In the realm of mathematics, there are concepts that serve as the bedrock for countless real-world applications, guiding everything from financial decisions to scientific research. Among these foundational ideas are averages, ratios, and proportions. These mathematical tools offer invaluable insights into understanding and analyzing data, making sense of relationships, and solving practical problems. Whether you’re deciphering stock market trends, scaling recipes in the kitchen, or interpreting demographic statistics, a firm grasp of averages, ratios, and proportions opens doors to a deeper understanding of the world around us. In this blog, we embark on a journey through these fundamental concepts, exploring their definitions, applications, and significance in various contexts. Let’s delve into the world of averages, ratios, and proportions to uncover their power and relevance in everyday life.

**Ratio & Proportion**

Fractions play a significant role in explaining ratio and proportion. When a fraction is stated as a:b, it is a ratio, whereas a proportion indicates that two ratios are equal. Here, a and b might be any two integers. The two most significant notions are ratio and proportion, which serve as the foundation for understanding many other concepts in mathematics and science.

**Meaning of Ratio**

In mathematics, a ratio is a quantitative comparison between two or more quantities, typically expressed as the quotient of one quantity divided by another. Ratios are used to represent the relationship in size, magnitude, or quantity between different entities or parts of a whole.

Ratios can be expressed in various forms, such as fractions, decimals, or as a ratio of two numbers separated by a colon (:). For example, if we have 2 red apples and 3 green apples, the ratio of red apples to green apples is 2:3.

**Meaning of Proportion**

In mathematics, proportion refers to the equality of two ratios or fractions. Specifically, it signifies the relationship between quantities or magnitudes that maintain the same relative size when compared.

Proportions are often expressed in the form of an equation where two ratios are equated. For instance, π/π=π/π. *ba*β=*dc*β, where π*a* and π*b*, as well as π*c* and π*d*, represent pairs of quantities that are compared. In simple words, it compares two ratios. **Proportions are denoted by the symbol β::β or β=β.**

The proportion can be classified into the following categories, such as:

- Direct Proportion
- Inverse Proportion
- Continued Proportion

**Direct Proportion**

The direct proportion explains the relationship between two quantities such that if one quantity increases, so does the other. Similarly, as one quantity declines, so does the other. Therefore, if “a” and “b” are two quantities, the direction proportion is expressed as aβb.

**Inverse Proportion**

The inverse proportion indicates a relationship between two quantities in which a rise in one causes a reduction in the other. Similarly, if one quantity decreases, another quantity increases. Thus, aβ(1/b) represents the inverse proportion of two quantities, say “a” and “b”.

**Continued Proportion**

Consider the ratios a:b and c:d. Then, to calculate the continuous percentage for the two supplied ratio terms, we transform the means to a single term/number. This is generally referred to as the LCM of means.

For the given ratio, the LCM of b and c is bc.

So, multiplying the first ratio by c and the second ratio by b, we get

First ratio, ca:bc.

Second ratioβbc:bd

Thus, the continuous percentage can be expressed as ca: bc: bd.

**Ratio & Proportion Formula**

Let’s hop to ratio and proportion formulas here:

**Ratio Formula**

The ratio formula is quite simple. It’s expressed as the comparison of two quantities or numbers, usually denoted as π and *b*, and is written as

**a:b= a/b**

For example, if you have 2 red apples and 3 green apples, the ratio of red apples to green apples can be expressed as 2/3 or 2:3.

In some cases, you might also need to simplify ratios. To simplify a ratio, you find the greatest common divisor (GCD) of the two numbers and divide both parts of the ratio by this GCD.

So, the basic ratio formula is:

Ratio=π/bβ

Where:

- π is the first quantity or number.
- π is the second quantity or number.

**Proportion Formula**

In mathematics, a proportion is an equation that states that two ratios are equal. The basic form of a proportion can be expressed as:

π/π=π/dββ

Where:

- π
*a*and π*b*are two quantities or numbers. - π
*c*and π*d*are two other quantities or numbers.

This equation essentially means that the ratio of π to π is equal to the ratio of π to *d*.

The proportion formula can also be rearranged to solve for any of the four variables involved. For example:

- If you want to find π
*a*, you can rearrange the proportion to get π=πc/d - If you want to find π
*b*, you can rearrange the proportion to get π=ππ/c - If you want to find π
*c*, you can rearrange the proportion to get π=ππ/b - If you want to find π
*d*, you can rearrange the proportion to get π=ππ/a

**Important Properties of Proportion**

The following are the important properties of proportion:

- Addendo β If a : b = c : d, then a + c : b + d
- Subtrahendo β If a : b = c : d, then a β c : b β d
- Dividendo β If a : b = c : d, then a β b : b = c β d : d
- Componendo β If a : b = c : d, then a + b : b = c+d : d
- Alternendo β If a : b = c : d, then a : c = b: d
- Invertendo β If a : b = c : d, then b : a = d : c
- Componendo and dividendo β If a : b = c : d, then a + b : a β b = c + d : c β d

**Ratio & Proportion: Solved Examples**

**Q.1. . If A:B=2:3 and B:C=4:5, then A:B:C is:**

**Solution. **

A:B=2:3- (1) x 4

B:C=4:5 (2) x 3

Multiply by 4 in equation (1) and multiply by 3 in equation (2)

