Ratio & Proportion — Class 6 IMO

Ratio & Equivalent Ratios

What is a Ratio?

A ratio is a comparison of two quantities. It tells us how much of one thing there is compared to another.

Example: In a fruit bowl with 3 apples and 2 oranges:

The ratio of apples to oranges is 3 to 2

Three Ways to Write a Ratio:

Form	Example	How to Read
Colon form	3:2	"3 to 2"
Fraction form	3/2	"3 halves" or "3 to 2"
Word form	3 to 2	"3 to 2"

Part-to-Part vs. Part-to-Whole:

Type	Meaning	Example (3 apples, 2 oranges, total 5 fruits)
Part-to-Part	Compare one part to another part	Apples to oranges = 3:2
Part-to-Whole	Compare one part to the total	Apples to total fruits = 3:5

RATIO VISUAL REPRESENTATION:

    Apples:  🍎 🍎 🍎
    Oranges: 🍊 🍊
    
    Apples : Oranges = 3 : 2
    
    ┌─────────────────────────────────────┐
    │  Part-to-Part:     Part-to-Whole:   │
    │                                     │
    │  Apples : Oranges   Apples : Total  │
    │     3    :   2         3    :   5   │
    │                                     │
    │  Oranges : Apples   Oranges : Total │
    │     2    :   3         2    :   5   │
    └─────────────────────────────────────┘


RATIO BAR MODEL:

    Part-to-Part (Apples : Oranges = 3:2)
    
    Apples   │■■■│
    Oranges  │■■│
    
    Part-to-Whole (Apples : Total = 3:5)
    
    Apples   │■■■│
    Total    │■■■■■│


RATIO IN REAL LIFE:

    Recipe: 2 cups flour : 1 cup sugar
    
    Paint: 4 parts blue : 1 part white
    
    Class: 5 boys : 7 girls
    
    Map: 1 inch : 10 miles

Example 1: Write 8 : 12 in simplest form.

Divide both by 4: 2 : 3.

Example 2: Write a ratio equivalent to 1 : 2.

3 : 6 (multiply both by 3).

Example 3:

In a bag, there are 6 red marbles, 4 blue marbles, and 10 green marbles. Write the ratio of red to blue in all three forms.

Red = 6, Blue = 4
Colon: 6:4 (simplifies to 3:2)
Fraction: 6/4 (simplifies to (3)/(2))
Word: 6 to 4 (or 3 to 2)
Answer: 6:4, 6/4, 6 to 4

Example 4:

In the same bag, write the ratio of blue marbles to total marbles.

Blue = 4, Total = 6+4+10 = 20
Part-to-whole = 4:20
Simplify: divide both by 4 = 1:5
Answer: 4:20 or 1:5

Example 5:

A class has 12 girls and 8 boys. Write the ratio of boys to girls in simplest form.

Boys = 8, Girls = 12
Ratio = 8:12
Divide both by 4 = 2:3
Answer: 2:3

Quick recap

A ratio a : b compares two quantities.
Simplify by dividing both parts by their HCF.
Ratio compares two quantities
Three forms: colon (:), fraction (/), word (to)
Part-to-part = one part vs. another part
Part-to-whole = one part vs. total
Always simplify ratios like fractions

✓ Quick check

What is 10 : 15 in simplest form?

Divide both by 5: 10 : 15 = 2 : 3.

Which ratio is equivalent to 2 : 3?

Multiply both by 2: 2 : 3 = 4 : 6.

Proportion (Means & Extremes)

What are Equivalent Ratios?

Equivalent ratios are ratios that name the same comparison. They are like equivalent fractions!

Example: 1:2 = 2:4 = 3:6 = 4:8

Finding Equivalent Ratios:

Multiply or divide both terms by the same number
Just like finding equivalent fractions!

The Proportion Method:

When two ratios are equal, they form a proportion:

3/5 = ?/20

Cross-multiply: 3 × 20 = 5 × ?
             60 = 5 × ?
             ? = 60 ÷ 5 = 12

So 3/5 = 12/20

EQUIVALENT RATIOS - TAPE DIAGRAM:

    Ratio 1:2
    
    │■│  :  │■■│
    
    Ratio 2:4 (multiply both by 2)
    
    │■■│  :  │■■■■│
    
    Ratio 3:6 (multiply both by 3)
    
    │■■■│  :  │■■■■■■│


CROSS-MULTIPLICATION METHOD:

    3   =   ?
    5       20
    
    Step 1: Cross-multiply
    3 × 20 = 5 × ?
    60 = 5 × ?
    
    Step 2: Divide to find ?
    ? = 60 ÷ 5 = 12
    
    Check: 3/5 = 12/20 ✓


FINDING MISSING TERM:

    ?   =   8
    7       28
    
    Cross-multiply: ? × 28 = 7 × 8
                  ? × 28 = 56
                  ? = 56 ÷ 28 = 2
    
    Check: 2/7 = 8/28 ✓ (both simplify to ~0.2857)

Example 1: Is 2 : 3 :: 4 : 6 a proportion?

2 × 6 = 12 and 3 × 4 = 12, so yes.

Example 2: Find x: 3 : 4 :: 9 : x.

3 × x = 4 × 9 = 36, so x = 12.

