Ratios, Rates, Proportional Relationships, and Units

Ratios, Rates, Proportional Relationships, and Units

SAT Math | Problem-Solving and Data Analysis | Foundation to Advanced

Ratios, rates, and proportions help us compare quantities and solve real-world problems involving measurements, prices, distances, time, recipes, maps, and scientific data.

On the Digital SAT, you must understand how quantities relate to one another, use proportional reasoning, convert between units, and correctly interpret rates with different units.

1. Understanding Ratios

A ratio compares two or more quantities.

For example, if a classroom contains 12 boys and 18 girls, the ratio of boys to girls is:

12:18

Divide both parts by 6:

2:3

This means that for every 2 boys, there are 3 girls.

A ratio can be written in three ways:

2:3

2 to 3

2/3

All three forms represent the same comparison.

Part-to-part and part-to-whole ratios

A part-to-part ratio compares two separate groups.

Example:

Boys to girls = 2:3

A part-to-whole ratio compares one group with the total.

Because there are 2 + 3 = 5 total parts:

Boys to all students = 2:5

Girls to all students = 3:5

Important: Do not confuse the number of ratio parts with the number of actual objects.

Worked Example 1 — Ratio of two groups

Difficulty: Easy

The ratio of red marbles to blue marbles in a container is 3:5.

If there are 24 red marbles, how many blue marbles are there?

Step 1: Identify the ratio.

Red : Blue = 3 : 5

Step 2: Find the scale factor.

24 ÷ 3 = 8

Step 3: Multiply the blue part by 8.

5 × 8 = 40

Final answer: 40 blue marbles

SAT tip

When a ratio is given, label the quantities before calculating.

The ratio 3:5 is different from 5:3 because the order of the quantities matters.

2. Dividing a Total Using Ratios

SAT questions may provide a total quantity and a ratio.

To find one group, first determine the total number of ratio parts.

General method

If the ratio of A to B is a and the total is T:

Total parts = a + b

Quantity A = [a/(a + b)] × T

Quantity B = [b/(a + b)] × T

Worked Example 2 — Dividing a total

Difficulty: Medium

The ratio of adults to children at an event is 4:7.

There are 165 people altogether.

How many children are present?

Step 1: Add the ratio parts.

4 + 7 = 11

Step 2: Find the value of one part.

165 ÷ 11 = 15

Step 3: Multiply by the number of child parts.

7 × 15 = 105

Final answer: 105 children

Ratios involving three quantities

Suppose the ratio A:B is 2:3:5.

The total number of parts is:

2 + 3 + 5 = 10

If the total quantity is 200:

A = (2/10) × 200 = 40

B = (3/10) × 200 = 60

C = (5/10) × 200 = 100

Common mistake: Using only two parts of a three-part ratio when calculating the total.

3. Understanding Rates and Unit Rates

A rate compares two quantities measured in different units.

Examples include:

60 miles per hour

15 dollars per notebook

8 liters per minute

250 people per square kilometer

A unit rate is a rate expressed per one unit of the second quantity.

Finding a unit rate

Unit rate = Total quantity ÷ Number of units

For example:

Five notebooks cost $17.50.

Cost per notebook:

17.50 ÷ 5 = 3.50

The unit rate is $3.50 per notebook.

Worked Example 3 — Unit price

Difficulty: Easy

A store sells 8 identical notebooks for $28.

What is the cost of 13 notebooks at the same rate?

Step 1: Find the price per notebook.

28 ÷ 8 = 3.50

Step 2: Multiply by 13.

13 × 3.50 = 45.50

Final answer: $45.50

Comparing unit rates

Suppose:

Store A sells 6 notebooks for $21.

Store B sells 8 notebooks for $26.

Store A:

21 ÷ 6 = $3.50 per notebook

Store B:

26 ÷ 8 = $3.25 per notebook

Store B offers the lower unit price.

SAT tip: When comparing two prices, speeds, or other rates, convert them to the same units before comparing.

4. Proportional Relationships

Two quantities are in a proportional relationship when one is always a constant multiple of the other.

The general equation is:

y = kx

where k is the constant of proportionality.

For example:

y = 4x

This means y is always four times x.

The ratio y/x is always 4 for every nonzero x in the table.

Therefore, the relationship is proportional.

