
Does the relationship stay constant?
Units
Do the units lead to the requested unit?
Remember: A complicated-looking context can still contain a simple proportional relationship. Strip away the story, identify the quantities, and let the units guide the mathematics.
Official references: SAT Problem-Solving and Data Analysis · Student Question Bank: Math · Assessment Framework
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