Ratio and Proportion

What is Ratio

A ratio is a mathematical expression that compares two or more quantities of the same unit.

It shows how the size of one quantity relates to another.

Ratios can be expressed in three different ways: by using the words “to” or “is to,” by separating the numbers with a colon (:), or by writing them as a fraction.

For example, the ratio of 5 hens to 7 cows can be expressed as:

  • 5 to 7 or 5 is to 7
  • 5 : 7
  • 5/7

Each of these is read as “5 to 7” or “5 is to 7”

 

Terms of a Ratio

In ratios, the numbers being compared are called terms. In a two-term ratio, the first number is known as the antecedent, and the second number is known as the consequent.

In our earlier example 5 is the antecedent while 7 is the consequent.

Non Commutability of Ratios

Ratios are non-commutative, meaning the terms in a ratio cannot always be swapped. In a two-term ratio, the antecedent and consequent form an ordered pair, and their positions are not interchangeable.

Two Types of Ratios

1. Part-to-Part Ratio

2. Part-to-Whole Ratio

There are 15 students in a class, with 10 boys and 5 girls.

  • Ratio A: The ratio of boys to girls is 10:5.
  • Ratio B: The ratio of boys to the total number of students is 10:15.

Ratio A compares one part of the whole (boys) with another part (girls). This is known as a part-to-part ratio because it compares two different parts of the whole.

On the other hand, Ratio B compares one part of the whole (boys) with the entire class (all students). This is called a part-to-whole ratio

Simplifying Ratios

1.  Ratios can be simplified in the same way as fractions. When a ratio cannot be reduced any further, it is called a simplified ratio, meaning it’s in its simplest form.

2.  To simplify a ratio, identify the greatest common factor (GCF) of all the terms. Then, divide each term by the GCF to obtain the simplified ratio. Let’s explore another example to see how this process works.

Example 1.

You have a collection of marbles consisting of green and yellow marbles. If you have 14 green marbles and 21 yellow marbles, the ratio of green marbles to yellow marbles is 14:21. How would you express this ratio in its simplest form?

14 : 21 or 2 : 3

Example 2. Let’s say you have 120 minutes to complete two tasks: reading and writing. If you spend 80 minutes reading and 40 minutes writing, the ratio of time spent on reading to writing is 80:40. How would you express this ratio in its simplest form?

80 : 40 or 2:1

Example 3: Let’s say you have 45 fruits consisting of apples and bananas. If you have 40 apples and 5 bananas, the ratio of apples to bananas is 40:5. How would you express this ratio in its simplest form?

40 : 5 or 8 : 1

 

Proportions

A proportion is an equation that shows two ratios are equal, establishing the equality between them. Proportions are typically represented by the equal sign (=).

Example : 5 : 10 = 1 : 2

                    5 : 10 = 25 : 50

                   5 : 10 = 3 : 6

If you multiply or divide each term of a ratio by the same non-zero number, the ratio remains equivalent.

The first and fourth terms (the outermost terms) in a proportion are known as the extremes. The second and third terms (the innermost terms) are referred to as the means

Thus, 5 and 2 are extremes while  10 and are the means.

 

Finding the Missing Terms in a Proportion

When solving for a missing term in a proportion:

  • If the missing term is a mean, multiply the two extremes and then divide the product by the known mean.
  • If the missing term is an extreme, multiply the two means and then divide the product by the known extreme.

Example 1. Find the value of y in the following proportion: 5 : 18 = 20 : y

y = 18 x 20 ÷ 5

360 ÷ 5 = 72

Example 2: Find the value of z in the following proportion: 7 : 35 = z : 25

z = 7 x 25 ÷ 35

175 ÷ 35

5

Difference Between Ratio and Proportion

RatioProportion
Used to compare two quantities with the same unit.Used to indicate that two ratios are equivalent.
Expressed using a colonExpressed using either an equal sign (=) or a double colon (::).
An expressionAn equation

Direct Proportion vs Indirect Proportion

Direct Proportion: When one quantity increases, the other quantity increases at the same rate, or when one quantity decreases, the other decreases at the same rate. This relationship is known as direct proportion.

  • Quantity and Price: The more items you buy, the higher the total price. If you buy double the quantity, the price doubles as well.

  • Exchange Rate and Money Received: If the exchange rate increases, the amount of money you receive for a given amount of foreign currency also increases proportionally. For example, if the exchange rate doubles, the amount of money you receive will also double.

Example 1.

If 5 liters of milk cost 150 pesos, how much will 8 liters of milk cost?

Since the cost is directly proportional to the quantity of milk:

5:  15 = 8 : x

5x = 15 x 8

5x =120

x = 120 ÷ 5 = 24

Example 2.

