Ratios and Proportions

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We use ratios and proportions every day in our lives such as calculating the appropriate proportions of ingredients for recipes and estimating the duration of a potential road trip.

Ratios and Proportions Definition

Ratio is an ordered pair of numbers a and b, represented as a / b, where b is not equal to 0. 

A proportion is an equation that sets two ratios at the same value. For instance, you could express the ratio as follows: 2: 5 if there are two roses and 5 lilies. Thus for every two roses there are 5 lillies and 2.5 lilies for every rose (5 divided by 2).

What are Ratios?

Mathematicians use the term “ratio” to compare two or more numbers. It serves as a comparison tool to show how big or little an amount is in relation to another. Two quantities are compared using division in a ratio. In this case, the dividend is referred to as the “antecedent” and the divisor as the “consequent.” 

For example, in a group of 20 people, 13 of them prefer to eat mangoes and 7 of them prefer to eat apples. To represent this information as a ratio, we write it as 13: 7. Here, the symbol ‘: ‘ is read as “is to”.

Ratio Formula

When evaluating the relationship between two numbers or quantities, we employ the ratio formula. In general, a: b, which can be interpreted as “a is to b,” is used to denote a ratio between two quantities, let’s say “a” and “b.”

Ratio Formula

Let’s use an example to better grasp this.

A class of 30 pupils, for instance, has 12 girls and the remaining are boys. How many boys are there in relation to how many girls?

There are 30 pupils overall, with 12 girls.

Total boys equals the sum of all students minus all girls.

= 30 – 12 = 18

The ratio is therefore (Number of Boys: Number of Girls), which is 18:12.

Simplified ratio = 3:2

What are the Different Types of Ratios?

There are two categories of ratios:

1. Traditional Classification

Traditional ratios are those accounting ratios that are based on the balance sheet, profit and loss account, and other financial statements.

  1. Income Statement Ratio (Ex- Gross Profit Ratio)
  2. Balance Sheet ratio (Ex- Current Ratio, Debt Equity Ratio)
  3. Composite Ratio (Ex- Debtors Turnover Ratio)

2. Functional Classification

This categorization represents the necessity and intent of ratio computation.

  1. Liquidity Ratios (calculated to assess short-term solvency)
  2. Long-Term solvency (calculated to assess long-term solvency)
  3. Activity Ratio (computed to assess the effectiveness and efficiency of processes)
  4. Profitability Ratio (calculated to evaluate the company’s financial performance)

What is the Difference Between a Ratio and a Proportion?

Ratio

Proportion

The ratio expresses the numerical relationship between two amounts by showing how often one value contains the other. 

Proportion is the component that explains how one component compares to another. 

Denoted by colon (:) sign. 

Denoted by double colon (::) or equal to (=) sign. 

Quantitative relationship between two categories.

Quantitative relationship of a category and the total. 

When comparing the numbers of two separate categories, such as the ratio of males to women in a city, the ratio is utilized. Men and women are the two distinct groups in this context.

When determining the quantity of one group relative to the total, such as the percentage of men among all city residents, the proportion is utilized. 

It is an expression. 

It is an equation.

 

How to Calculate a Ratios and Proportions?

Calculating Ratios

We can use the subsequent techniques to determine the ratio of two quantities. 

Let’s use an example to better grasp this. Let’s figure out the proportion of milk and sugar used in the recipe, for instance, if rice pudding requires 4 cups of milk and 3 cups of sugar.

  • Step 1: Find the quantities for each of the two scenarios for which we are calculating the ratio. It is 4 and 3 in this instance.
  • Step 2: It should be expressed as a fraction, a/b. Thus, we represent it as 4/3.
  • Step 3: If you can, further simplify the fraction. The ultimate ratio will be shown by the simple fraction. The ratio in this case is 4/3, and since this ratio cannot be further simplified, it will remain as is.
  • Step 4: As a result, the flour to sugar ratio can be written as 4: 3.

Calculating Proportions

The proportion of black current to chocolate ice cream cones in an ice cream parlor is 2/5. If there are 20 chocolate ice cream cones in this parlor, how many black-current ice cream cones are there as well?

This means that if there are 2 black-current ice cream cones in this parlor, then there are 5 chocolate ice cream cones.

Keep in mind that the number of chocolate ice-cream cones is at the bottom and the number of black-current ice-cream cones is at the top.

Consequently, 2/5

Next, since 20 represents the number of chocolate ice-cream cones and this number was at the bottom in the ratio. 

Therefore, 2/5 = ?/20

Let’s assume ? be x; 

2/5 = x/20

x= 2/5 * 20

x= 8

There are eight black-current ice cream cones as a result.

Ratio and Proportion Examples

Solution: 

Given ratios are 2/3, 3/4, 5/6, 1/5 

The L.C.M. of 3, 4, 6, 5 is 2 × 2 × 3 × 5 = 60 

Now, 2/3 = (2 × 20)/(3 × 20) = 40/60 

3/4 = (3 × 15)/(4 × 15) = 45/60 

5/6 = (5 × 10)/(6 × 10) = 50/60 

1/5 = (1 × 12)/(5 × 12) = 12/60 

Clearly, 50/60 > 45/60 > 40/60 > 12/60 

5/6>3/4>2/3>1/5.

Solution:

Sum of the terms of the ratio = 3 + 4 = 7

Sum of numbers = 63

Therefore, first number = 3/7 * 63 = 27

Second number = 4/7 * 63 = 36

Therefore, the two numbers are 27 and 36.

