Proportions

For Elementary Students

What Is a Proportion?

A proportion is when two ratios are equal!

Think about it like this: If 2 cookies cost $4, then 4 cookies should cost $8, right? That's a proportion—the relationship stays the same!

2 cookies   4 cookies
────────  = ────────
   $4          $8

Both equal the same ratio!

Writing Proportions

A proportion looks like this:

a   c
─ = ─
b   d

or

a:b = c:d

Example: 2/3 = 4/6 is a proportion

2   4
─ = ─
3   6

Both simplify to 2/3!

Are These Equal? The Cross Check!

To check if two ratios make a proportion, use the cross-multiplication trick!

Step 1: Multiply diagonally (cross-multiply) Step 2: If the products are equal, it's a proportion!

a   c
─ = ─
b   d

Check: a × d = b × c

Example 1: Checking a Proportion

Question: Is 3/4 = 9/12 a proportion?

Cross-multiply:

  3     9
  ─  =  ─
  4    12

3 × 12 = 36
4 × 9  = 36

36 = 36 ✓ YES, it's a proportion!

Example 2: Not a Proportion

Question: Is 2/5 = 3/8 a proportion?

Cross-multiply:

  2     3
  ─  =  ─
  5     8

2 × 8 = 16
5 × 3 = 15

16 ≠ 15 ✗ NO, not a proportion!

Finding Missing Numbers

You can use cross-multiplication to find a missing number!

Example: What is x?

x   5
─ = ──
6   10

Step 1: Cross-multiply

x × 10 = 6 × 5
10x = 30

Step 2: Divide to find x

x = 30 ÷ 10
x = 3

Answer: x = 3! ✓

Visual Way to Think About It

Problem: 2 pizzas cost $10. How much do 6 pizzas cost?

2 pizzas = $10
6 pizzas = ?

Set up proportion:
2    6
── = ─
10   x

Cross-multiply:
2 × x = 10 × 6
2x = 60
x = 30

Answer: $30!

Example: Recipe Problem

Problem: A recipe for 4 people needs 6 eggs. How many eggs for 10 people?

Step 1: Set up the proportion

 4 people    10 people
────────  = ──────────
 6 eggs       x eggs

Step 2: Cross-multiply

4 × x = 6 × 10
4x = 60

Step 3: Solve

x = 60 ÷ 4
x = 15 eggs

Answer: 15 eggs! ✓

Example: Map Scale

Problem: On a map, 1 cm = 5 km. Two cities are 3.5 cm apart on the map. How far apart in real life?

Set up:

1 cm     3.5 cm
────  =  ──────
5 km      x km

Cross-multiply:

1 × x = 5 × 3.5
x = 17.5 km

Answer: 17.5 km apart! ✓

The Cross-Multiplication Song

"Cross multiply, then divide,
Find the missing number inside!"

Steps to remember:

1. Cross-multiply (multiply diagonally)
2. Divide by the number with the variable
3. That's your answer!

Real-Life Proportions

Shopping: "If 3 apples cost $6, how much do 7 apples cost?"

Speed: "If you drive 60 km in 1 hour, how far in 3 hours?"

Recipes: "If 2 cups make 8 cookies, how many cups for 20 cookies?"

Quick Tips

Tip 1: Always check which numbers go together!

Tip 2: Keep the same units in the same position (top or bottom)

Tip 3: Cross-multiply to check OR to solve

For Junior High Students

Understanding Proportions

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

Definition: For ratios a/b and c/d, a proportion is:

a/b = c/d  or  a:b = c:d

Fundamental property: In a proportion, the product of the extremes equals the product of the means.

If a/b = c/d, then ad = bc

Terminology:

• a and d are the extremes (outer terms)
• b and c are the means (inner terms)

Verifying Proportions

Method: Cross-multiplication

Principle: Two ratios form a proportion if and only if their cross products are equal.

Example 1: Verify 3/4 = 9/12

Cross products:
3 × 12 = 36
4 × 9 = 36

Since 36 = 36, the proportion is valid ✓

Example 2: Verify 2/5 = 3/7

Cross products:
2 × 7 = 14
5 × 3 = 15

Since 14 ≠ 15, this is not a proportion ✗

Alternative method: Simplify both fractions to lowest terms

3/4 = 3/4
9/12 = 3/4 (divide by 3)
Both equal 3/4, so they form a proportion ✓

Solving Proportions

Objective: Find the value of an unknown variable in a proportion.

