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Ratios & Rates

Interactive study guide — Reveal Math Unit 3 · Standard 6.RP

3.1–3.2 — Understand Ratios & Unit Rates

Write ratios in three forms, identify part-to-part vs. part-to-whole relationships, and calculate unit rates.

Key Ideas

A ratio compares two quantities. It can be written three ways: a : b, a to b, or a/b.

A part-to-part ratio compares one part of a group to another part (e.g., red marbles to blue marbles). A part-to-whole ratio compares one part to the total (e.g., red marbles to all marbles).

A unit rate is a rate with a denominator of 1. Divide to find it.

Ratio example: A bag has 3 red and 4 blue marbles.
Part-to-part: 3 : 4 (red to blue)
Part-to-whole: 3 : 7 (red to total)
Unit rate example: $15 for 3 notebooks
$15 ÷ 3 = $5 per notebook

Practice

A class has 12 boys and 18 girls. What is the ratio of boys to girls in simplest form?

12 : 18
3 : 2
2 : 3

You drive 150 miles in 3 hours. What is the unit rate?

3 miles per hour
50 miles per hour
450 miles per hour

A bag has 5 red and 8 blue marbles. What is the part-to-whole ratio of red marbles to total marbles?

5 : 13
5 : 8
8 : 13
Score
0/3

3.3 — Equivalent Ratios

Generate equivalent ratios using multiplication and division, and verify with cross products.

Key Ideas

Equivalent ratios are ratios that express the same relationship. Multiply or divide both terms by the same number to find them.

A ratio table organizes equivalent ratios in rows or columns. A double number line shows two quantities lined up at equivalent values.

Cross products can verify equivalence: if a/b = c/d, then a × d = b × c.

Example: 4 : 5 → multiply both by 2 → 8 : 10
Check: 4 × 10 = 40 and 5 × 8 = 40 — cross products are equal, so 4 : 5 = 8 : 10
Ratio table:
Apples: 2, 4, 6, 8
Oranges: 3, 6, 9, 12
Every pair is equivalent to 2 : 3

Practice

Which ratio is equivalent to 3 : 5?

6 : 8
9 : 15
5 : 3

In a ratio table, if the first row is 4 and 10, what pair comes next when you multiply by 2?

8 and 20
6 and 12
8 and 10

Are 6 : 9 and 10 : 15 equivalent? Use cross products.

No, because 6 × 15 = 80
No, because 9 × 10 = 100
Yes, because 6 × 15 = 90 and 9 × 10 = 90
Score
0/3

3.4 — Graph Ratio Relationships

Plot equivalent ratios on a coordinate plane, interpret the slope as a unit rate, and connect graphs to tables and equations.

Key Ideas

Each equivalent ratio can be plotted as an ordered pair (x, y) on the coordinate plane. All equivalent ratios form a straight line through the origin.

The slope (steepness) of the line equals the unit rate. In the equation y = kx, k is the constant of proportionality (unit rate).

Example: A car travels at 1.5 miles per minute.
Equation: y = 1.5x
Table: (1, 1.5), (2, 3), (4, 6)
All three points lie on the same line through (0, 0).
Reading a graph: If the point (4, 10) is on the line, the unit rate is 10 ÷ 4 = 2.5.

Practice

A graph of equivalent ratios always passes through which point?

(0, 0)
(1, 1)
(0, 1)

The equation y = 3x represents a proportional relationship. What is the unit rate?

1/3
x
3

The point (6, 9) is on a proportional line through the origin. What is the constant of proportionality?

6
1.5
54
Score
0/3

3.5 — Compare Ratios & Percents

Compare rates to find better buys, understand percents as ratios per 100, and convert between fractions, decimals, and percents.

Key Ideas

To find the better buy, calculate the unit rate for each option and compare.

A percent means "per 100." It is a ratio that compares a number to 100.

Converting: Fraction → Decimal → Percent
Divide numerator by denominator to get a decimal, then multiply by 100 to get a percent.

Better buy: Store A: $6 for 4 apples ($1.50 each). Store B: $10 for 8 apples ($1.25 each).
Store B is the better buy.
Percent conversion: 25% = 25/100 = 0.25
3/4 = 0.75 = 75%

Practice

Pack A: 6 pencils for $3. Pack B: 10 pencils for $4. Which is the better buy?

Pack A ($0.50 each)
Pack B ($0.40 each)
They cost the same

What is 40% written as a fraction in simplest form?

2/5
4/5
40/10

Convert 3/8 to a percent.

38%
30%
37.5%
Score
0/3

3.6 — Measurement Conversions

Convert units within and between measurement systems using ratio reasoning and conversion factors.

Key Ideas

A conversion factor is a ratio equal to 1 that relates two units (e.g., 12 inches / 1 foot).

Multiply by the conversion factor so the unwanted unit cancels out. For multi-step conversions, chain conversion factors.

Example: Convert 48 inches to feet.
48 in × (1 ft / 12 in) = 4 feet
Multi-step: Convert 3 yards to inches.
3 yd × (3 ft / 1 yd) = 9 ft
9 ft × (12 in / 1 ft) = 108 inches

Practice

How many feet are in 36 inches?

3 feet
4 feet
6 feet

Convert 5 kilometers to meters. (1 km = 1,000 m)

500 meters
50 meters
5,000 meters

How many inches are in 2 yards? (1 yard = 3 feet, 1 foot = 12 inches)

36 inches
72 inches
24 inches
Score
0/3

Vocabulary

Tap a card to reveal the definition.

Ratio
A comparison of two quantities, written as a : b, a to b, or a/b.
Rate
A ratio that compares two quantities with different units, such as miles per hour or dollars per item.
Unit Rate
A rate with a denominator of 1. Found by dividing both terms so the second quantity equals 1.
Equivalent Ratio
Ratios that express the same relationship. Found by multiplying or dividing both terms by the same number.
Proportion
An equation that states two ratios are equal. Example: 2/3 = 4/6.
Percent
A ratio that compares a number to 100. The symbol % means "per hundred."
Conversion Factor
A ratio equal to 1 used to convert from one unit to another. Example: 12 in / 1 ft.
Constant of Proportionality
The constant value k in the equation y = kx. It is the unit rate in a proportional relationship.
Double Number Line
A diagram that shows two number lines aligned so that corresponding values represent equivalent ratios.
Tape Diagram
A visual model using rectangular bars to represent parts of a ratio or relationship between quantities.