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Ratio Proportion and Partnership


Concept 1: Ratio

As we already know, that ratio is the comparison of two Quantities by division or the relation that one quantity bears to another with respect to magnitude.

If a and b are two numbers, then the ratio a to b is a/b or a÷b and is denoted by a:b. The two quantities that are being compared are called terms. 

The first is called antecedent and the second term is called consequent. 

As always, we would be focusing on the underlying concepts, rather than memorizing formulas, following are the types of ratios you would encounter.

i.) Duplicate Ratio: The ratio of squares of two numbers is called the duplicate ratio of the two numbers.

25 is the inverse ratio of 52 .

iv.) Compound Ratio: When we compound two or more ratios with each other through product or multiplication, the result is simply a compound ratio. Thus, the product of two or more ratios; i.e, ab:cd is a ratio compounded of the simple ratios a:c and b:d. Required compounded ratio = (2/3 x 6/11 x 11/2) = 2/1.

These terms are just for informative purposes, so that you do not get confused on not being familiar with the terminology.

Now, Let’s see some examples and test our learning till now.

Ex.1.) The ratio of boys to girls in a class is 2:3. If total students are 120.Find the number of boys in the class.

Ans.) Here, we see that the ratio of boys to girls is 2:3. Therefore we can deduce that for every 2 boys in the class, we have 3 girls. So, if we consider 2 boys and 3 girls as a group, we have 5 students in total in that group.

Concept 2: Proportion: 

Now, let’s discuss the terms related to proportion, and eventually we will see some examples at the end.

But, an important concept, that needs to be discussed is as follows :

If four quantities are in proportion, then Product of means = Product of extremes, for example, in proportion a:b::c:d, we have b x c= a x d.

Revision Notes for Maths Chapter 12 - Ratio and Proportion (Class 6th) |  askIITians

please explain the componendo and dividendo rule i am unable to understand  or use it effectively - Mathematics - TopperLearning.com | onstw4kk

# For componendo, we are just adding 1 to both sides of the equation, and for dividendo, 1 is being subtracted from both sides of the equation.

Concept 3 – Partnership: 

We have discussed already that partnership requires partners, whether it is cricket or partners in business.

To run a business or any organisation, we need to have partnerships. There are two types of partners.

i.) Working partners: Those partners who invest the money and manage the business or affairs of an organisation are called working partners.

ii.) Sleeping partners: Those partners who merely invest money and do not involve themselves in business affairs are called sleeping partners.

In partnerships, we have mainly three types of conditions related to distribution of profit in business.

Condition 1: 

When time period of investment is constant, then profit of partners will be divided into ratio C1 : C2 : C3 : ……Cn,

Where, C1, C2, C3, …..Cn are investments of respective partners.

 

Condition 2 : 

When amount invested is same for different partners but time period is different, then profit will be divided in the ratio of their time invested, that is

t1 : t2 : t3…… : tn .

Condition 3:

When the amount of investment and time period both are different for different partners, then profit will be divided into the following ratio.

         C1t1 : C2t2 : C3t3 : …………. : Cntn.

In order to have a clear understanding of the above mentioned cases, lets take an example,

Ex.1.) Three partners undergo a partnership. Their investments are Rs.30000, Rs.45000 and Rs.50000 respectively for the time periods of 4 years, 2 years and  1 year respectively. Find the ratio of their profits.

Ans.)  Here we see that the investments and time period of investments, both are different for all the partners, so we would are provide with Condition 3.

Using the same concept, the ratio of their profits would be,

 

C1t1 : C2t2 : C3t3

  • 30000 x 4 : 45000 x 2 : 50000 x 1

  • 30 x 4 : 45 x 2 : 50 x 1

  • 12 : 9 : 5 

Ex.2.) A and B started a business with initial investments in the ratio 5:7 . If after one year their profits were in the ratio 1:2 and the period of A’s investment was 7 months, B invested the money for ?

Ans.) Let investments of A and B respectively be 5x and 7x and period of B’s investment be “t” months.

Then , 5x x 77x x t = 12

            Therefore, we get, t = 10.

1. 

A and B together have Rs. 1210. If  https://www.indiabix.com/_files/images/aptitude/1-div-4by15.gif of A’s amount is equal to  https://www.indiabix.com/_files/images/aptitude/1-div-2by5.gif of B’s amount, how much amount does B have?

A.

Rs. 460

B.

Rs. 484

C.

Rs. 550

D.

Rs. 664

2. 

