A ratio is a comparison of two quantities

by division

a : b

Read as "a to b" or written as a/b

Using a colon: 3:4

Using the word "to": 3 to 4

As a fraction: 3/4

Antecedent: The first term (a in a:b)

Consequent: The second term (b in a:b)

Equivalent ratios are ratios that express

the SAME relationship between quantities

• They have the same simplified form

• Example: 2:3 = 4:6 = 6:9 = 8:12

Method 1: Multiply

a:b = (a×n):(b×n)

Multiply both terms by the SAME number

Method 2: Divide

a:b = (a÷n):(b÷n)

Divide both terms by the SAME number

A unit rate is a rate with a

denominator of 1

• Shows the amount per ONE unit

• Example: miles per hour, cost per item, words per minute

Unit Rate = a/b = a ÷ b

Divide the first quantity by the second

Example 1: 150 miles in 3 hours. Find the unit rate.

Example 2: $12 for 4 pounds. Find cost per pound.

When dealing with fractions:

Step 1: Write as a complex fraction

Step 2: Divide (multiply by reciprocal)

Example: 1/2 mile in 1/4 hour. Find miles per hour.

Method 1: Convert to same second term (find LCD)

Method 2: Convert to decimals and compare

Method 3: Find unit rates and compare

A proportion is an equation stating that

two ratios are EQUAL

a:b = c:d

or a/b = c/d

Cross Products Method

If a/b = c/d, then a×d = b×c

Multiply diagonally (cross multiply)

If products are equal → Proportion!

To solve: a/b = c/x

Step 1: Cross multiply: a × x = b × c

Step 2: Solve for the variable

Step 3: Divide both sides to isolate variable

a/b = c/d → a×d = b×c

Step 1: READ carefully and identify what you're comparing

Step 2: SET UP the ratio or proportion

Step 3: SOLVE using cross multiplication

Step 4: CHECK your answer - does it make sense?

Problem: A recipe uses 2 cups of flour for 3 dozen cookies. How many cups are needed for 9 dozen cookies?

Problem: A biologist tags 50 fish and releases them. Later, she catches 80 fish and finds 8 are tagged. Estimate the total fish population.

✓ Ratio notation: Can be written as a:b, a to b, or a/b

✓ Order matters: 3:4 is different from 4:3

✓ Equivalent ratios: Multiply or divide both terms by same number

✓ Unit rate: Always has denominator of 1

✓ To find unit rate: Divide first quantity by second

✓ Proportion test: Cross products must be equal

✓ Solving proportions: Cross multiply then divide

✓ Keep units consistent: Same units in numerator and denominator positions

✓ Check your work: Plug answer back into original problem

✓ Word problems: Identify what's being compared first

Understanding Ratios:

"A ratio shows relation - comparing with division!"

Equivalent Ratios:

"Same relationship, different look - multiply or divide, that's the book!"

Unit Rates:

"Per means divide - unit rate is your guide!"

Proportions:

"Two ratios equal and true - that's a proportion for you!"

Cross Multiplication:

"Multiply across in an X - that's the cross product test!"

"Butterfly wings help you fly - multiply diagonally, don't be shy!"

Solving Proportions:

"Cross multiply to make it clean - divide to find what x does mean!"

Word Problems:

"Read it twice and think it through - set up the ratio, solve for you!"