Ratios, Rates, Proportions & Units – SAT Math

📖 What Are Ratios, Rates, and Proportions?

Ratio

$3:5$ or $\frac{3}{5}$

A comparison of two quantities.

Rate

$60$ miles/hour

A ratio with different units.

Proportion

$\frac{a}{b} = \frac{c}{d}$

Two ratios set equal to each other.

Unit Rate

$\$2.50$ per pound

A rate with denominator 1.

1. Ratios

📌 RULE: Writing Ratios

A ratio compares two quantities. It can be written in three ways:

\[ a:b \quad \text{or} \quad \frac{a}{b} \quad \text{or} \quad a \text{ to } b \]

Example: There are 5 boys and 7 girls in a class.
\[ \text{Ratio of boys to girls} = 5:7 \]
\[ \text{Ratio of girls to boys} = 7:5 \]
\[ \text{Ratio of boys to total students} = 5:12 \]

📌 RULE: Simplifying Ratios

Divide both terms by their greatest common factor (GCF).

Example: $12:18$
\[ \text{GCF} = 6 \quad \rightarrow \quad 12:18 = 2:3 \]

💡 Strategy — Ratios

1. Identify what two quantities are being compared.
2. Write them in the correct order.
3. Simplify by dividing by the GCF.

📝 SOLVED EXAMPLE 1 — Ratio

In a bag of marbles, there are 8 red marbles and 12 blue marbles. What is the ratio of red marbles to blue marbles?

Step 1: Red marbles = 8, Blue marbles = 12

Step 2: Ratio = $8:12$

Step 3: Simplify: GCF = 4 → $\color{var(--math)}{2:3}$

✅ $2:3$

💡 Tip: Always simplify ratios to their simplest form.

2. Rates

📌 RULE: Rates

A rate is a ratio that compares different units.

Examples:
\[ \text{Speed} = \frac{\text{distance}}{\text{time}} = 60 \text{ miles/hour} \]
\[ \text{Price} = \frac{\text{cost}}{\text{quantity}} = \$2.50 \text{ per pound} \]
\[ \text{Density} = \frac{\text{mass}}{\text{volume}} = 1.5 \text{ g/cm}^3 \]

💡 Strategy — Rates

1. Identify the two different units being compared.
2. Write as a fraction: $\frac{\text{unit 1}}{\text{unit 2}}$.
3. Simplify by dividing the numerator and denominator by the same factor.

📝 SOLVED EXAMPLE 2 — Rate

A car travels 240 miles in 4 hours. What is the speed in miles per hour?

Step 1: Speed = $\frac{\text{distance}}{\text{time}}$

Step 2: Speed = $\frac{240 \text{ miles}}{4 \text{ hours}}$

Step 3: $\color{var(--math)}{60 \text{ miles/hour}}$

✅ $60$ miles per hour

💡 Tip: "Per" means "for each" — it indicates division.

3. Proportions

📌 RULE: Proportions

A proportion states that two ratios are equal:

\[ \frac{a}{b} = \frac{c}{d} \]

Cross-Multiplication: \[ a \times d = b \times c \]

Example: If $\frac{3}{4} = \frac{x}{12}$, then $3 \times 12 = 4 \times x$ → $36 = 4x$ → $x = 9$

📌 RULE: Direct Proportion

When two quantities are directly proportional, they increase or decrease together at the same rate.

\[ y = kx \quad \text{or} \quad \frac{y}{x} = k \]
where $k$ is the constant of proportionality.

Example: The cost of apples is directly proportional to the weight. If 2 pounds cost \$4, then 5 pounds cost \$10.

💡 Strategy — Proportions

1. Set up the proportion with corresponding values in the same positions.
2. Cross-multiply to solve for the unknown.
3. Check that the units are consistent.

📝 SOLVED EXAMPLE 3 — Proportion

If 6 pounds of apples cost \$9, how much would 10 pounds cost?

Step 1: Set up proportion: $\frac{6}{9} = \frac{10}{x}$

Step 2: Cross-multiply: $6x = 90$

Step 3: $x = \color{var(--math)}{15}$

✅ $\$15$

💡 Tip: You can also find the unit price: \$9 ÷ 6 = \$1.50 per pound × 10 = \$15.

4. Unit Conversions

📌 RULE: Unit Conversion

Use conversion factors to change from one unit to another.

\[ \text{Original unit} \times \frac{\text{new unit}}{\text{original unit}} = \text{new unit} \]

Common Conversions:
$1 \text{ mile} = 5280 \text{ feet}$
$1 \text{ hour} = 60 \text{ minutes} = 3600 \text{ seconds}$
$1 \text{ kilogram} = 1000 \text{ grams}$
$1 \text{ gallon} = 4 \text{ quarts} = 8 \text{ pints}$

💡 Strategy — Unit Conversions

1. Identify the starting unit and the target unit.
2. Find the conversion factor (how many of one unit equals the other).
3. Multiply by the conversion factor so that the old unit cancels out.