Two numbers are respectively 20% and 50% more than a third number. The ratio of the two numbers is:

A.

2 : 5

B.

3 : 5

C.

4 : 5

D.

6 : 7

3. 

A sum of money is to be distributed among A, B, C, D in the proportion of 5 : 2 : 4 : 3. If C gets Rs. 1000 more than D, what is B’s share?

A.

Rs. 500

B.

Rs. 1500

C.

Rs. 2000

D.

None of these

4. 

Seats for Mathematics, Physics and Biology in a school are in the ratio 5 : 7 : 8. There is a proposal to increase these seats by 40%, 50% and 75% respectively. What will be the ratio of increased seats?

A.

2 : 3 : 4

B.

6 : 7 : 8

C.

6 : 8 : 9

D.

None of these

5. 

In a mixture 60 litres, the ratio of milk and water 2 : 1. If this ratio is to be 1 : 2, then the quantity of water to be further added is:

A.

20 litres

B.

30 litres

C.

40 litres

D.

60 litres

MODERATE

1. 

The ratio of the number of boys and girls in a college is 7 : 8. If the percentage increase in the number of boys and girls be 20% and 10% respectively, what will be the new ratio?

A.

8 : 9

B.

17 : 18

C.

21 : 22

D.

Cannot be determined

2. 

Salaries of Ravi and Sumit are in the ratio 2 : 3. If the salary of each is increased by Rs. 4000, the new ratio becomes 40 : 57. What is Sumit’s salary?

A.

Rs. 17,000

B.

Rs. 20,000

C.

Rs. 25,500

D.

Rs. 38,000

3. 

If 0.75 : x :: 5 : 8, then x is equal to:

A.

1.12

B.

1.2

C.

1.25

D.

1.30

4. 

The sum of three numbers is 98. If the ratio of the first to second is 2 :3 and that of the second to the third is 5 : 8, then the second number is:

A.

20

B.

30

C.

48

D.

58

5. 

If Rs. 782 be divided into three parts, proportional to https://www.indiabix.com/_files/images/aptitude/1-div-1by2.gif : https://www.indiabix.com/_files/images/aptitude/1-div-2by3.gif : https://www.indiabix.com/_files/images/aptitude/1-div-3by4.gif, then the first part is:

A.

Rs. 182

B.

Rs. 190

C.

Rs. 196

D.

Rs. 204

HOTS-HIGH ORDER THINKING SKILLS

1. 

A and B invest in a business in the ratio 3 : 2. If 5% of the total profit goes to charity and A’s share is Rs. 855, the total profit is:

A.

Rs. 1425

B.

Rs. 1500

C.

Rs. 1537.50

D.

Rs. 1576

2. 

A, B and C jointly thought of engaging themselves in a business venture. It was agreed that A would invest Rs. 6500 for 6 months, B, Rs. 8400 for 5 months and C, Rs. 10,000 for 3 months. A wants to be the working member for which, he was to receive 5% of the profits. The profit earned was Rs. 7400. Calculate the share of B in the profit.

A.

Rs. 1900

B.

Rs. 2660

C.

Rs. 2800

D.

Rs. 2840

3. 

A, B and C enter into a partnership in the ratio https://www.indiabix.com/_files/images/aptitude/1-div-7by2.gif : https://www.indiabix.com/_files/images/aptitude/1-div-4by3.gif : https://www.indiabix.com/_files/images/aptitude/1-div-6by5.gif. After 4 months, A increases his share 50%. If the total profit at the end of one year be Rs. 21,600, then B’s share in the profit is:

A.

Rs. 2100

B.

Rs. 2400

C.

Rs. 3600

D.

Rs. 4000

4. 

A, B, C subscribe Rs. 50,000 for a business. A subscribes Rs. 4000 more than B and B Rs. 5000 more than C. Out of a total profit of Rs. 35,000, A receives:

A.

Rs. 8400

B.

Rs. 11,900

C.

Rs. 13,600

D.

Rs. 14,700

5. 

Three partners shared the profit in a business in the ratio 5 : 7 : 8. They had partnered for 14 months, 8 months and 7 months respectively. What was the ratio of their investments?

A.

5 : 7 : 8

B.

20 : 49 : 64

C.

38 : 28 : 21

D.