Solution:

Length of ribbon originally = 30 cm

Let the original length be 5x and reduced length be 3x.

But 5x = 30 cm

x = 30/5 cm = 6 cm

Therefore, reduced length = 3 cm

= 3 * 6 cm = 18 cm

Solution:

Let the fourth term be x.

Thus 42, 36, 35, x are in proportion.

Product of extreme terms = 42 * x

Product of mean terms = 36 * 35

Since, the numbers make up a proportion

Therefore, 42 * x = 36 × 35

or, x = (36 * 35)/42

or, x = 30

Therefore, the fourth term of the proportion is 30.

What are the Properties of Ratios and Proportions?

A few useful properties of Ratio are as follows

  1. A ratio is only expressed when two quantities are measured in the same unit. 
  2. We use the notation “:” to represent ratios.
  3. If the corresponding fractions of two ratios are equal, we refer to them as being equivalent.
  4. The first term, a, is known as the antecedent and the second word, b, is known as the consequent for ratios expressed as a: b.
  5. The places of antecedent and consequent are not interchangeable, hence the order of the terms in ratios is crucial.
  6. The outcome is known as a Continued Ratios or a Compassion between the Quantities if more than one like quantity is expressed in a ratio format. It can be written as a: b: c: d, for example.

A few useful properties of Proportion are as follows

  1. The ratio of the first two quantities must match the ratio of the last two quantities for the numbers a, b, c, and d to be considered proportional, which is expressed as “a is to b as c is to d.” “Is as” is represented by the symbol “::”.
  2. A proportion’s constituent quantities are referred to as its terms or proportional.
  3. The first and last terms in a proportion are known as the extremes, and the second and third terms are known as the means.
  4. When four numbers—a, b, c, and d—are proportional (i.e., a: b: c: d), the extreme and middle terms—a and d—are referred to as such.
  5. Fourth proportional refers to a proportion’s fourth phrase.
  6. The product of the extremes always equals the product of the means for any proportion, i.e., a: b: c: d if and only if ad = bc.
  7. We can create three additional proportions from the terms of a given proportion.
  8. It is said that x, y, and z are in continuing proportion if x: y = y: z.
  9. Y is the mean proportional between x and z if x, y, and z are in continuing proportion (i.e., x: y: y: y: z).
  10. The third quantity is referred to as the third proportionate to the first and second, i.e., z is the third proportional to x and y, if x, y, and z are in continuing proportion (i.e., x: y:: y: z).

Ratios and Proportions Math Problems

  1. 2/3
  2. 4/5
  3. 8/15
  4. 4/3

Solution: 

Total number of students in the class = 40 + 35 = 75

40/75

8/15 :Ans (C)

  1. Equal
  2. Not Equal

 

Solution:

Let’s simplify this: 32/48 

2/3

Let’s simplify this: 70/210

1/3

As it is ⅔ and ⅓, hence it is not equal.  Ans (B)

  1. Rs.10,000, Rs.1,000
  2. Rs.11,250, Rs.2,250
  3. Rs.11,000, Rs.2,250
  4. Rs.8500, Rs.1,000

 

Solution:

5/4 = Monthly income/Savings

Let monthly income be x

5/4 = x/9000

x= 9000 * 5/4

X = 11250 = Monthly Income

Monthly Income = Savings + Expenditure

11250 = 9000 +  Expenditure

Expenditure = 2250 ; Monthly Income = 11250  Ans (B)

  1. 400, 300, 240
  2. 500, 300, 200
  3. 300, 400, 500
  4. 320, 400, 220

 

Solution:

Let the amount that each one carries be x.

x/3 + x/4 + x/5 = 940

LCM of 3. 4 and 5 = 60

20x + 15x + 12x = 940

60

X = 1200

A = 1/3 = 1200/3 = 400

B = 1/4 = 1200/4 = 300

C = 1/5 = 1200/5 = 240 Ans (A)

  1. 13 and 15
  2. 12 and 20
  3. 20 and 28
  4. 15 and 25

 

Solution: 

Let their present ages be 3x and 5x

Ages after 4 years = (3x +4)+ (5x + 4) 

(3x +4)+ (5x + 4) = 48

3x + 4 + 4 + 5x = 48

8x + 8 = 48

8x = 40

Since, x = 5

Their Present ages:-

3x = 3×5

x = 15

5x = 5×5 = 25

15 and 25 Ans (D)

FAQ'S

The proportional formula can be expressed as a: b, which can be interpreted as “a is to b,” which is used to denote a ratio between two quantities.

When the numerator and denominator are both natural numbers of any size and are coprime, the ratio is straightforward (i.e. they share no common factors, hence the ratio is simple).

Such as : 

2:3

23:44

The Symbol “∝” denotes “is proportionate to.” Two values x and y are expressed as x ∝ y when they have an inverse relationship with each other.

 

When one quantity’s value rises relative to another’s decline or vice versa, it is said that the two are inversely proportional. It implies that the two quantities behave inherently differently. For instance, time and speed have an inverse relationship. The time decreases as you increase the speed.  

 

Conclusion

There are many daily calculations that employ ratio and proportions. While the proportion is an equation that proves two ratios or fractions are equal, the ratio aids in comparing two or more quantities. For more questions on ratio and proportions visit {igebra.ai}. 

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