Identifying proportional relationships

A relationship is proportional if:

• The ratio y/x is constant for nonzero inputs.

• Its equation can be written as y = kx.

• Its graph is a straight line passing through the origin.

A linear relationship such as:

y = 4x + 3

is not proportional because the starting value is 3 rather than 0.

Worked Example 4 — Constant of proportionality

Difficulty: Medium

The variables x and y are proportional.

When x = 6, y = 42.

What is y when x = 11?

Step 1: Find the constant of proportionality.

k = y/x

k = 42/6 = 7

Step 2: Write the relationship.

y = 7x

Step 3: Substitute x = 11.

y = 7(11) = 77

Final answer: 77

Direct variation

The expression direct variation is another name for a proportional relationship.

For example:

Distance = Speed × Time

When speed remains constant, distance is directly proportional to time.

Doubling the travel time doubles the distance.

Tripling the travel time triples the distance.

5. Solving Proportions

A proportion is an equation stating that two ratios are equal.

For example:

3/5 = x/20

A common method is cross-multiplication.

Cross-multiplication

For:

a/b = c/d

where b and d are nonzero:

ad = bc

Worked Example 5 — Solving a proportion

Difficulty: Easy

Solve:

4/7 = x/35

Step 1: Cross-multiply.

4 × 35 = 7x

Step 2: Simplify.

140 = 7x

Step 3: Divide by 7.

x = 20

Final answer: 20

SAT shortcut

If the denominator changes from 7 to 35, it has been multiplied by 5.

Therefore, the numerator must also be multiplied by 5:

4 × 5 = 20

This is often faster than cross-multiplication.

When proportions are useful

Proportions commonly appear in problems involving:

• Recipes

• Maps

• Similar figures

• Exchange rates

• Scale drawings

• Production rates

• Measurement conversions

6. Ratios with Changing Quantities

Harder SAT questions may involve a ratio that changes after something is added or removed.

Worked Example 6 — Changing a ratio

Difficulty: Hard

The ratio of red beads to blue beads is 2:3.

After 12 red beads are added, the ratio becomes 4:3.

How many blue beads were originally present?

Step 1: Represent the original quantities.

Red = 2k

Blue = 3k

Step 2: Write the new ratio.

(2k + 12)/(3k) = 4/3

Step 3: Cross-multiply.

3(2k + 12) = 4(3k)

Step 4: Expand.

6k + 36 = 12k

Step 5: Solve.

36 = 6k

k = 6

Step 6: Find the original blue beads.

Blue = 3k

Blue = 18

Final answer: 18 blue beads

SAT tip

When the original ratio is a, represent the quantities as ak and bk.

This is more reliable than assuming the original quantities are exactly a and b.

7. Speed, Distance, and Time

Speed is a rate comparing distance with time.

The main formula is:

Speed = Distance / Time

Rearranging:

Distance = Speed × Time

Time = Distance / Speed

Worked Example 7 — Average speed

Difficulty: Easy

A vehicle travels 180 miles in 3 hours at a constant speed.

What is its speed?

Speed = 180/3

Speed = 60

Final answer: 60 miles per hour

Worked Example 8 — Finding travel time

Difficulty: Medium

A train travels at a constant speed of 72 kilometers per hour.

How long will it take to travel 198 kilometers?

Step 1: Use:

Time = Distance / Speed

Step 2: Substitute.

Time = 198/72

Time = 2.75 hours

Step 3: Interpret the decimal.

0.75 hour = 45 minutes

Final answer: 2 hours 45 minutes

Average speed

Average speed is:

Total distance / Total time

It is not necessarily the simple average of two given speeds.

For example, if a vehicle travels equal distances at two different speeds, the slower portion takes more time and affects the overall average.

Common mistake: Averaging two speeds without considering time or distance.

8. Converting Units of Length, Mass, and Volume

Unit conversion changes how a quantity is expressed without changing its actual magnitude.

Common length conversions

1 kilometer = 1,000 meters

1 meter = 100 centimeters

1 centimeter = 10 millimeters

1 mile = 5,280 feet

1 foot = 12 inches

1 inch = 2.54 centimeters

Common mass conversions

1 kilogram = 1,000 grams

1 gram = 1,000 milligrams

Common volume conversions

1 liter = 1,000 milliliters

1 milliliter = 1 cubic centimeter

1 cubic meter = 1,000 liters

Worked Example 9 — Length conversion

Difficulty: Easy

A road is 3.6 kilometers long.