None of these

SOLUTIONS FOUNDATION

1. Answer: Option B

Explanation:

4

A

=

2

B

15

5

https://www.indiabix.com/_files/images/aptitude/1-sym-imp.gif A =

https://www.indiabix.com/_files/images/aptitude/1-sym-oparen-h1.gif

2

x

15

https://www.indiabix.com/_files/images/aptitude/1-sym-cparen-h1.gifB

5

4

https://www.indiabix.com/_files/images/aptitude/1-sym-imp.gif A =

3

B

2

https://www.indiabix.com/_files/images/aptitude/1-sym-imp.gif

A

=

3

B

2

https://www.indiabix.com/_files/images/aptitude/1-sym-imp.gif A : B = 3 : 2.

https://www.indiabix.com/_files/images/aptitude/1-sym-tfr.gif B’s share = Rs.

https://www.indiabix.com/_files/images/aptitude/1-sym-oparen-h1.gif

1210 x

2

https://www.indiabix.com/_files/images/aptitude/1-sym-cparen-h1.gif

= Rs. 484.

5

Answer: Option B

Explanation:

4

A

=

2

B

15

5

https://www.indiabix.com/_files/images/aptitude/1-sym-imp.gif A =

https://www.indiabix.com/_files/images/aptitude/1-sym-oparen-h1.gif

2

x

15

https://www.indiabix.com/_files/images/aptitude/1-sym-cparen-h1.gifB

5

4

https://www.indiabix.com/_files/images/aptitude/1-sym-imp.gif A =

3

B

2

https://www.indiabix.com/_files/images/aptitude/1-sym-imp.gif

A

=

3

B

2

https://www.indiabix.com/_files/images/aptitude/1-sym-imp.gif A : B = 3 : 2.

https://www.indiabix.com/_files/images/aptitude/1-sym-tfr.gif B’s share = Rs.

https://www.indiabix.com/_files/images/aptitude/1-sym-oparen-h1.gif

1210 x

2

https://www.indiabix.com/_files/images/aptitude/1-sym-cparen-h1.gif

= Rs. 484.

5

2. Answer: Option C

Explanation:

Let the third number be x.

Then, first number = 120% of x =

120x

=

6x

100

5

Second number = 150% of x =

150x

=

3x

100

2

Ratio of first two numbers =

https://www.indiabix.com/_files/images/aptitude/1-sym-oparen-h1.gif

6x

:

3x

https://www.indiabix.com/_files/images/aptitude/1-sym-cparen-h1.gif

= 12x : 15x = 4 : 5.

5

2

3. Answer: Option C

Explanation:

Let the shares of A, B, C and D be Rs. 5x, Rs. 2x, Rs. 4x and Rs. 3x respectively.

Then, 4x – 3x = 1000

https://www.indiabix.com/_files/images/aptitude/1-sym-imp.gif x = 1000.

https://www.indiabix.com/_files/images/aptitude/1-sym-tfr.gif B’s share = Rs. 2x = Rs. (2 x 1000) = Rs. 2000.

4. Answer: Option A

Explanation:

Originally, let the number of seats for Mathematics, Physics and Biology be 5x, 7x and 8x respectively.

Number of increased seats are (140% of 5x), (150% of 7x) and (175% of 8x).

https://www.indiabix.com/_files/images/aptitude/1-sym-imp.gif

https://www.indiabix.com/_files/images/aptitude/1-sym-oparen-h1.gif

140

x 5x

https://www.indiabix.com/_files/images/aptitude/1-sym-cparen-h1.gif

,

https://www.indiabix.com/_files/images/aptitude/1-sym-oparen-h1.gif

150

x 7x

https://www.indiabix.com/_files/images/aptitude/1-sym-cparen-h1.gif

and

https://www.indiabix.com/_files/images/aptitude/1-sym-oparen-h1.gif

175

x 8x

https://www.indiabix.com/_files/images/aptitude/1-sym-cparen-h1.gif

100

100

100

https://www.indiabix.com/_files/images/aptitude/1-sym-imp.gif 7x,

21x

and 14x.

2

https://www.indiabix.com/_files/images/aptitude/1-sym-tfr.gif The required ratio = 7x :

21x

: 14x

2

https://www.indiabix.com/_files/images/aptitude/1-sym-imp.gif 14x : 21x : 28x

https://www.indiabix.com/_files/images/aptitude/1-sym-imp.gif 2 : 3 : 4.

5. Answer: Option D

Explanation:

Quantity of milk =

https://www.indiabix.com/_files/images/aptitude/1-sym-oparen-h1.gif

60 x

2

https://www.indiabix.com/_files/images/aptitude/1-sym-cparen-h1.giflitres = 40 litres.

