Today we will be providing short notes and tricks to solve question from** Ratio & Proportion**.It is an important topic from the quantitative aptitude section **.**These formulas and shortcuts will be helpful for your upcoming Exam .

**Ratio and Proportion**

- The number of times one quantity contains another quantity of the same kind is called the
**ratio**of the two quantities. - Observe carefully that the two quantities must be of the same kind. There can be a ratio between Rs.20 and Rs 30, but there can be no ratio between Rs 20 and 30 mangoes.

The ratio 2 to 3 is written **as 2 : 3 or 2/3.** 2 and 3 are called the **terms of the ratio**. 2 is the first term and 3 is the second term.

**Consequent**

- In the
**ratio 2 :3**, 2 is the antecedent and 3 is the consequent. **Note:**- The word ‘
**consequent**’ literally means ‘that which goes after’. - Since the quotient obtained on dividing one concrete quantity by another of the same kind is an abstract number, the ratio between two concrete quantities of the same kind is an abstract number. Thus, the ratio between Rs 5 and 7 is 5:7.

- The word ‘

**Compound Ratio**

- Ratios are
**compound**by multiplying together the antecedents for a new antecedent, and the consequent for a new consequent.

**Eg: find the ratio compound of the ratio:** 4:3, 9: 13, 26 : 5 and 2:15

**Solution-**

The required ratio =

**Inverse Ratio**

- If 2:3 be the given ratio, then 1/2: 1/3 or 3 :2 is called its
**inverse**or**reciprocal ratio.** - If the antecedent = the consequent, the ratio is called the
**ratio of equality**, such as 3:3. - If the antecedent >the consequent, the ratio is called the
**ratio of greater inequality**, as 4 :3 - If the antecedent < the consequent, the ratio is called the
**ratio of less inequality**, as 3 : 4.

**Ex. **Divide 1458 into two parts such that one may be to the other as 2: 7**.**

**Solution:**

1^{st} part =

2^{nd} part =

**Proportion**

**Consider the two ratios:**

1^{st} ratio 2^{nd} ratio

6 : 18 8 : 24

Since 6 is one-third of 18, and 8 is one –third of 24, the two ratios are equal. The equality of ratio is called **proportion.**

The number 6, 18, 8 and 24 are said to be in **proportion.**

The proportion may be written as

**6 : 18 :: 8 : 24** (6 is to 18 as 8 is to 24)

Or, 6 :18 = 8:24 or

- The numbers 6, 18, 8 and 24 are called the
**terms**. - 6 is the
**first terms**, 18 the**second,**8 the**third,**and 24**the fourth**. - The first and fourth terms, i.e. 6 and 24 are called the
**extremes**(end terms), and the second and the third terms, i.e., 18 and 8 are called the**means**(middle terms). 24 is called the**fourth**proportional. *If your quantities be in***proportion**, the product of the extremes is equal to the product of the means.

Let the four quantities 3, 4, 9 and 12 be in proportion.

We have ,

*Three quantities of the same kind are said to be in***continued proportion**when the ratio of the first to the second is equal to the ratio of the second the third.*The second quantity is called the***mean proportional**between the first and the third ; and the**third quantity**is called the third proportional to the first and second.

Thus, 9, 6 and 4 are in continued proportion for 9 :6 ::6 :4.

Hence, 6 is the mean proportional between 9 and 4, and 4 is the third proportional to 9 and 6.

**Eg **Find the fourth proportional to the numbers 6, 8 and 15.

**Solution:**

If x be the fourth proportional, then 6 : 8 = 15:x

**Eg2.** Find the third proportional to 15 and 20.

**Solution**:

Here, we have to find a fourth proportional to 15, 20 and 20. If x be the fourth proportional, we have 15 :20 = 20:x

**Direct Proportion:**

- Consider the following example .

**Ex. 1:** If 5 ball cost Rs 8, what do 15 balls cost?

**Solution:**

It will be seen at once that if the number of balls be increased 2, 3, 4,….times, the price will also be increased 2, 3, 4… times.

Therefore, 5 balls is the same fraction of 15 balls that the cost of balls is of the cost of 15 balls.

5 balls : 15 balls :: Rs8 : required cost

the required cost = Rs=Rs 24

This, example is an illustration of what is called direct proportion. In this case, the two given quantities are so related to each other that if one of them is multiplied (or divided) by any number, the other is also multiplied (or divided) by the same number.

**Inverse Proportion:** Consider the following example

**Ex. 1**: If 15 men can reap a filed in 28 days, in how many days will 10 men reap it?

**Solution:**

Here, it will be seen that if the number of men be increased 2, 3,4,….times, the number of days will be decreased 2, 3, 4…times. Therefore, the inverse ratio of the number of men is equal to the ratio of the corresponding number of days.

Or , 10: 15 :: 28 : the required number of days

the required number of days=

The above example is an illustration of what is called inverse proportion. In this case, the two quantities are so related that if one of them is multiplied by any number, the other is divided by the same number, and *vice versa.*

**Ex 2: **The employer decreases the number of his employees in the ratio 10 :9 and increase their ways in the ratio 11:12. What is the ratio of his two expenditures?

** Solution:**

The required ratio = 10×11 : 9 ×12 = 55 : 54

**Ex3:** A vessel contains liquid A and B in ratio 5 :3. If 16 liters of the mixture are removed and the same quantity of liquid B is added, the ratio becomes 3 :5. What quantity does the vessel hold?