What is its length in meters?

Step 1: Use the conversion.

1 kilometer = 1,000 meters

Step 2: Multiply.

3.6 × 1,000 = 3,600

Final answer: 3,600 meters

Dimensional analysis

A reliable conversion method is to multiply by a conversion fraction equal to 1.

For example:

3.6 km × (1,000 m / 1 km) = 3,600 m

The kilometer units cancel, leaving meters.

SAT tip: Write units throughout multi-step calculations to make sure the correct units cancel.

9. Converting Rates and Compound Units

Rates contain two units, so both units may need conversion.

Examples:

Miles per hour

Meters per second

Liters per minute

Grams per cubic centimeter

Worked Example 10 — Speed conversion

Difficulty: Medium

A car travels at 72 kilometers per hour.

What is its speed in meters per second?

Step 1: Convert kilometers to meters.

72 kilometers = 72,000 meters

Step 2: Convert hours to seconds.

1 hour = 3,600 seconds

Step 3: Divide.

72,000/3,600 = 20

Final answer: 20 meters per second

Shortcut

To convert kilometers per hour to meters per second:

Divide by 3.6

To convert meters per second to kilometers per hour:

Multiply by 3.6

Worked Example 11 — Miles per hour to feet per minute

Difficulty: Medium

A vehicle travels at 55 miles per hour.

What is its speed in feet per minute?

Step 1: Convert miles to feet.

55 × 5,280 = 290,400 feet per hour

Step 2: Convert hours to minutes.

290,400 ÷ 60 = 4,840

Final answer: 4,840 feet per minute

Common mistake: Multiplying by 60 instead of dividing when converting a rate from per hour to per minute.

10. Converting Area and Volume Units

Area units are squared.

Volume units are cubed.

This is important because converting area or volume requires squaring or cubing the corresponding length conversion factor.

Area conversion

Since:

1 meter = 100 centimeters

then:

1 square meter = 100² square centimeters

1 m² = 10,000 cm²

Volume conversion

Since:

1 meter = 100 centimeters

then:

1 cubic meter = 100³ cubic centimeters

1 m³ = 1,000,000 cm³

Worked Example 12 — Area conversion

Difficulty: Medium

A surface has an area of 2.5 square meters.

What is its area in square centimeters?

Step 1: Use:

1 m² = 10,000 cm²

Step 2: Multiply.

2.5 × 10,000 = 25,000

Final answer: 25,000 cm²

Worked Example 13 — Volume conversion

Difficulty: Medium

A container has a volume of 0.018 cubic meters.

What is its volume in liters?

Step 1: Use:

1 m³ = 1,000 liters

Step 2: Multiply.

0.018 × 1,000 = 18

Final answer: 18 liters

SAT tip: Never use a simple length conversion factor for area or volume without accounting for the squared or cubed units.

11. Scale Drawings and Maps

A scale drawing represents a real object using proportional dimensions.

For example:

1 centimeter on a map represents 5 kilometers in reality.

This means:

1 cm : 5 km

Worked Example 14 — Map scale

Difficulty: Medium

On a map, 1 centimeter represents 12 kilometers.

Two cities are 5.5 centimeters apart on the map.

What is the actual distance between the cities?

Step 1: Identify the scale.

1 cm = 12 km

Step 2: Multiply.

5.5 × 12 = 66

Final answer: 66 kilometers

Scale factor

If a figure is enlarged by a scale factor of k:

• Every length is multiplied by k.

• Its area is multiplied by k².

• Its volume is multiplied by k³.

Worked Example 15 — Area under scaling

Difficulty: Hard

A rectangular drawing measures 3 centimeters by 5 centimeters.

Every centimeter on the drawing represents 0.8 kilometer in reality.

What is the real area represented by the rectangle?

Step 1: Convert the first dimension.

3 × 0.8 = 2.4 km

Step 2: Convert the second dimension.

5 × 0.8 = 4 km

Step 3: Find the area.

2.4 × 4 = 9.6

Final answer: 9.6 square kilometers

SAT tip

When a problem asks for actual area, convert both dimensions before multiplying, or square the scale factor.

12. Density and Other Derived Rates

A derived unit combines two or more basic units.