3

Quantity of water in it = (60- 40) litres = 20 litres.

New ratio = 1 : 2

Let quantity of water to be added further be x litres.

Then, milk : water =

https://www.indiabix.com/_files/images/aptitude/1-sym-oparen-h1.gif

40

https://www.indiabix.com/_files/images/aptitude/1-sym-cparen-h1.gif

.

20 + x

Now,

https://www.indiabix.com/_files/images/aptitude/1-sym-oparen-h1.gif

40

https://www.indiabix.com/_files/images/aptitude/1-sym-cparen-h1.gif

=

1

20 + x

2

https://www.indiabix.com/_files/images/aptitude/1-sym-imp.gif 20 + x = 80

https://www.indiabix.com/_files/images/aptitude/1-sym-imp.gif x = 60.

https://www.indiabix.com/_files/images/aptitude/1-sym-tfr.gif Quantity of water to be added = 60 litres.

MODERATE

1. Answer: Option C

Explanation:

Originally, let the number of boys and girls in the college be 7x and 8x respectively.

Their increased number is (120% of 7x) and (110% of 8x).

https://www.indiabix.com/_files/images/aptitude/1-sym-imp.gif

https://www.indiabix.com/_files/images/aptitude/1-sym-oparen-h1.gif

120

x 7x

https://www.indiabix.com/_files/images/aptitude/1-sym-cparen-h1.gif

and

https://www.indiabix.com/_files/images/aptitude/1-sym-oparen-h1.gif

110

x 8x

https://www.indiabix.com/_files/images/aptitude/1-sym-cparen-h1.gif

100

100

https://www.indiabix.com/_files/images/aptitude/1-sym-imp.gif

42x

and

44x

5

5

https://www.indiabix.com/_files/images/aptitude/1-sym-tfr.gif The required ratio =

https://www.indiabix.com/_files/images/aptitude/1-sym-oparen-h1.gif

42x

:

44x

https://www.indiabix.com/_files/images/aptitude/1-sym-cparen-h1.gif

= 21 : 22.

5

5

2. Answer: Option D

Explanation:

Let the original salaries of Ravi and Sumit be Rs. 2x and Rs. 3x respectively.

Then,

2x + 4000

=

40

3x + 4000

57

https://www.indiabix.com/_files/images/aptitude/1-sym-imp.gif 57(2x + 4000) = 40(3x + 4000)

https://www.indiabix.com/_files/images/aptitude/1-sym-imp.gif 6x = 68,000

https://www.indiabix.com/_files/images/aptitude/1-sym-imp.gif 3x = 34,000

Sumit’s present salary = (3x + 4000) = Rs.(34000 + 4000) = Rs. 38,000.

3. Answer: Option B

Explanation:

(x x 5) = (0.75 x 8)   https://www.indiabix.com/_files/images/aptitude/1-sym-imp.gif x =

https://www.indiabix.com/_files/images/aptitude/1-sym-oparen-h1.gif

6

https://www.indiabix.com/_files/images/aptitude/1-sym-cparen-h1.gif

= 1.20

5

4. Answer: Option B

Explanation:

Let the three parts be A, B, C. Then,

A : B = 2 : 3 and B : C = 5 : 8 =

https://www.indiabix.com/_files/images/aptitude/1-sym-oparen-h1.gif

5 x

3

https://www.indiabix.com/_files/images/aptitude/1-sym-cparen-h1.gif

:

https://www.indiabix.com/_files/images/aptitude/1-sym-oparen-h1.gif

8 x

3

https://www.indiabix.com/_files/images/aptitude/1-sym-cparen-h1.gif

= 3 :

24

5

5

5

https://www.indiabix.com/_files/images/aptitude/1-sym-imp.gif A : B : C = 2 : 3 :

24

= 10 : 15 : 24

5

https://www.indiabix.com/_files/images/aptitude/1-sym-imp.gif B =

https://www.indiabix.com/_files/images/aptitude/1-sym-oparen-h1.gif

98 x

15

https://www.indiabix.com/_files/images/aptitude/1-sym-cparen-h1.gif

= 30.