Method: Cross-multiplication and algebraic manipulation

General form: Given a/b = c/d, solve for unknown

Example 1: Solve x/6 = 5/10

Step 1: Cross-multiply

x × 10 = 6 × 5
10x = 30

Step 2: Isolate variable

x = 30/10
x = 3

Verification: 3/6 = 1/2 and 5/10 = 1/2 ✓

Example 2: Solve 4/7 = 20/n

Step 1: Cross-multiply

4 × n = 7 × 20
4n = 140

Step 2: Solve for n

n = 140/4
n = 35

Verification: 4/7 ≈ 0.571 and 20/35 ≈ 0.571 ✓

Example 3: Solve (x+2)/5 = 6/10

Step 1: Cross-multiply

(x+2) × 10 = 5 × 6
10(x+2) = 30

Step 2: Distribute and solve

10x + 20 = 30
10x = 10
x = 1

Verification: (1+2)/5 = 3/5 = 0.6 and 6/10 = 0.6 ✓

Properties of Proportions

Property 1: Reciprocal property If a/b = c/d, then b/a = d/c

Property 2: Means-extremes property If a/b = c/d, then ad = bc

Property 3: Addition property If a/b = c/d, then (a+b)/b = (c+d)/d

Property 4: Alternation property If a/b = c/d, then a/c = b/d

Example of Property 4:

If 3/4 = 6/8, then 3/6 = 4/8
Verify: 3/6 = 1/2 and 4/8 = 1/2 ✓

Applications: Direct Variation

Definition: Two quantities are in direct proportion if their ratio is constant.

Form: y/x = k (constant), or y = kx

Example: Distance varies directly with time at constant speed

If you travel 120 km in 2 hours, how far in 5 hours?

Rate = 120/2 = 60 km/h (constant)

Set up proportion:
120/2 = d/5

Cross-multiply:
2d = 600
d = 300 km

Scale Problems

Maps, models, and drawings use proportions.

Example: A map scale is 1:50,000 (1 cm represents 50,000 cm = 500 m)

Two cities are 7.5 cm apart on the map. Find actual distance.

1/500 = 7.5/x

x = 7.5 × 500 = 3,750 m = 3.75 km

Similar Figures

Proportions are fundamental to similarity in geometry.

Example: Two similar triangles have corresponding sides in proportion.

If sides are 3, 4, 5 and 6, 8, 10:
3/6 = 4/8 = 5/10 = 1/2

All ratios equal 1/2, confirming similarity

Recipe and Mixture Problems

Example: A concrete mix uses cement:sand:gravel in ratio 1:2:3

For 12 kg of cement, find sand and gravel amounts.

Cement:Sand = 1:2
12:x = 1:2
x = 24 kg sand

Cement:Gravel = 1:3
12:y = 1:3
y = 36 kg gravel

Rate Problems

Example: A printer prints 45 pages in 3 minutes. How many pages in 8 minutes?

45/3 = x/8

3x = 360
x = 120 pages

Unit rate approach:

Rate = 45/3 = 15 pages/min
Pages in 8 min = 15 × 8 = 120 pages

Shopping and Unit Price

Example: Brand A: 6 cans for $8.40. Brand B: 8 cans for $10.40. Which is cheaper per can?

Using proportions:

Brand A: 6/8.40 = 1/x → x = $1.40/can
Brand B: 8/10.40 = 1/y → y = $1.30/can

Brand B is cheaper

Percent Problems as Proportions

Percent means "per hundred," so it's a proportion with denominator 100.

Example: What is 35% of 80?

35/100 = x/80

100x = 2800
x = 28

Common Mistakes

Mistake 1: Incorrect setup—mixing up corresponding values

❌ "2 apples cost $3, 5 apples cost x" → 2/5 = 3/x ✓ Keep ratios consistent: 2/3 = 5/x or 2/5 = 3/x

Mistake 2: Arithmetic errors in cross-multiplication

❌ If 3/4 = x/8, then 3 × 8 = 4x → 24 = 4x → x = 8 ✓ 24 = 4x → x = 6

Mistake 3: Forgetting to simplify or verify

Always check your answer makes sense in context

Mistake 4: Switching numerator and denominator

Maintain consistency in what goes on top vs. bottom

Mistake 5: Not identifying the correct relationship

Ensure you're setting up a valid proportion (constant ratio)

Tips for Success

Tip 1: Label your ratios clearly (what quantity over what quantity)

Tip 2: Keep the same units in corresponding positions

Tip 3: Cross-multiply carefully and show all steps

Tip 4: Check reasonableness of answer (if 2 costs $3, then 5 shouldn't cost $1)

Tip 5: Simplify fractions when possible to make calculations easier

Tip 6: Use unit rates as an alternative method to verify

Tip 7: Draw diagrams for complex problems to visualize relationships

Extension: Continued Proportions

Three or more ratios can be equal:

a/b = c/d = e/f

Example: In triangle similarity, all corresponding sides are proportional

3/6 = 4/8 = 5/10 = 1/2

Summary

Key concepts:

• Proportion: equation of equal ratios
• Cross-multiplication: ad = bc if a/b = c/d
• Solving: isolate variable after cross-multiplying
• Applications: recipes, maps, rates, similarity, percentages

Formula: If a/b = c/d, then ad = bc

Practice