Example 3:

Find the missing term: (2)/(3) = ?/15

Cross-multiply: 2 × 15 = 3 × ?
30 = 3 × ?
? = 30 ÷ 3 = 10
Answer: 10 ((2)/(3) = 10/15)

Example 4:

Find the missing term: (5)/(8) = 20/?

Cross-multiply: 5 × ? = 8 × 20
5 × ? = 160
? = 160 ÷ 5 = 32
Answer: 32 ((5)/(8) = 20/32)

Example 5:

A recipe uses 3 cups of flour for every 2 cups of sugar. How much sugar for 9 cups of flour?

Ratio: flour : sugar = 3 : 2
Set up proportion: (3)/(2) = 9/x
Cross-multiply: 3x = 18
x = 6
Answer: 6 cups of sugar

Example 6:

Are the ratios 4:6 and 8:12 equivalent?

Simplify 4:6 = divide by 2 = 2:3
Simplify 8:12 = divide by 4 = 2:3
Both simplify to 2:3, so YES they are equivalent
Answer: Yes

Quick recap

a : b :: c : d means a × d = b × c.
Extremes are a and d; means are b and c.
Equivalent ratios = same value when simplified
Multiply or divide both terms by same number
Use cross-multiplication to find missing terms
Proportion: two equal ratios
Check equivalence by simplifying or cross-multiplying

✓ Quick check

Find the missing term: 1 : 2 :: 5 : ___ ?

1 × ? = 2 × 5 = 10, so the missing term is 10.

Are 2 : 5 and 4 : 10 in proportion?

2 × 10 = 20 and 5 × 4 = 20, so yes.

Unitary Method & Word Problems

What is a Ratio Table?

A ratio table is an organized way to list equivalent ratios. It helps you see patterns and solve problems.

Example: Ratio of apples to oranges = 3:2

Apples	3	6	9	12	15
Oranges	2	4	6	8	10

What is a Double Number Line?

A double number line shows two quantities on parallel lines. It's great for visualizing ratios!

Example: 3 cups flour : 2 cups sugar

Flour:    0    3    6    9    12   15
          ├────┼────┼────┼────┼────┤
Sugar:    0    2    4    6    8    10

RATIO TABLE - BUILDING EQUIVALENTS:

 Start with 3:2
 
 ×2 → 6:4
 ×3 → 9:6
 ×4 → 12:8
 ×5 → 15:10
 
 As a table:
 
 ┌─────────┬────┬────┬────┬────┬────┐
 │ Apples │ 3 │ 6 │ 9 │ 12 │ 15 │
 ├─────────┼────┼────┼────┼────┼────┤
 │ Oranges │ 2 │ 4 │ 6 │ 8 │ 10 │
 └─────────┴────┴────┴────┴────┴────┘
 
 +3 each time +2 each time


DOUBLE NUMBER LINE - STEP BY STEP:

 Problem: A car travels 30 miles per 1 hour.
 How far in 2.5 hours?
 
 Miles: 0 15 30 45 60 75
 ├─────┼─────┼─────┼─────┼─────┤
 Hours: 0 0.5 1 1.5 2 2.5
 
 Answer: At 2.5 hours → 75 miles


FINDING MISSING VALUES USING RATIO TABLE:

 Problem: 5 tickets cost 20. How much for 8 tickets?
 
 ┌─────────┬────┬────┬────┬────┐
 │ Tickets │ 5 │ 1 │ 8 │ │
 ├─────────┼────┼────┼────┼────┤
 │ Cost() │ 20 │ 4 │ 32 │ │
 └─────────┴────┴────┴────┴────┘
 
 Step 1: Find unit rate (÷5): 4 per ticket
 Step 2: Multiply for 8 tickets (×8): 32

Example 1: 5 pens cost ₹50. What do 8 pens cost?

One pen = ₹10, so 8 pens = ₹80.

Example 2: A car covers 120 km in 2 hours. How far in 5 hours at the same speed?

Speed = 60 km/hr, so 60 × 5 = 300 km.

Example 3:

Complete the ratio table for the ratio 4:7

First	4	8	12	16	20
Second	7	?	21	?	35

Multiply both by 2: 8 → 14
Multiply both by 4: 16 → 28
Answer: 14 and 28

Example 4:

Use a double number line to solve: A recipe uses 2 eggs for every 3 cups of flour. How many eggs for 9 cups of flour?

``` Eggs: 0 2 4 6 ├────┼────┼────┼────┤ Flour: 0 3 6 9 ```

Answer: 6 eggs

Example 5:

A store sells 3 shirts for 45. Use a ratio table to find the cost of 7 shirts.

Shirts	3	1	7
Cost	45	15	105

Unit rate: 15 per shirt
7 shirts: 7 × 15 = 105
Answer: 105

Quick recap

Find the value of one unit, then multiply.
Works for shopping, mixing, scaling and speed.
Ratio tables list equivalent ratios in rows/columns
Multiply or divide both quantities by same number
Double number lines show two quantities on parallel lines
Both methods help find missing values
Unit rate = value of 1 unit (divide by first term)

✓ Quick check

3 kg of sugar cost ₹90. What is the cost of 1 kg?

90 ÷ 3 = ₹30.

4 books cost ₹200. What do 6 books cost?

One book = ₹50, so 6 books = ₹300.