49

5. Answer: Option D

Explanation:

Given ratio = https://www.indiabix.com/_files/images/aptitude/1-div-1by2.gif : https://www.indiabix.com/_files/images/aptitude/1-div-2by3.gif : https://www.indiabix.com/_files/images/aptitude/1-div-3by4.gif = 6 : 8 : 9.

https://www.indiabix.com/_files/images/aptitude/1-sym-tfr.gif 1st part = Rs.

https://www.indiabix.com/_files/images/aptitude/1-sym-oparen-h1.gif

782 x

6

https://www.indiabix.com/_files/images/aptitude/1-sym-cparen-h1.gif

= Rs. 204

23

HOTS-HIGH ORDER THINKING SKILLS

1. Answer: Option B

Explanation:

Let the total profit be Rs. 100.

After paying to charity, A’s share = Rs.

https://www.indiabix.com/_files/images/aptitude/1-sym-oparen-h1.gif

95 x

3

https://www.indiabix.com/_files/images/aptitude/1-sym-cparen-h1.gif

= Rs. 57.

5

If A’s share is Rs. 57, total profit = Rs. 100.

If A’s share Rs. 855, total profit =

https://www.indiabix.com/_files/images/aptitude/1-sym-oparen-h1.gif

100

x 855

https://www.indiabix.com/_files/images/aptitude/1-sym-cparen-h1.gif

= 1500.

57

2. Answer: Option B

Explanation:

For managing, A received = 5% of Rs. 7400 = Rs. 370.

Balance = Rs. (7400 – 370) = Rs. 7030.

Ratio of their investments = (6500 x 6) : (8400 x 5) : (10000 x 3)

   = 39000 : 42000 : 30000

   = 13 : 14 : 10

https://www.indiabix.com/_files/images/aptitude/1-sym-tfr.gif B’s share = Rs.

https://www.indiabix.com/_files/images/aptitude/1-sym-oparen-h1.gif

7030 x

14

https://www.indiabix.com/_files/images/aptitude/1-sym-cparen-h1.gif

= Rs. 2660.

37

3. Answer: Option D

Explanation:

Ratio of initial investments =

https://www.indiabix.com/_files/images/aptitude/1-sym-oparen-h1.gif

7

:

4

:

6

https://www.indiabix.com/_files/images/aptitude/1-sym-cparen-h1.gif

= 105 : 40 : 36.

2

3

5

Let the initial investments be 105x, 40x and 36x.

https://www.indiabix.com/_files/images/aptitude/1-sym-tfr.gif A : B : C =

https://www.indiabix.com/_files/images/aptitude/1-sym-oparen-h1.gif

105x x 4 +

150

x 105x x 8

https://www.indiabix.com/_files/images/aptitude/1-sym-cparen-h1.gif

: (40x x 12) : (36x x 12)

100

   = 1680x : 480x : 432x = 35 : 10 : 9.

Hence, B’s share = Rs.

https://www.indiabix.com/_files/images/aptitude/1-sym-oparen-h1.gif

21600 x

10

https://www.indiabix.com/_files/images/aptitude/1-sym-cparen-h1.gif

= Rs. 4000.

54

4. Answer: Option D

Explanation:

Let C = x.

Then, B = x + 5000 and A = x + 5000 + 4000 = x + 9000.

So, x + x + 5000 + x + 9000 = 50000

https://www.indiabix.com/_files/images/aptitude/1-sym-imp.gif 3x = 36000

https://www.indiabix.com/_files/images/aptitude/1-sym-imp.gif x = 12000

A : B : C = 21000 : 17000 : 12000 = 21 : 17 : 12.

https://www.indiabix.com/_files/images/aptitude/1-sym-tfr.gif A’s share = Rs.

https://www.indiabix.com/_files/images/aptitude/1-sym-oparen-h1.gif

35000 x

21

https://www.indiabix.com/_files/images/aptitude/1-sym-cparen-h1.gif

= Rs. 14,700.

50

5. Answer: Option B

Explanation:

Let their investments be Rs. x for 14 months, Rs. y for 8 months and Rs. z for 7 months respectively.

Then, 14x : 8y : 7z = 5 : 7 : 8.

Now,

14x

=

5

   https://www.indiabix.com/_files/images/aptitude/1-sym-bim.gif    98x = 40y    https://www.indiabix.com/_files/images/aptitude/1-sym-bim.gif    y =

49

x

8y

7

20

And,

14x

=

5

   https://www.indiabix.com/_files/images/aptitude/1-sym-bim.gif    112x = 35z    https://www.indiabix.com/_files/images/aptitude/1-sym-bim.gif    z =

112

x =

16

x.

7z

8

35

5

https://www.indiabix.com/_files/images/aptitude/1-sym-tfr.gif x : y : z = x :

49

x

:

16

x

= 20 : 49 : 64.

20

5