Examples include:

Grams per cubic centimeter

Kilometers per liter

People per square kilometer

Kilowatt-hours

Density

Density compares mass with volume.

Density = Mass / Volume

Rearranging:

Mass = Density × Volume

Volume = Mass / Density

Worked Example 16 — Density

Difficulty: Medium

A material has a mass of 378 grams and a volume of 140 cubic centimeters.

What is its density?

Step 1: Use the formula.

Density = Mass / Volume

Step 2: Substitute.

Density = 378/140

Step 3: Calculate.

Density = 2.7

Final answer: 2.7 grams per cubic centimeter

Population density

Population density is:

Population / Area

If a region has 12,000 residents and covers 40 square kilometers:

Density = 12,000/40

Density = 300 residents per square kilometer.

Fuel efficiency

Fuel efficiency can be measured as:

Miles per gallon = Distance / Fuel used

If a vehicle travels 240 miles using 8 gallons:

240/8 = 30 miles per gallon

Electric energy

Electrical energy consumption can be measured in kilowatt-hours (kWh).

Energy (kWh) = Power (kW) × Time (hours)

For example, a 2-kilowatt appliance running for 3 hours uses:

2 × 3 = 6 kWh

Important: A kilowatt-hour is a unit of energy, not power.

13. Working with Multiple Rates

Some SAT questions combine several rates or require determining a total based on repeated production.

For example:

A machine produces 20 items per hour.

In 6 hours, it produces:

20 × 6 = 120 items

If three identical machines operate independently at the same rate, their combined production is:

3 × 120 = 360 items

Worked Example 17 — Production rates

Difficulty: Hard

Three identical machines produce 480 items in 8 hours.

At the same rate, how many items can five machines produce in 6 hours?

Step 1: Find the total machine-hours in the first situation.

3 × 8 = 24 machine-hours

Step 2: Calculate production per machine-hour.

480 ÷ 24 = 20 items

Step 3: Find the new machine-hours.

5 × 6 = 30 machine-hours

Step 4: Calculate production.

30 × 20 = 600

Final answer: 600 items

SAT tip

When multiple machines, workers, or identical devices operate at constant independent rates, determine the output per machine-hour or worker-hour first.

14. Currency Conversion and Multistep Unit Problems

Unit conversions can involve currencies and several stages of calculation.

A problem will provide any exchange rates needed.

Worked Example 18 — Currency exchange

Difficulty: Medium

Suppose the exchange rates are:

1 British pound = 1.25 US dollars

1 US dollar = 145 Japanese yen

How many Japanese yen are equivalent to 80 British pounds?

Step 1: Convert pounds to dollars.

80 × 1.25 = 100 dollars

Step 2: Convert dollars to yen.

100 × 145 = 14,500 yen

Final answer: 14,500 Japanese yen

Multistep unit conversions

Suppose a pump moves 3,600 liters of water in 2.5 hours.

What is its rate in milliliters per second?

Step 1: Convert liters to milliliters.

3,600 × 1,000 = 3,600,000 mL

Step 2: Convert hours to seconds.

2.5 × 3,600 = 9,000 seconds

Step 3: Calculate the rate.

3,600,000 ÷ 9,000 = 400

Final answer: 400 milliliters per second

SAT tip: When a problem contains multiple conversions, write the units at every stage. The final units should match exactly what the question asks for.

15. SAT Strategies and Common Mistakes

Strategy 1 — Label every quantity

In ratio problems, write what each ratio part represents.

For example:

Red : Blue = 3 : 5

Do not accidentally reverse the groups.

Strategy 2 — Find the unit rate first

When comparing prices or rates, calculate the amount per one unit.

This often simplifies complicated problems.

Strategy 3 — Recognize proportionality

A proportional relationship has a constant ratio and can be expressed as:

y = kx

A nonzero starting value means the relationship is not directly proportional.

Strategy 4 — Use dimensional analysis

Write units as part of the calculation.

Cancel units to verify that the result is in the required measurement system.

Strategy 5 — Convert area and volume correctly

If lengths are scaled by k, area changes by k² and volume by k³.

Strategy 6 — Watch for multistep conditions

Some questions require combining ratios, unit conversions, and rates.

Read all conditions before calculating.

Strategy 7 — Check units in your answer

If the question asks for meters per second, an answer in kilometers per hour is not the requested final answer.

