Ratio and Proportion

A ratio is a fractional representation of how one quantity compares to another. This fraction consists of two or more quantities of the same kind and in the same units. The symbol used to denote a ratio is ":" and it shows how many times one quantity is contained within the other.

If we have two numbers, a and b (where b is not equal to 0), the ratio of a to b is written as a : b.

Where 'a' is the antecedent and 'b' is the consequent.

Ratio in Simplest Form

A ratio is considered in its simplest form or lowest form when the two terms of the ratio (namely the antecedent and consequent) do not share any common factors other than 1.

Reducing a ratio to its simplest terms:

The simplest form of a ratio also known as a ratio in its lowest terms can be achieved by converting the ratio into a fraction and then dividing both the numerator and denominator by their highest common factor (HCF).

Examples:

1. The ratio 6 : 7 is already in its simplest form because the HCF of 6 and 7 is 1.
2. The ratio 28 : 24 is not in its simplest form because the HCF of 28 and 24 is 4, not 1. To simplify it, we divide both terms of the ratio by 4, resulting in 7 : 6. Thus, the simplest form of 28 : 24 is 7:6 which has HCF 1.

Example: The height of a pencil is 12 cm and the height of a pen is 8 cm. What is the ratio of their heights?

a) 2 : 3
b) 3 : 2
c) 4 : 2
d) 4 : 3

Answer: b) 2 : 3

Explanation: Height of a pencil = 12 cm

Height of a pen = 8 cm

Ratio of their heights = 12 : 8 = 3 : 2

Equivalent Ratios

Ratios are considered equivalent when their simplest forms are identical. To create a ratio that is equivalent to a given ratio, you can either multiply or divide all the terms of the ratio by the same number.

Example:

2 : 3, 4 : 6 and 6 : 9 are equivalent ratios.

Ratio of Two Quantities

1. The concept of a ratio between two quantities is applicable only when these quantities share the same unit of measurement.
   Example: Ratio of 30 litres to 45 litres = 30L/45L
                                                            = 2/3 or 2:3
2. When quantities are in a ratio which is measured in different units, it is essential to convert them to a common unit. Typically, the larger unit is converted into the smaller one.
   Example: Ratio of 600 millilitres to 1 litre = 600 mL/1 L
                                                                 = 600 mL/1000 L
                                                                  = 3/5 or 3:5
3. The order in which the quantities are presented in a ratio such as a : b holds significance. Changing the order, i.e., reversing the antecedent and consequent results in a different ratio b : a.
   Example: Ratio 3 : 5 is not the same as 5 : 3.

Comparison of Ratios

To compare two ratios, the approach involves converting each of them into a fraction and then the comparison of fractions is performed so that both fractions have the same denominators.
Example: Compare the ratios 3 : 5 and 7 : 9.

Therefore, 2 : 5 is less than 7 : 9 (2 : 5 < 7 : 9).

Proportion

A proportion is an equation that states that two ratios are equal. The symbol of proportional is “::”. In a proportion, you have four quantities (a, b, c, and d) and it asserts that the ratio of a to b is equal to the ratio of c to d. Mathematically, this is represented as

Here a, b, c and d are called first, second, third and fourth terms, respectively. First and fourth terms (a and d) are known as extreme terms and the second and third terms are known as means.

For example → 2 : 3 = 4 : 6 is written as 2 : 3 :: 4 : 6.

The product of the extreme terms is equal to the product of the means. This relationship is expressed as

a : b = c : d
⇒ a/b = c/d
⇒ ad = bc
⇒ Product of the extreme terms = Product of the means

Three quantities a, b, and c are in continued proportion when the ratio of a to b is equal to the ratio of b to c which is expressed as:

In this arrangement, b is termed as mean proportional between a and c while c is termed as the third proportional.

Proportions are used in various mathematical and real-world contexts to solve problems involving comparisons, scaling and relationships between quantities.

Word Problems on Proportion

Word problems involve various real-life situations and mathematical concepts related to proportions. You can set up proportions to solve them by finding the appropriate ratios and solving for the unknown quantities.

Example: The ratio of tin and copper is used to make a steel bottle is 7 : 13. If the weight of copper in a steel bottle is 7.8 kg, What the weight of tin in the steel bottle?

a) 3.2 kg
b) 3.6 kg
c) 4.2 kg
d) 4.6 kg

Answer: c) 4.2 kg

Explanation:

Unitary Method

The Unitary Method is an approach where we determine the value of a single unit first and then calculate the value of the desired number of units based on this single unit. This method follows two fundamental rules:

1. To increase the quantity, we use multiplication.
2. To decrease the quantity, we apply division.

Word Problem on Unitary Method

The unitary method is a technique that is used in various real-life situations to solve mathematical problems.

Example:  What is the price of twenty-four greeting cards if the price of six cards is $2.3?

a) $7.2
b) $8.2
c) $9.2
d) $10.2

Answer: c) $9.2

Explanation: Price of 6 cards = $2.3
Price of 1 card = $^2.3⁄₆
Price of 24 cards = $^2.23×24⁄₆
                          = $9.2