Tag: ratio and proportions

Questions Related to ratio and proportions

After applying invertendo to $30:50::80:20$  we get $50:a::b:80$ 
Find the value of $a-b$

properties of proportion ratio and proportions ratio and proportion maths

  1. $10$

  2. $20$

  3. $30$

  4. $40$

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Correct Option: A
Explanation:

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

We have $30:50::80:20$
After applying invertendo we get
$50:30::20:80\equiv 50:a::b:20\ a-b=30-20=10$
Option A is correct

After applying invertendo to $1:2::3:4$ we get

properties of proportion ratio and proportions ratio and proportion maths

  1. $1:2::3:4$

  2. $2:1::4:3$

  3. $1:3::2:4$

  4. $4:2::3:1$

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Correct Option: B
Explanation:

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

We have $1:2::3:4$
After applying invertendo we get
$2:1::4:3$
Option B is correct

After applying invertendo to $9:12::13:40$ we get

properties of proportion ratio and proportions ratio and proportion maths

  1. $9:13::12:40$

  2. $40:12::13:9$

  3. $9:12::13:40$

  4. $12:9::40:13$

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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 $9:12::13:40$
After applying invertendo we get
$12:9::40:13$
Option D is correct

If $\dfrac { a }{ b } =\dfrac { b }{ c } =\dfrac { c }{ d }$ then $\dfrac { { b }^{ 3 }+{ c }^{ 3 }+{ d }^{ 3 } }{ { a }^{ 3 }+{ b }^{ 3 }+{ c }^{ 3 } }$ will be equal to

properties of proportion ratio and proportions ratio and proportion maths

  1. $\dfrac{a}{b}$

  2. $\dfrac{b}{c}$

  3. $\dfrac{c}{d}$

  4. $\dfrac{d}{a}$

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Correct Option: D

If $\cfrac{{x}^{3}+{x}^{2}+x+1}{{x}^{3}-{x}^{2}+x-1}=\cfrac{{x}^{2}+x+1}{{x}^{2}-x+1}$, then the number of real-value of $x$ satisfying are

properties of proportion ratio and proportions ratio and proportion maths

  1. 0

  2. 1

  3. 2

  4. 3

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Correct Option: A
Explanation:

Given that


$\dfrac{x^3+x^2+x+1}{x^3-x^2+x+1}=\dfrac{x^2+x+1}{x^2-x+1}$

Performing Componendo Dividendo, we get

$\Rightarrow \dfrac{x^3+x^2+x+1+x^3-x^2+x+1}{x^3+x^2+x+1-x^3+x^2-x+1}=\dfrac{x^2+x+1+x^2-x+1}{x^2+x+1-x^2+x-1}$

$\Rightarrow \dfrac{2x^3+2x}{2x^2+2}=\dfrac{2x^2+2}{2x} $

$\Rightarrow \dfrac{x^3+x}{x^2+1}=\dfrac{x^2+1}{x} $

$\Rightarrow x(x^3+x)=(x^2+1)^2$

$\Rightarrow x^4+x^2=x^4+1+2x^2$

$\Rightarrow x^2=-1$

Hence no real values of $x$ satisfy this equation.

If $\cfrac{a+3d}{a+9d}=\cfrac{a+d}{a+5d}=k$, then $k$ is equal to $(a,d> 0)$

properties of proportion ratio and proportions ratio and proportion maths

  1. $\dfrac{1}{2}$

  2. $2$

  3. $6$

  4. $0$

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Correct Option: A
Explanation:

$\dfrac{a+3d}{a+9d}=\dfrac{a+d}{a+5d}$


Applying Componendo-Dividendo

$\Rightarrow \dfrac{a+3d+a+9d}{a+3d-a-9d}=\dfrac{a+d+a+5d}{a+d-a-5d}$ 

$\Rightarrow \dfrac{2a+12d}{-6d}=\dfrac{2a+6d}{-4d}$

$\Rightarrow \dfrac{a+6d}{-3}=\dfrac{a+3d}{-2}$

$\Rightarrow -2a-12d=-3a-9d$

$\Rightarrow a=3d$

Now, given that 

$\Rightarrow k=\dfrac{a+3d}{a+9d}$

$\Rightarrow k=\dfrac{3d+3d}{3d+9d}$

$\Rightarrow k=\dfrac{6d}{12d}$

$\Rightarrow k=\dfrac{1}{2}$

Solve for $x$:

