Tag: ratio and proportions

Questions Related to ratio and proportions

Three numbers $A,B$ and $C$ are in the ratio $12\colon15\colon25$. If the sum of these numbers is $312$, find the ratio between the difference of $A$ and $B$ and the difference of $C$ and $B$.

properties of proportion ratio and proportions ratio and proportion maths

  1. $\;3\colon7$

  2. $\;10\colon3$

  3. $\;3\colon10$

  4. $\;7\colon3$

Show answer
Correct Option: C
Explanation:

The ratio of $A,B,C =12:15:25$
Then, total of the ratio is $12+15+25=52$
Then, $ A=\displaystyle \frac{12}{52}\times 312=72$

$B=\displaystyle \frac{15}{52}\times 312=90$

$C=\displaystyle \frac{25}{52}\times 312=150$
Then, $B-A=18$ and $C-B=60$
Hence, the required ratio $=18:60\Rightarrow 3:10$

Mark the correct alternative of the following.
If $x : y =1 : 1$, then $\dfrac{3x+4y}{5x+6y}=?$

properties of proportion ratio and proportions ratio and proportion maths

  1. $\dfrac{7}{11}$

  2. $\dfrac{17}{11}$

  3. $\dfrac{17}{23}$

  4. $\dfrac{4}{5}$

Show answer
Correct Option: A
Explanation:

Given, $x:y=1:1$

or, $\dfrac{x}{y}=\dfrac{1}{1}$
or,  $x=y$......(1)

Now,
$\dfrac{3x+4y}{5x+6y}$
$=\dfrac{7x}{11x}$ [ Using (1)]
$=\dfrac{7}{11}$.

The given property $a : b :: c : d$ then $(a - b) : b :: (c - d) : d.$ is known as

properties of proportion ratio and proportions ratio and proportion maths

  1. Alternendo property

  2. Dividendo property

  3. Componendo property

  4. Invertendo Property

Show answer
Correct Option: B
Explanation:

If $\dfrac{a}{b}=\dfrac{c}{d},$ then by dividendo property

$\dfrac{a-b}{b}=\dfrac{c-d}{d}\Rightarrow (a-b):b::(c-d):d$

After applying invertendo to $1:2::8:9$ we get:

properties of proportion ratio and proportions ratio and proportion maths

  1. $2:1::9:8$

  2. $1:8::2:9$

  3. $8:9::1:2$

  4. $1:9::8:2$

Show answer
Correct Option: A
Explanation:

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

We have $1:2::8:9$
After applying invertendo we get
$2:1::9:8$
Option A is correct

The given property $a : b :: c : d$ then $a : c :: b : d$ is known as:

properties of proportion ratio and proportions ratio and proportion maths

  1. Componendo property

  2. Dividendo property

  3. Invertendo property

  4. Alternendo property

Show answer
Correct Option: D
Explanation:
If $\dfrac{a}{b}=\dfrac{c}{d}$ then, by alternedo

$\dfrac{a}{c}=\dfrac{b}{d}\Rightarrow a:c::b:d$

If $\displaystyle \frac{x}{y}=\frac{6}{5}$ then $\displaystyle \frac{x^{2}+y^{2}}{x^{2}-y^{2}}$ is:

properties of proportion ratio and proportions ratio and proportion maths

  1. $\displaystyle \frac{36}{25}$

  2. $\displaystyle \frac{25}{36}$

  3. $\displaystyle \frac{61}{11}$

  4. $\displaystyle \frac{11}{61}$

Show answer
Correct Option: C
Explanation:

$\displaystyle \frac{x}{y}=\frac{6}{5}$

$x = 6k, y = 5k$

$\displaystyle \frac{x^{2}+y^{2}}{x^{2}-y^{2}}=\frac{\left ( 6k \right )^{2}+\left ( 5k \right )^{2}}{\left ( 6k \right )^{2}-\left ( 5k \right )^{2}}=\frac{61k^{2}}{11k^{2}}=\frac{61}{11}$

If $x=\cfrac { 4ab }{ a+b } $ then value of $\cfrac { x+2a }{ x-2a } +\cfrac { x+2b }{ x-2b } $

properties of proportion ratio and proportions ratio and proportion maths

  1. $a$

  2. $b$

  3. $0$

  4. $2$

Show answer
Correct Option: D
Explanation:

