Tag: ratio and proportion

Questions Related to ratio and proportion

If two ratios are equal then their inverse ratios are equal. This property is known as

properties of proportion ratio and proportions ratio and proportion maths

  1. Dividendo property

  2. Invertendo property

  3. Componendo property

  4. Alternendo property

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Correct Option: B
Explanation:

If $a : b :: c : d$ then $b : a :: d : c \Rightarrow $ invertendo property

If $\dfrac {x}{y}=\dfrac {3}{4}$ and $\dfrac {x}{2z}=\dfrac {3}{2}$, then $\dfrac {2x+z}{x-2z}+\left (\dfrac {6}{7}+\dfrac {y-x}{y+x}\right )$ will be equivalent to

properties of proportion ratio and proportions ratio and proportion maths

  1. 8

  2. 9

  3. 11

  4. 12

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Correct Option: A
Explanation:

$\because \dfrac {x}{y}=\dfrac {3}{4}\Rightarrow \dfrac {y-x}{y+x}=\dfrac{4-3}{4+3}=\dfrac {1}{7}$
$\because \dfrac {x}{2z}=\dfrac {3}{2}\Rightarrow 2x=6z$ or, $x=3z$
$\therefore \dfrac {2x+z}{x-2z}+\left (\dfrac {6}{7}+\dfrac {y-x}{y+x}\right )=\dfrac {6z+z}{3z-2z}+\left (\dfrac {6}{7}+\dfrac {1}{7}\right )$
$=\dfrac {7z}{z}+\dfrac {7}{7}=7+1=8$.

If $a=2+\sqrt 3$ then the value of $a+\frac {1}{a}$ is

properties of proportion ratio and proportions ratio and proportion maths

  1. 1

  2. 2

  3. 3

  4. 4

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Correct Option: D
Explanation:

$\because a=2+\sqrt 3$
$\therefore \frac {1}{a}=\frac {1}{2+\sqrt 3}\times \frac {2-\sqrt 3}{2-\sqrt 3}=\frac {2-\sqrt 3}{4-3}=2-\sqrt 3$
$a+\frac {1}{a}=2+\sqrt 3+2-\sqrt 3=4$

After applying invertendo to $3:7::2:9$ we get

properties of proportion ratio and proportions ratio and proportion maths

  1. $3:7::2:9$

  2. $3:2::7:9$

  3. $9:7::2:3$

  4. $7:3::9:2$

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Correct Option: D
Explanation:

If $a : b :: c : d$ then $b : a :: d : c$ is invertendo property of ratios

We have $3:7::2:9$
After applying invertendo we get
$7:3::9:2$
Option D is correct

If $x=7-4\sqrt 3$, the value of $x^2+\displaystyle\frac{1}{x^2}$ will be

properties of proportion ratio and proportions ratio and proportion maths

  1. $146$

  2. $148$

  3. $194$

  4. $196$

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Correct Option: C
Explanation:

Given,
$x=7-4\sqrt 3$
$\therefore \displaystyle\frac{1}{x}=\displaystyle\frac{1}{(7-4\sqrt 3)}\times \displaystyle\frac{7+4\sqrt 3}{7+4\sqrt 3}$
$=\displaystyle\frac{7+4\sqrt 3}{(49-48)}$
$=7+4\sqrt 3$
Now,
$x+\displaystyle\frac{1}{x}=(7-4\sqrt 3)+(7+4\sqrt 3)$
$=14$
$=> \left( x+\displaystyle\frac{1}{x}\right)^2=(14)^2$
Using $(a+b)^2=a^2+b^2+2ab$ we get,
$=> x^{ 2 }+\frac { 1 }{ x^{ 2 } } +2(x)(\frac { 1 }{ x } )=196$
$=> x^2+\displaystyle\frac{1}{x^2}=196-2$
$\therefore x^2+\displaystyle\frac{1}{x^2}=194$

If $x=3+\sqrt 8$ then the value of $x^2+\frac {1}{x^2}$ is

properties of proportion ratio and proportions ratio and proportion maths

  1. 30

  2. 32

  3. 34

  4. 36

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Correct Option: C
Explanation:

$x=3+\sqrt 8\Rightarrow \frac {1}{x}=\frac {1}{3+\sqrt 8}\times \frac {3-\sqrt 8}{3-\sqrt 8}=\frac {3-\sqrt 8}{9-8}$
$=3-\sqrt 8$
$x^2+\frac {1}{x^2}=\left (x+\frac {1}{x}\right )^2-2=36-2=34$

If $a : b :: c : d$ then $b : a :: d : c$. This property is known as :

properties of proportion ratio and proportions ratio and proportion maths

  1. Invertendo property

  2. Alternendo property

  3. Componendo property

  4. Dividendo property

Show answer
Correct Option: A
Explanation:

If $a : b :: c : d$ then $b : a :: d : c\Rightarrow $invertendo property.

proof
$\dfrac{a}{b}=\dfrac{c}{d}$

inverting both sides, we get
$\Rightarrow \dfrac{b}{a}=\dfrac{d}{c}$

After applying invertendo to $31:15::18:19$  we get $15:a::b:18$ 
What is the value of $a+b$

properties of proportion ratio and proportions ratio and proportion maths

  1. $40$

  2. $50$

  3. $30$

  4. $60$

Show answer
Correct Option: B
Explanation:

If $a : b :: c : d$ then $b : a :: d : c$ is invertendo property of ratios

We have $31:15::18:19$
After applying invertendo we get
$15:31::19:18\equiv  15:a::b:18\ a+b=31+19=50$
Option B is correct

Find the value of $a$ and $b$ respectively
After applying invertendo to $2:5::3:9$  we get $a:2::9:b$ 

properties of proportion ratio and proportions ratio and proportion maths

  1. $9$ and $2$

  2. $2$ and $9$

  3. $3$ and $5$

  4. $5$ and $3$

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Correct Option: D
Explanation:

If $a : b :: c : d$ then $b : a :: d : c$ is invertendo property of ratios

We have $2:5::3:9$
After applying invertendo we get
$5:2::9:3\equiv a:2::9:b$
$\Rightarrow a=5,b=3$
Option D is correct

Here 'x' in the following is : $\dfrac{\sqrt{a+x}+\sqrt{a-x}}{\sqrt{a+x}-\sqrt{a-x}}=b$

properties of proportion ratio and proportions ratio and proportion maths

  1. $\dfrac{2ab}{(b^2+1)}$

  2. $\dfrac{2ab}{a+b}$

  3. $\dfrac{a+b}{2ab}$

  4. $\dfrac{b^2+1}{2ab}$

Show answer
Correct Option: A
Explanation:

$\dfrac{\sqrt{a+x}+\sqrt{a-x}}{\sqrt{a+x}-{\sqrt{a-x}}}=\dfrac{b}{1}$

$\Rightarrow \dfrac{\sqrt{a+x}}{\sqrt{a-x}}=\dfrac{b+1}{b-1}$

$\Rightarrow \dfrac{a+x}{a-x}=\dfrac{(b+1)^2}{(b-1)^2}$

$\Rightarrow \dfrac{(a+x)+(a-x)}{(a+x)-(a-x)}=\dfrac{(b+1)^2+(b-1)^2}{(b+1)^2-(b-1)^2}$

$\Rightarrow \dfrac{2a}{2x}=\dfrac{2(b^2+1)}{4b}$

$\Rightarrow x=\dfrac{2ab}{b^2+1}$