Strategy 8 — Use Desmos when helpful

The Digital SAT includes a built-in Desmos calculator.

Use it to evaluate proportions, calculate conversions, and check multistep calculations.

However, understanding the units and relationships is essential.

Common Mistake 1 — Reversing ratio order

A ratio of 2:5 is not the same as 5:2 when the quantities are ordered.

Common Mistake 2 — Confusing part-to-part and part-to-whole ratios

If boys = 2:3, the fraction of all students who are boys is 2/5, not 2/3.

Common Mistake 3 — Ignoring the total number of parts

For a three-part ratio of 2:3:5, the total number of parts is 10.

Common Mistake 4 — Assuming every linear relationship is proportional

y = 3x is proportional.

y = 3x + 5 is not proportional.

Common Mistake 5 — Using different units without converting

A rate measured in kilometers per hour cannot be compared directly with a rate measured in meters per second without converting.

Common Mistake 6 — Reversing a conversion factor

When changing from larger units to smaller units, the numerical value usually increases.

For example:

2 kilometers = 2,000 meters.

Common Mistake 7 — Forgetting squared and cubed units

1 m² is not 100 cm².

It is 10,000 cm².

Common Mistake 8 — Averaging rates incorrectly

Average speed is total distance divided by total time.

It is not always the arithmetic mean of the speeds for separate parts of a trip.

Common Mistake 9 — Ignoring the number of machines or workers

For identical machines operating at the same rate, total production depends on both the number of machines and the operating time.

Common Mistake 10 — Using the wrong units for density

Density may be expressed in grams per cubic centimeter, kilograms per cubic meter, or other equivalent units.

Check the required units before answering.

16. Check Your Understanding

Question 1 — Ratio

The ratio of cats to dogs at a shelter is 3:4.

If there are 21 cats, how many dogs are there?

A. 24

B. 28

C. 32

D. 36

Correct answer: B — 28

The scale factor is:

21 ÷ 3 = 7

Dogs = 4 × 7 = 28.

Question 2 — Unit rate

Six identical notebooks cost $21.

At the same rate, what is the cost of 10 notebooks?

A. $30

B. $32

C. $35

D. $40

Correct answer: C — $35

Price per notebook:

21 ÷ 6 = $3.50

Cost for 10:

10 × 3.50 = $35.

Question 3 — Proportional relationship

The variables x and y are proportional.

When x = 4, y = 18.

What is y when x = 10?

A. 40

B. 45

C. 50

D. 54

Correct answer: B — 45

Constant of proportionality:

18/4 = 4.5

Therefore:

y = 4.5(10) = 45.

Question 4 — Unit conversion

A vehicle travels at 90 kilometers per hour.

What is its speed in meters per second?

A. 20

B. 25

C. 30

D. 32

Correct answer: B — 25

Convert:

90 × 1,000 / 3,600 = 25

Question 5 — Map scale

A map uses a scale of 1 centimeter to 8 kilometers.

Two towns are 6.5 centimeters apart.

What is their actual distance?

A. 40 km

B. 48 km

C. 52 km

D. 56 km

Correct answer: C — 52 km

Actual distance:

6.5 × 8 = 52 km.

Question 6 — Density

A substance has a mass of 450 grams and a volume of 150 cubic centimeters.

What is its density?

Correct answer: 3 grams per cubic centimeter

Density = Mass / Volume

450/150 = 3

17. Lesson Summary

A ratio compares quantities using division.

Part-to-part ratios compare separate groups, while part-to-whole ratios compare a group with the total.

A rate compares quantities measured in different units.

A unit rate expresses a rate per one unit of another quantity.

A proportional relationship has the form y = kx.

The constant k is called the constant of proportionality.

Proportions can be solved using scaling or cross-multiplication.

Speed is distance divided by time.

Unit conversions change the measurement units without changing the actual quantity.

Area conversions use squared length factors.

Volume conversions use cubed length factors.

Scale drawings preserve proportional lengths.

Density is mass divided by volume.

Derived units combine measurements, such as kilometers per hour, grams per cubic centimeter, and kilowatt-hours.

Multistep SAT problems may combine ratios, rates, conversions, and real-world constraints.

You are now ready to complete Ratios, Rates, Proportional Relationships, and Units practice questions in Baloo Prep.