Given, $x = 4ab/(a+b)$

=> $x = 2a\times 2b / (a+b)$
=> $x/2a = 2b / (a+b)$
Using componendo and dividendo 
=> $x+2a / x-2a = 2b+a+b / 2b-a-b$
=> $x+2a / x-2a = 3b+a / b-a$

Similarly, $x+2b / x-2b = 3a+b / a-b$

Now, adding both,
$[x+2a / x-2a] + [x+2b / x-2b ] = [3b+a / b-a ] + [ 3a +b / a-b ]$
RHS => $[3b+a / b-a ] - [ 3a+b / b-a ]$
=> $3b+a-3a-b / b-a$
=> $2b-2a / b-a$
=> $2(b-a)/b-a$
=> $2$

Hence, LHS $= 2$ 

If $a : b = c : d = e : f$, then the value of each ratio is $(a + c + e) : (b + d + f)$
This property is called as 

properties of proportion ratio and proportions ratio and proportion maths

  1. Componendo property

  2. Convertendo property

  3. Addendo property

  4. Dividendo property

Show answer
Correct Option: C
Explanation:

$\dfrac{a}{b}=\dfrac{c}{d}=\dfrac{e}{f}=k$

$a=bk,c=dk,e=fk\ \Rightarrow \dfrac{a+c+e}{b+d+f}=\dfrac{bk+dk+fk}{b+d+f}=k\ \Rightarrow a:b=c:d=e:f=(a+c+e):(b+d+f) $
addendo property

 $\dfrac{a+be^y}{a-be^y} = \dfrac{b+ce^y}{b-ce^y}  =  \dfrac{c+de^y}{c-de^y}$, then $a,b,c,d$  are  in

properties of proportion ratio and proportions ratio and proportion maths

  1. $A.P.$

  2. $G.P.$

  3. $H.P.$

  4. $A.G.P.$

Show answer
Correct Option: B

If $\cfrac { { a }^{ 3 }+3a{ b }^{ 2 } }{ 3{ a }^{ 2 }b+{ b }^{ 3 } } =\cfrac { { x }^{ 3 }+3x{ y }^{ 2 } }{ 3{ x }^{ 2 }y+{ y }^{ 3 } } $ then

properties of proportion ratio and proportions ratio and proportion maths

  1. $bx=ay$

  2. $by=ax$

  3. ${ b }^{ 2 }y={ a }^{ 2 }x$

  4. ${ b }^{ 2 }x={ a }^{ 2 }y$

Show answer
Correct Option: A
Explanation:

$\dfrac{{{a^3} + 3a{b^2}}}{{3{a^2}b + {b^3}}} = \dfrac{{{x^3} + 3x{y^2}}}{{3{x^2}y + {y^3}}}$


Use of compodendo and Devidendo

$\dfrac{{{a^3} + 3a{b^2} + 3{a^2}b + {b^3}}}{{3{a^2}b + {b^3} - 3{a^2}b + {b^3}}} = \dfrac{{{x^3} + 3x{y^2} + 3{x^2}y + {y^3}}}{{3{x^2}y + {y^3} - 3{x^2}y + {y^3}}}$

$ \Rightarrow \dfrac{{{{\left( {a + b} \right)}^3}}}{{{{(a - b)}^3}}} = \dfrac{{{{\left( {x + y} \right)}^3}}}{{{{(x - y)}^3}}}$

$ \Rightarrow {\left( {\dfrac{{a + b}}{{a + b}}} \right)^3} = {\left( {\dfrac{{x + y}}{{x - y}}} \right)^3}$

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

$ \Rightarrow \left( {a + b} \right)\left( {x + y} \right) = \left( {x + y} \right)\left( {a - b} \right)$

$ \Rightarrow ax{\text{ }} - {\text{ }}ay{\text{ }} + {\text{ }}bx{\text{ }} - {\text{ }}by{\text{ }} = {\text{ }}xa{\text{ }} - {\text{ }}xb{\text{ }} + {\text{ }}ay{\text{ }} - {\text{ }}yb$

$ \Rightarrow bx{\text{ }} + {\text{ }}xb{\text{ }} = {\text{ }}ay{\text{ }} + {\text{ }}ay$

$ \Rightarrow 2bx{\text{ }} = {\text{ }}2ay$

$ \Rightarrow bx{\text{ }} = {\text{ }}ay$