📝 SOLVED EXAMPLE 4 — Unit Conversion

Convert 3 miles to feet. (1 mile = 5280 feet)

Step 1: $3 \text{ miles} \times \frac{5280 \text{ feet}}{1 \text{ mile}}$

Step 2: $3 \times 5280 = \color{var(--math)}{15840 \text{ feet}}$

✅ $15,840$ feet

💡 Tip: "Miles" cancels out, leaving "feet."

5. Speed, Distance, and Time

📌 RULE: Speed-Distance-Time

The fundamental relationship:

\[ \text{Distance} = \text{Speed} \times \text{Time} \]
\[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} \]
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]

Check units! If speed is in miles/hour, time must be in hours, distance in miles.

📝 SOLVED EXAMPLE 5 — Speed, Distance, Time

A car travels at 65 miles per hour for 3 hours. How far does it travel?

Step 1: Use formula: Distance = Speed × Time

Step 2: Distance = $65 \times 3$

Step 3: $\color{var(--math)}{195 \text{ miles}}$

✅ $195$ miles

💡 Tip: Always check that units match (miles/hour × hours = miles).

🧪 Practice Questions

Solve each problem using the rules above. Click "Show Answer" to see the full solution.

Question 1

There are 15 boys and 25 girls in a class. What is the ratio of boys to girls in simplest form?

A) $3:5$

B) $5:3$

C) $3:5$

D) $5:8$

✓ Answer: C

15:25 → divide by 5 → 3:5

📝 Solution: $15:25 = 3:5$

Question 2

A car travels 180 miles in 3 hours. What is the speed in miles per hour?

A) 50 mph

B) 60 mph

C) 70 mph

D) 80 mph

✓ Answer: B

Speed = 180 ÷ 3 = 60 mph

📝 Solution: $180/3 = 60$ mph

Question 3

If 4 pounds of bananas cost \$2.80, how much would 7 pounds cost?

A) \$4.20

B) \$4.40

C) \$4.80

D) \$4.90

✓ Answer: D

Unit price = 2.80 ÷ 4 = 0.70 per pound. 7 × 0.70 = 4.90

📝 Solution: $2.80/4 = 0.70$ → $7 \times 0.70 = 4.90$

Question 4

Convert 5 miles to feet. (1 mile = 5280 feet)

A) 26,400 feet

B) 26,400 feet

C) 52,800 feet

D) 10,560 feet

✓ Answer: B

5 × 5280 = 26,400 feet

📝 Solution: $5 \times 5280 = 26400$

Question 5

If $\frac{3}{x} = \frac{12}{20}$, what is the value of $x$?

A) 3

B) 4

C) 5

D) 6

✓ Answer: C

$3 \times 20 = 12x$ → $60 = 12x$ → $x = 5$

📝 Solution: $60 = 12x$ → $x = 5$

Question 6

A train travels at 80 miles per hour. How long will it take to travel 240 miles?

A) 2 hours

B) 3 hours

C) 4 hours

D) 5 hours

✓ Answer: B

Time = Distance ÷ Speed = 240 ÷ 80 = 3 hours

📝 Solution: $240/80 = 3$ hours

Question 7

The ratio of cats to dogs in a shelter is 4:7. If there are 16 cats, how many dogs are there?

A) 20

B) 24

C) 27

D) 28

✓ Answer: D

$\frac{4}{7} = \frac{16}{x}$ → $4x = 112$ → $x = 28$

📝 Solution: $4x = 112$ → $x = 28$

Question 8

Convert 2.5 hours to minutes.

A) 120 minutes

B) 150 minutes

C) 180 minutes

D) 200 minutes

✓ Answer: B

2.5 × 60 = 150 minutes

📝 Solution: $2.5 \times 60 = 150$

Question 9

A recipe calls for 2 cups of flour for every 3 eggs. If you use 9 eggs, how many cups of flour do you need?

A) 4 cups

B) 5 cups

C) 6 cups

D) 7 cups

✓ Answer: C

$\frac{2}{3} = \frac{x}{9}$ → $2 \times 9 = 3x$ → $18 = 3x$ → $x = 6$

📝 Solution: $2 \times 9 = 3x$ → $x = 6$

Question 10

A car travels 300 miles in 5 hours. What is the average speed in miles per hour?

A) 50 mph

B) 60 mph

C) 70 mph

D) 75 mph

✓ Answer: B

Speed = 300 ÷ 5 = 60 mph

📝 Solution: $300/5 = 60$ mph

🎉 WELL DONE!

You've completed the Ratios, Rates, Proportions & Units lesson. You now know how to work with ratios, rates, proportions, unit conversions, and speed-distance-time problems.

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