A:B:C= 8:12:15

**Q.2. Find a:b:c, if 6a = 9b = 10c**

**Solution.**

Let 6a = 9b = 10c = k

a=k/6; b=k/9; c=k/10

L.C.M. of 6,9,10= 180

a= k/6*180; b=k/9*180; c=k/10*180

a= 30k; b-= 20k; c=18k

a:b:c= 30k:20k:18k

a:b:c= 30:20:18 (dividing all by 2)

a:b:c= 15:10:9

**Q.3. A, B and C enter into a partnership and their shares are in the ratio 1/2:1/3:1/4 After 2 months A withdraws half of his capital and after 10 months a profit of Rs 378 is divided among them. What is Bs share?**

**Solution.**

A:B:C = LCM of 1/2: 1/3: 1/4

L.C.M. of 2,3, &4 = 12

= (1/2×12):(1/3Γ12): (1/4Γ12)

= 6:4:3

Total profit=Rs. 378

For 2 monthβ A:B:C=6:4:3

After 2 months A withdraw half of his capital

For next 10 monthsβA: B: C=3:4:3

Total money ratioβA: B:C=6:4:3

βA: B: C = {(6Γ2)+(3Γ10)}: ((4Γ2)+(4Γ10)}: {(3×2) + (3 Γ 10) = 42:48:36

= 7:8:6

Sum of ratio=7+8+6=21

B’s share = 8/21* Rs 378=Rs144

**Q.4. Seats of Physics, Chemistry and Mathematics in a school are in the ratio 4:5:6. There is a proposal to increase these seats by 75 in each department. What was the total number of seats in the school finally?**

**Solution.**

Seats of physics, Chemistry and Mathematics are in the ratio 4: 5: 6.

4K, 5K & 6K

Total = 4K + 5K + 6K = 15K

Seats Increased in each department = 75

Seats increased = 75+75 +75=225

Total Seats = 15K + 225 = 15(K +15)

Number of seats can be 600 if K = 25

Number of seats can be 750 if K = 35

Number of seals can be 900 il K = 45 Value Depend upon value of K

Hence can not determine

CAN NOT BE DETERMINED will be the correct answer.

**Q.5. Rs.4536 is divided among 4 men. 5 women and 2 boys, so that the share of a man, a woman and a boy are in the proportion of 7:4:3. The share of a woman is?**

**Solution.**

Let the share of a man, a woman, and a boy be 7x, 4x, and 3x respectively.

According to the question,

(4 x 7x) + (5 x 4x) + (2 x 3x) = 4536

28x + 20x + 6×4536

β 54x = 4536

4536/54 = 84

.. Share of a woman = 4x

4 x 84

β‘ Rs.336

Hence, the share of a woman is Rs.336.

**Income & Expense Based Questions**

**Q.6.The income of A, B and C are in the ratio of 7: 9: 12 and the ratio of their ratio expenditures are 8: 9:15. If A saves (1/4)th of his income then the savings of A, B and C are in the ratio of?**

**Solution.**

Let the Income of A, B and C be 7X, 9X, 12X respectively

And Expenditure of A, B and C be 8Y, 9Y, and 15Y respectively

Income of A x (1/4) = Savings of A

7Xx (1/4)=7X-8Y

28X-32Y = 7X

21X = 32Y

X:Y32:21

The ratio of savings of A, B, C

(7X-8Y): (9X9Y): (12X-15Y) = ((7×32)-(8×21)): ((9×32) (9×21)): ((12Γ32)-(15 x 21))

= 56:99:69

**Q.7. The income of Riya and Priya are in the ratio of 4: 5 and their expenditure is in the ratio of 2: 3. If each of them saves Rs. 10,000, then find the sum of their incomes.**

**Solution.**

a:b=c:d

Product of Extremes = Product of Means

Calculation:

Income Ratio = 4:5

Raj income be Rs. 4x and Prem income be Rs. 5x

Expenditure = Salary – Saving

Raj expenditure = 4x – 10000

Prem’s expenditure = 5x-10000

According to question,

(4x-10000): (5x-10000) 2:3

12x-30000 = 10x-20000

2x = 10000

β x = 5000

So, Raj’s Income = 4x = 4 x 5000 = Rs.20,000

Prem’s Income 5x 5 x 5000 = Rs.25,000

Sum of the incomes = Rs.(20000 + 25000) = Rs.45,000

Rs.45,000 is the sum of their incomes.

**Q.8. The mean proportional between A and B is 9 and the third proportional of A & B is 243. Find the larger of the two numbers.**

**Solution. **

Let, the two numbers be x and y respectively,

Mean proportional between x and y = 9

x: 9 = 9: y

= xy = 81

x=81/y —-(1)

Third proportional to x and y = 243

x:y=y: 243

From equation(1),

yΒ²= (243) x (81/y)

yΒ³ = 243 x 81

yΒ³ = 27 x 27 x 27 = (27)3

β y = 27

Again, From equation(1), we get,

x = 81/y = 81/27 = 3

Now,

Average (27+ 3)/2 = 15

.: The average of those numbers is 15

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