$\dfrac{{2 - 7x}}{{1 - 5x}} = \dfrac{{3 + 7x}}{{4 + 5x}}$

properties of proportion ratio and proportions ratio and proportion maths

  1. $\dfrac{3}{2}$

  2. $\dfrac{1}{2}$

  3. $\dfrac{3}{4}$

  4. $\dfrac{1}{4}$

Show answer
Correct Option: B
Explanation:

$\dfrac{2-7x}{1-5x}=\dfrac{3+7x}{4+5x}$


Applying Alternendo,

$\Rightarrow \dfrac{2-7x}{3+7x}=\dfrac{1-5x}{4+5x}$

Now applying Componendo and Dividendo,
$\Rightarrow\dfrac{(2-7x)+(3+7x)}{(2-7x)-(3+7x)}=\dfrac{(1-5x)+(4+5x)}{(1-5x)-(4+5x)}$

$\Rightarrow\dfrac{5}{-1-14x}=\dfrac{5}{-3-10x}$

$\Rightarrow-1-14x=-3-10x\\$
$\Rightarrow 14x-10x=3-1\\$
$\Rightarrow 4x=2\\$
$\Rightarrow x=\dfrac{2}{4}=\dfrac{1}{2}$

A naughty student breaks the pencil in such a way that the ratio of two broken parts is same as that of the original length of the pencil to one of the larger part of the pencil, The ratio of the other part to the original length of pencil is:

properties of proportion ratio and proportions ratio and proportion maths

  1. $1 :2 \sqrt{5}$

  2. $2 : (3+\sqrt{5})$

  3. $2 : \sqrt{5}$

  4. $can't\ be\ determined $

Show answer
Correct Option: B
Explanation:

Suppose that,

The length of larger part be$=x$

And the smaller part be $=y$


So, their ratio is $x$ is to

$ \dfrac{x}{y}=\dfrac{kx+ky}{kx} $

$ \Rightarrow \dfrac{x}{y}=\dfrac{x+y}{x} $

$ \Rightarrow {{x}^{2}}=xy+{{y}^{2}} $

$ \Rightarrow {{x}^{2}}-{{y}^{2}}=xy $


Let $y=1$

$ {{x}^{2}}-{{1}^{2}}=x $

$ \Rightarrow {{x}^{2}}-x-1=0 $

$ \Rightarrow x=\dfrac{1\pm \sqrt{5}}{2}\,\,\,\,\left( \text{On}\,\text{solving}\,\text{by}\,\text{second}\,\text{degree}\,\text{equation}\,\text{rule} \right) $


Negative value cannot be considered

So,

$x=\dfrac{1+\sqrt{5}}{2}$ so, the ratio

$ \dfrac{x}{y}=\dfrac{\dfrac{1+\sqrt{5}}{2}}{1} $

$ \Rightarrow x:y=\left( 1+\sqrt{5} \right):2 $


Therefore,

$ \dfrac{y}{x+y}=\dfrac{2}{\left( 1+\sqrt{5}+2 \right)} $

$ \Rightarrow \dfrac{y}{x+y}=\dfrac{2}{\left( 3+\sqrt{5} \right)} $


Hence, this is the answer.

$\dfrac{7a-3b}{7c-3d}=1, then\ \dfrac{a}{b}=\dfrac{d}{c}$

properties of proportion ratio and proportions ratio and proportion maths

  1. True

  2. False

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Correct Option: B

If  a/b = x/y = p/q , then $\dfrac{6a + 9x + 2p}{6b + 9y + 2q}$ = _________. 

properties of proportion ratio and proportions ratio and proportion maths

  1. a/b

  2. x/y

  3. p/q

  4. All of these

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Correct Option: D
Explanation:

All of these.


Given,

$\dfrac{a}{b}=\dfrac{x}{y}=\dfrac{p}{q}$

Let, $\dfrac{a}{b}=\dfrac{x}{y}=\dfrac{p}{q}=k$

$\Rightarrow a=bk, x=yk, p=qk$

$\Rightarrow 6a=(6b)k, 9x=(9y)k, 2p=(2q)k$

Now,

$\dfrac{6a+9x+2p}{6b+9y+2q}$

$=\dfrac{(6b)k+(9y)k+(2q)k}{6b+9y+2q}$

$=k\dfrac{6b+9y+2q}{6b+9y+2q}$

$=k$

$=\dfrac{a}{b}=\dfrac{x}{y}=\dfrac{p}{q}$