If b is the mean proportional of a and c then prove that a 2 + b 2 a b = a + c b .

Given, b is the mean proportional of a and c . We have to prove that a 2 + b 2 a b = a + c b Two varying quantities are said to be in a relation of proportionality, multiplicatively connected to a constant; that is, when either their ratio or their product yields a constant. So, if a , b , c , d are in proportion, we can write it as a : b : : c : d , then a d = b c According to question, a : b : : b : c So, b 2 = a c ...... (1) Now, let us consider, L.H.S. = a 2 + b 2 a b = a 2 + a c a b [Substituting (1)] = a ( a + c ) a b = a + c b = R.H.S Hence, a 2 + b 2 a b = a + c b proved.

EASY

Earn 100

If the distance between Kanpur and Lucknow and the distance between Lucknow and Allahabad are in the ratio of 3:8, and the distance between Lucknow and Allahabad is 200 km, then what will be the distance between Lucknow and Kanpur?

Important Questions on Ratio and Proportion

EASY

Mathematics>Arithmetic>Ratio and Proportion>Proportion

What is the $4th$ proportional of $3, 8, 12$ ?

1. $36$

2. $26$

3. $32$

4. $16$

EASY

Mathematics>Arithmetic>Ratio and Proportion>Proportion

In a class of

80

students and

5

teachers, each student got sweets that are

15 %

of the total number of students and each teacher got sweets that are

25 %

of the total number of students. How many sweets were there?

EASY

Mathematics>Arithmetic>Ratio and Proportion>Proportion

A person carries $₹ 165$ in the form of currency notes of $₹ 5$ , $₹ 10$ and $₹ 20$ in the ratio of $3 : 2 : 1$ . What is the value of currency notes of $₹ 20$ denomination?

EASY

Mathematics>Arithmetic>Ratio and Proportion>Proportion

What

x

is added to each of

10, 16, 22, 32

the numbers so obtained in this order are in proportion? What is the mean proportion between the numbers

(x + 1)

and

(3 x + 1)

EASY

Mathematics>Arithmetic>Ratio and Proportion>Proportion

The two positive numbers are in ratio

3 : 4

. The difference between their squares is

28

. Find the greater number.

MEDIUM

Mathematics>Arithmetic>Ratio and Proportion>Proportion

x

is the mean proportional between

12.8

and

64.8

and

y

is the third proportional to

38.4

and

57.6

, then

2 x : y

is equal to:

MEDIUM

Mathematics>Arithmetic>Ratio and Proportion>Proportion

20 litre

mixture contains milk and water in the ratio

3 : 2

. Then

10 litres

of the mixture is removed and replaced with pure milk and the operation is repeated once more. At the end of the two removals and replacements, what is the ratio of milk and water in the resultant mixture?

EASY

Mathematics>Arithmetic>Ratio and Proportion>Proportion

What is the ratio of mean proportional between

1.8

and

3.2

and the third proportional of

5

and

3

MEDIUM

Mathematics>Arithmetic>Ratio and Proportion>Proportion

Rs.

730

were divided among

A

B

and

C

in such a way that if

A

gets Rs.

3

, then

B

gets Rs.

4

and if

B

gets Rs.

3.50

, then

C

gets Rs.

3

. The share of

B

exceeds that of

C

by:

EASY

Mathematics>Arithmetic>Ratio and Proportion>Proportion

Gita, Shweta and Sita invested

Rs . 4200

Rs . 8400

and

Rs . 5400

respectively while starting a business. At the end of the year, they earned a profit of

Rs . 24, 000

. Sita invested

32 %

of her share of the profit in a savings scheme. How much amount is left with her?

MEDIUM

Mathematics>Arithmetic>Ratio and Proportion>Proportion

Two numbers are in the ratio

3 : 5

. If

13

is subtracted from each, the new numbers are in the ratio

10 : 21

. If

15

is added to each of the original number, then the ratio becomes:

EASY

Mathematics>Arithmetic>Ratio and Proportion>Proportion

Third proportional of

8, 12

is _____.

MEDIUM

Mathematics>Arithmetic>Ratio and Proportion>Proportion

A sum is divided among

A, B, C

and

D

such that the ratio of the shares of

A

and

B

2 : 3

, that of

B

and

C

1 : 2

and that of

C

and

D

3 : 4

. If the difference between the shares of

A

and

D

₹ 648

, then the sum of their shares is:

EASY

Mathematics>Arithmetic>Ratio and Proportion>Proportion

2 x + 1, x + 2, 2

and

5

are in proportion, then what is the mean proportional between

3.5 (1 - x)

and

8 (1 + x) ?

MEDIUM

Mathematics>Arithmetic>Ratio and Proportion>Proportion

(5 x + 2 y) : (10 x + 3 y) = 5 : 9

, then

(2 x^{2} + 3 y^{2}) : (4 x^{2} + 9 y^{2}) = ?

MEDIUM

Mathematics>Arithmetic>Ratio and Proportion>Proportion

Alloy

A

contains copper and zinc in the ratio of

4 : 3

and alloy

B

contains copper and zinc in the ratio of

5 : 2

A

and

B

are taken in the ratio of

5 : 6

and melted to form a new alloy. The percentage of zinc on the new alloy is close to:

MEDIUM

Mathematics>Arithmetic>Ratio and Proportion>Proportion

A sum of

₹ x

is divided among

A

B

,and

C

such that the ratio of the shares of

A

and

B

6 : 7

and that of

B

and

C

3 : 2

. If the difference between the shares of

A

and

C

₹ 540

, then the value of

x

is;

MEDIUM

Mathematics>Arithmetic>Ratio and Proportion>Proportion

A certain sum is divided among A, B and C in such way that A gets Rs.

40

more than half of the sum. B gets Rs.

120

less than three-eighths of the sum and C gets Rs.

200

. What is the total sum?

EASY

Mathematics>Arithmetic>Ratio and Proportion>Proportion

In a mixture the ratio of milk and water is

3 : 2

. If there is

5

litre milk more than water the quantity of milk in the mixture is:

EASY

Mathematics>Arithmetic>Ratio and Proportion>Proportion

If $b$ is the mean proportional of $a$ and $c$ then prove that $\frac{a^{2} + b^{2}}{a b} = \frac{a + c}{b}$ .

If the distance between Kanpur and Lucknow and the distance between Lucknow and Allahabad are in the ratio of 3:8, and the distance between Lucknow and Allahabad is 200 km, then what will be the distance between Lucknow and Kanpur?

Important Questions on Ratio and Proportion

Learn and Prepare for any exam you want !