Complete Guide to Ratios

Introduction to Ratios

A ratio is a comparison of two or more numbers showing how many times one value contains or is contained within the other. Ratios express the relationship between quantities and help us understand proportional relationships.

Key Points:

A ratio compares quantities with the same unit of measurement
Written as a:b, a/b, or "a to b"
Can be simplified like fractions
Order matters in ratios (a:b is different from b:a)

Types of Ratios

1. Part-to-Part Ratio

Compares one part of a whole to another part of the same whole.

Example: In a classroom with 15 boys and 12 girls, the ratio of boys to girls is 15:12 or 5:4 (simplified).

2. Part-to-Whole Ratio

Compares one part of a whole to the total or whole amount.

Example: In the same classroom with 15 boys and 12 girls (27 students total), the ratio of boys to total students is 15:27 or 5:9 (simplified).

3. Whole-to-Part Ratio

Compares the total or whole amount to one part.

Example: Continuing with the classroom example, the ratio of total students to girls is 27:12 or 9:4 (simplified).

4. Rate Ratio

Compares quantities with different units of measurement.

Example: If a car travels 240 miles in 4 hours, the rate ratio is 240 miles : 4 hours or 60 miles per hour.

5. Continued Ratio

Compares three or more quantities in order.

Example: If a mixture contains ingredients A, B, and C in quantities of 2, 3, and 5 cups respectively, the continued ratio is 2:3:5.

Equivalent Ratios

Equivalent ratios express the same relationship between numbers, similar to equivalent fractions. You can create equivalent ratios by multiplying or dividing both terms by the same non-zero number.

Example: The ratios 1:2, 2:4, 3:6, and 5:10 are all equivalent (all simplify to 1:2).

Verification:
1:2 → 1 ÷ 1 = 1 and 2 ÷ 1 = 2, ratio = 1:2
2:4 → 2 ÷ 2 = 1 and 4 ÷ 2 = 2, ratio = 1:2
3:6 → 3 ÷ 3 = 1 and 6 ÷ 3 = 2, ratio = 1:2
5:10 → 5 ÷ 5 = 1 and 10 ÷ 5 = 2, ratio = 1:2

Finding Equivalent Ratios

Method 1: Multiply both terms by the same number.

To find an equivalent ratio to 3:4, multiply both terms by 2:

3 × 2 : 4 × 2 = 6:8

Method 2: Divide both terms by their GCD (Greatest Common Divisor).

To simplify 18:24 to its simplest form:

GCD of 18 and 24 is 6

18 ÷ 6 : 24 ÷ 6 = 3:4

Original Ratio	Equivalent Ratios
1:3	2:6, 3:9, 4:12, 5:15
2:5	4:10, 6:15, 8:20, 10:25
3:4	6:8, 9:12, 12:16, 15:20

Methods for Solving Ratio Problems

1. Finding the Value of One Part

If quantities A and B are in the ratio a:b, and the total value is T, then:

Value of A = (a / (a + b)) × T

Value of B = (b / (a + b)) × T

Example: If $900 is to be divided in the ratio 2:3, how much does each person get?

Solution:

Total ratio parts = 2 + 3 = 5

Value of one part = $900 ÷ 5 = $180

First person gets = 2 × $180 = $360

Second person gets = 3 × $180 = $540

Check: $360 + $540 = $900 ✓

2. Comparing Ratios

To compare ratios, convert them to the same format (usually fractions) and then compare.

Example: Compare the ratios 2:5 and 3:8

Solution:

Convert to fractions: 2:5 = 2/5 = 0.4 and 3:8 = 3/8 = 0.375

Since 0.4 > 0.375, we can conclude that 2:5 > 3:8

3. Finding the Missing Term in a Proportion

If a:b = c:d, and one value is missing, cross-multiply to find the missing term.

a × d = b × c

Example: If 4:5 = x:15, find the value of x.

Solution:

Using the proportion: 4:5 = x:15

Cross multiply: 4 × 15 = 5 × x

60 = 5x

x = 12

Verify: 4:5 = 12:15 (both simplify to 4:5) ✓

4. Combined Ratios

When combining quantities in the same ratio, add the corresponding parts.

Example: Mixture A has salt and water in the ratio 1:4. Mixture B has salt and water in the ratio 2:3. If equal amounts of both mixtures are combined, what is the ratio of salt to water in the final mixture?

Solution:

Let's say we have 5 units of each mixture.

Mixture A: 1 unit salt, 4 units water

Mixture B: 2 units salt, 3 units water

Combined: 1+2=3 units salt, 4+3=7 units water

Ratio of salt to water = 3:7

5. Using Unit Ratios

Convert the ratio to a unit ratio (where one term is 1) to find corresponding values easily.

Example: If 3 workers can complete a task in 4 hours, how long will it take 5 workers to complete the same task?

Solution:

Ratio of workers to time: 3:4

Unit ratio (1 worker): 1:(4/3) = 1:1.33...

For 5 workers: 5 × (4/3 ÷ 5) = 5 × (4/15) = 4/3 × 1/1 × 1/5 = 4/15 hours

But this isn't right! The relationship is inverse: more workers means less time. Let's correct:

3 workers × 4 hours = 12 worker-hours (total work)

For 5 workers: 12 worker-hours ÷ 5 workers = 2.4 hours

Real-world Applications of Ratios

1. Financial Applications

Example: Profit-sharing among partners in a business according to their investment ratio.

Three partners invested in a business in the ratio 2:3:5. If the annual profit is $50,000, how much does each partner receive?

Solution:

Total ratio parts = 2 + 3 + 5 = 10

Value of one part = $50,000 ÷ 10 = $5,000

First partner: 2 × $5,000 = $10,000

Second partner: 3 × $5,000 = $15,000

Third partner: 5 × $5,000 = $25,000

2. Cooking and Recipes

Example: A recipe for 4 servings requires 2 cups of flour and 3 cups of milk. How much flour and milk are needed for 6 servings?

Solution:

Original ratio: 4 servings : 2 cups flour : 3 cups milk

For 1 serving: 1 serving : 2/4 cups flour : 3/4 cups milk

For 6 servings: 6 servings : 6 × (2/4) cups flour : 6 × (3/4) cups milk

= 6 servings : 3 cups flour : 4.5 cups milk

3. Map Scales

Example: On a map with a scale of 1:100,000, a distance of 5 cm represents what actual distance in kilometers?

Solution:

Scale 1:100,000 means 1 cm on the map represents 100,000 cm in reality

5 cm on the map represents: 5 × 100,000 = 500,000 cm = 5 km in reality

4. Mixing and Dilution

Example: A solution contains acid and water in the ratio 3:7. If 4 liters of water are added, the ratio becomes 3:11. Find the initial volumes of acid and water.

Solution:

Let's say initially there are 3x liters of acid and 7x liters of water

After adding 4 liters of water, we have:

3x liters of acid and (7x + 4) liters of water

According to the new ratio: 3x : (7x + 4) = 3 : 11

11 × 3x = 3 × (7x + 4)

33x = 21x + 12

12x = 12

x = 1

Therefore, initially there were 3 liters of acid and 7 liters of water