If a car travels 120 kilometers in 2 hours, how far will it travel in 5 hours at the same speed?

Since the distance is directly proportional to the time:

120 : 2 = x : 5

2x =600

x = 600 ÷ 2 = 300

Indirect Proportion: When one quantity increases, the other quantity decreases, or when one quantity decreases, the other increases. This inverse relationship is known as indirect proportion.

  • Speed and Time: The faster you travel, the less time it takes to reach your destination. If you double your speed, the time required to cover a certain distance is halved.

  • Work Problems: The more people working on a task, the less time it will take to complete it. If you double the number of workers, the time to finish the task is cut in half.

Example 1.  If 10 workers can complete a project in 15 days, then how long will it take for 6 workers to complete the same project?
 
Since the number of workers and the time taken are inversely proportional:
 
10 workers x 15 days = 6 workers : x
 
x = 150 ÷ 6
 
x = 25 days
 
Example 2. Problem: If a car traveling at 80 km/h takes 3 hours to reach a destination, how long will it take if the car travels at 60 km/h?
 
Since speed and time are inversely proportional:
 
80km/hr : 3 hours = 60km/hr : x
 
80 x 3 = 60 x (y)
 
60y = 240
y = 240 ÷ 60
y = 4 hours

Partitive Proportion

Partitive proportion is used when you need to divide a given whole into several parts based on a specified ratio. It allows you to allocate portions of the whole according to the ratio provided, ensuring that each part is proportional to the others as dictated by the ratio. This method is commonly used in scenarios where a total amount or quantity needs to be split among different groups or categories in a way that reflects their relative sizes or importance.

Example 1.  A sum of 1000 pesos is to be divided among three friends in the ratio 5:3:2. How much will each friend receive?

Solution: First, calculate the total parts in the ratio:

5 + 3 + 2 = 10 parts

Now, divide the total amount (1000 pesos) by the total parts to find the value of one part:

Value of one part=1000 ÷ 10 = 100 pesos

Now, multiply the value of one part by each part of the ratio:

  • Friend 1: 5 × 100 = 500
  • Friend 2: 3 × 100= 300
  • Friend 3: 2 × 100= 200

So, the friends will receive 500 pesos, 300 pesos, and 200 pesos respectively.

Example 2.

A teacher has 40 pencils to distribute among five students in the ratio 3:5:4:6:2. How many pencils will each student get?

Solution: First, calculate the total parts in the ratio:

3 + 5 + 4 + 6 + 2 = 20 parts

 
Now, divide the total number of pencils (40) by the total parts to find the value of one part:
Value of one part= 40 ÷ 20 = 2 pencils
 

Now, multiply the value of one part by each part of the ratio:

  • Student 1: 3 × 2 = 6
  • Student 2: 5 × 2 = 10
  • Student 3: 4 × 2 = 8
  • Student 4: 6 × 2 = 12
  • Student 5: 2 × 2 = 4

So, the students will receive 6, 10, 8, 12, and 4 pencils respectively.

Exercises on Direct, Indirect and Partitive Proportion

Identify if its Direct Proportion, Indirect Proportion or Partitive Proportion, and solve.

1.  If 7 kilograms of rice cost 280 pesos, how much will 10 kilograms of rice cost?

2. A prize of 600 pesos is to be shared among three siblings in the ratio 4:3:5. How much will each sibling receive?

3. If a plane flying at 600 km/h takes 5 hours to reach its destination, how long will it take if the plane flies at 400 km/h?

4. If 3 dozen eggs cost 180 pesos, how much will 5 dozen eggs cost?

5. If a cyclist traveling at 20 km/h takes 4 hours to reach a certain point, how long will it take if the cyclist slows down to 15 km/h?

6. If a cyclist covers 80 kilometers in 4 hours, how far will the cyclist travel in 7 hours at the same speed?

7. If 4 meters of fabric cost 320 pesos, how much will 7 meters of fabric cost?

8. If a train travels 150 kilometers in 3 hours, how far will it travel in 8 hours at the same speed?

9. A total of 800 candies are to be distributed among four children in the ratio 2:3:4:1. How many candies will each child get?

10. If a train traveling at 90 km/h takes 8 hours to cover a certain distance, how long will it take if the train travels at 60 km/h?

Answers

1. Direct Proportion, PHP 400

2. Partitive Proportion, PHP 200, PHP 150, PHP 250

3. Indirect Proportion, 7.5 hours

4. Direct Proportion, PHP 300

5. Indirect Proportion, 5.33 hours

6. Direct Proportion, 140 km

7. Direct Proportion, PHP 560

8. Direct Proportion, 400 km

9. Partitive Proportion, 160, 240, 320, and 80 candies.

10. Indirect Proportion, 12 hours

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