**Solution:**

Quicker Method:

When the ratio is reversed (i.e., 5:3 becomes 3 :5), we can use the formula;

Total quantity = ×Quantity of A in the removed mixture

= liters.

__Quiz on ratio and proportion:-__

**1** If P : Q : R = 2 : 3 : 4 , Then P / Q : Q / R : R / P = ?

- 8: 9: 24
- 7: 9: 24
- 4: 6: 15
- 8: 11: 24
- None of these

**2:** If 2P = 3Q = 4R, Then P : Q : R = ?

- 2: 3: 5
- 2: 3: 4
- 3: 5: 6
- 1: 2: 3
- None of these

**3:** If P : Q = 2 : 3 , Q : R = 4 : 5 and R : S = 6 : 7 , then P : S = ?

- 18: 25
- 17: 35
- 16: 35
- 8: 11
- None of these

**4:**Rama distributes his pencil among his four friends Rakesh, Rahul, Ranjan, and Rohit in the ratio 1 / 2 : 1 / 3 : 1 / 4 : 1 / 5 . What is the minimum number of pencils that the person should have?

- 66
- 64
- 72
- 77
- None of these

**5:** Two numbers are respectively 40% and 60% more than third number. Find the ration of two numbers ?

- 8: 7
- 7: 9
- 9: 11
- 8: 13
- None of these

**6: **Rs 1210 were divided among three person P, Q, R so that P : Q = 5 : 4 and Q : R = 9 : 10. Then R gets the amount.

- 450
- 400
- 500
- 375
- None of these

**7: **Share Rs.4200 among joy, sanjay and bijoy in the ration 2 : 4 : 6.Find the amount received by sanjay.

- 1200
- 1300
- 1400
- 1500
- None of these

**8 :**Find the mean proportional between given two number that is 64 and 49.

- 45
- 52
- 54
- 56
- None of these

**9: **What number has to be added to each term of 3 : 5 to make the ratio 5 : 6 .

- 7
- 6
- 8
- 5
- None of these

**10:**Rs. 385 were divided among P , Q , R in such a way that P had Rs 20 more than Q and R had Rs 15 more than P . How much was R’s share?

- 156
- 145
- 152
- 150
- None of these

**Answers with Explanation:-**

**1. **P : Q : R = 2 : 3 : 4 .

Let P = 2k,

Q = 3k,

R = 4k.

Then,

P / Q = 2k / 3k = 2 / 3 ,

Q / R = 3k / 4k = 3 / 4

R / P = 4k / 2k = 2 / 1.

SO, P / Q : Q / R : R / P = 2 / 3 : 3 / 4 : 2 / 1 = 8 : 9 : 24.

- Let 2P = 3Q = 4R = k ,

Then ,

P = k / 2,

Q = k / 3 ,

R = k / 4.

SO , P : Q : R = k / 2 : k / 3 : k / 4 = 6 : 4 : 3.

- (C)

- Rakesh : Rahul : Ranjan : Rohit = 1 / 2 : 1 / 3 : 1 / 4 : 1 / 5

**Step 1:**At First we need to do is LCM of 2,3,4 and 5 is 60.

**Step 2:**Then pencil are distributed in ratio among friends,

Rakesh = ( 1 / 2 x 60 ) = 30.

Rahul = ( 1 / 3 x 60 ) = 20.

Ranjan = ( 1 / 4 x 60 ) = 15.

Rohit = ( 1 / 5 x 60 ) = 12.

**Step 3:** Total number of pencils are ( 30 x + 20 x + 15 x + 12 x) = 77 x.

For minimum number of pencils x = 1 .

The person should have atleast 77 pencils.

**5.Step 1:** Let the third number is A

Then first number is 140% of A = 140 x A / 100 = 7A / 5 and second number is 160% of B = 160 x B / 100 = 8B /5.

**Step 2:** now ratio of first and second number is 7A / 5 : 8B / 5 = 35A : 40B = 7 : 8.

- P : Q = 5 : 4, Q : R = 9 : 10 = ( 9 x 4 / 9 ) : ( 10 x 4 / 9 ) = 4 : 40 / 9.

So, P : Q : R = 5 : 4 : 40 /9 = 45 : 36 : 40

Sum of ratio terms is = ( 45 + 36 + 40 ) =121.

R share of amount is Rs (1210 x 40 / 121) = Rs. 400.

- Amount received by sanjay.

4 / 12 X 4200 = 1400= ( related ratio / sum of ratio ) x Total amount

So, the Amount received by sanjay is 1400.

- The mean proportion of two numbers is

Root of 64 and 49 is √8 x √ 7 = 8 x 7 = 56.

**So, the mean proportional is 56.**

**9.**Let the number to be added x , Then

3 + x / 5 + x = 5 / 6

6 ( 3 + x ) = 5 ( 5 + x )

x = ( 25 – 18 ) = 7

**So , the number to be added is 7 .**

- Let Q gets Rs x. Then We can say P gets Rs (x + 20 ) and R gets Rs ( x + 35) .

x + 20 + x + x + 35 = 385

3x = 330

x = 110 .

**R’s share = Rs ( 110 + 35 ) = Rs 145 .**

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