Using properties of proportion find x : y , given: x 2 + 2 x 2 x + 4 = y 2 + 3 y 3 y + 9

Using properties of proportion we have to find x : y , given: x 2 + 2 x 2 x + 4 = y 2 + 3 y 3 y + 9 Applying componendo and dividendo, ⇒ x 2 + 2 x + 2 x + 4 x 2 + 2 x - 2 x - 4 = y 2 + 3 y + 3 y + 9 y 2 + 3 y - 3 y - 9 ⇒ ( x + 2 ) 2 ( x - 2 ) ( x + 2 ) = ( y + 3 ) 2 ( y - 3 ) ( y + 3 ) ⇒ ( x + 2 ) 2 ( x - 2 ) 2 = ( y + 3 ) 2 ( y - 3 ) 2 [ because when we split ( x + 2 ) 2 and ( y + 3 ) 2 we will get, x 2 + 4 x + 4 and y 2 + 6 y + 9 ] ⇒ x + 2 x - 2 = y + 3 y - 3 Applying componendo and dividendo again, ⇒ x + 2 + x - 2 x + 2 - x + 2 = y + 3 + y - 3 y + 3 - y + 3 ⇒ 2 x / 4 = 2 y / 6 ⇒ x / 2 = y / 3 ⇒ x / y = 2 / 3 Hence, the ratio is x : y = 2 : 3 .

If x = 2 a + 1 + 2 a - 1 2 a + 1 - 2 a - 1 , prove that x 2 - 4 a x + 1 = 0

We can write the given expression as 2 a + 1 + 2 a - 1 2 a + 1 - 2 a - 1 = x 1 Apply Componendo and dividendo, we get, ⇒ 2 2 a + 1 2 2 a - 1 = x + 1 x - 1 ⇒ 2 a + 1 2 a - 1 = x + 1 x - 1 Squaring both sides, we get, 2 a + 1 2 a - 1 = x + 1 2 x - 1 2 Again applying componendo and dividendo. 2 a + 1 + 2 a - 1 2 a + 1 - 2 a + 1 = ( x + 1 ) 2 + ( x - 1 ) 2 ( x + 1 ) 2 - ( x - 1 ) 2 = 2 x 2 + 1 4 x ⇒ 2 a 1 = x 2 + 1 2 x ⇒ 4 ax = x 2 + 1 ⇒ x 2 - 4 ax + 1 = 0 Hence, proved.

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What is the formula for direct proportion?

Important Questions on Direct and Inverse Proportions

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Mathematics>Algebra>Direct and Inverse Proportions>Direct Proportion

Mayank can do a piece of work in

3

days. Sanjay can do the same work in

4

days. The wage for full work is

₹ 350

. If both work together to complete the work, then find out the earning of Sanjay if the wages are paid in proportion to work done.

EASY

Mathematics>Algebra>Direct and Inverse Proportions>Direct Proportion

If $p$ persons working $p$ hours a day for each of $p$ days produce $p$ units of work, then the units of the work produced by $q$ persons working $q$ hours a day for each $q$ day is:

EASY

Mathematics>Algebra>Direct and Inverse Proportions>Direct Proportion

x

varies as

y z

, then

y

varies inversely as

MEDIUM

Mathematics>Algebra>Direct and Inverse Proportions>Direct Proportion

A vessel contained a solution of acid and water in which water was

64 %

. Four litres of the solution were taken out of the vessel and the same quantity of water was added. If the resulting solution contains

30 %

acid, the quantity(in litres) of the solution, in the beginning in the vessel, was

HARD

Mathematics>Algebra>Direct and Inverse Proportions>Direct Proportion

The cost of milk per litre is $₹ 15$ . Draw the graph for the relation between the quantity and cost. Hence find the cost of $3$ litres of milk..

EASY

Mathematics>Algebra>Direct and Inverse Proportions>Direct Proportion

y = 3 + 1.9 x

, and mode of

x

15

, then the mode of

y

is:

MEDIUM

Mathematics>Algebra>Direct and Inverse Proportions>Direct Proportion

A set of farmers can completely harvest a crop in

10

days. However

12

farmers fell ill and now the remaining can do this job in

15

days. Find the original number of farmers.

EASY

Mathematics>Algebra>Direct and Inverse Proportions>Direct Proportion

If $(x + 3) : 27 : : 1 : 3$ , then find the value of $x$ .

HARD

Mathematics>Algebra>Direct and Inverse Proportions>Direct Proportion

The cost of milk per litre is $₹ 15$ . Draw the graph for the relation between the quantity and cost. Hence find the proportionality constant.

EASY

Mathematics>Algebra>Direct and Inverse Proportions>Direct Proportion

x α y

then

EASY

Mathematics>Algebra>Direct and Inverse Proportions>Direct Proportion

Using properties of proportion find $x : y$ , given:

$\frac{x^{2} + 2 x}{2 x + 4} = \frac{y^{2} + 3 y}{3 y + 9}$

MEDIUM

Mathematics>Algebra>Direct and Inverse Proportions>Direct 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?

MEDIUM

Mathematics>Algebra>Direct and Inverse Proportions>Direct Proportion

Using properties of proportion solve for $x$ , given

$\frac{\sqrt{5 x} + \sqrt{2 x - 6}}{\sqrt{5 x} - \sqrt{2 x - 6}} = 4$

EASY

Mathematics>Algebra>Direct and Inverse Proportions>Direct Proportion

A sum of

₹ 549

is to be divided among Priya, Preeti and Sonu in such a way that

3

times Priya's share,

4

times Preeti's share and

7

times Sonu's share are all equal. What is the share of Priya?

MEDIUM

Mathematics>Algebra>Direct and Inverse Proportions>Direct Proportion

Using properties of proportion, solve for $x$ , where $x$ is a positive number:

$\frac{2 x + \sqrt{4 x^{2} - 1}}{2 x - \sqrt{4 x^{2} - 1}} = 4$

MEDIUM

Mathematics>Algebra>Direct and Inverse Proportions>Direct Proportion

x = \frac{\sqrt{2 a + 1} + \sqrt{2 a - 1}}{\sqrt{2 a + 1} - \sqrt{2 a - 1}}

, prove that

x^{2} - 4 a x + 1 = 0

EASY

Mathematics>Algebra>Direct and Inverse Proportions>Direct Proportion

The third proportional of

18

and

6

will be_____

MEDIUM

Mathematics>Algebra>Direct and Inverse Proportions>Direct Proportion

How much time the minute hand of a clock will take to describe an angle of

\frac{2 π}{3}

radians ?

EASY

Mathematics>Algebra>Direct and Inverse Proportions>Direct Proportion

A car consumes

5.4 litres

of petrol to cover

60.48 km

, how many kilometres be covered with

22 litres

of petrol?

HARD

Mathematics>Algebra>Direct and Inverse Proportions>Direct Proportion

There are two candles of same length and same size. Both of them burn at uniform rate. The first, one burns in

5

hours and the second one burns in

3

hours. Both the candles are lit together. After how many minutes the length of the first candle is

3

times that of the other?

What is the formula for direct proportion?

Important Questions on Direct and Inverse Proportions

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