Tag: ratio and proportions

Questions Related to ratio and proportions

If $x=\cfrac { 2\sqrt { 5 }  }{ \sqrt { 3 } +\sqrt { 5 }  } $, then what is the value of $\cfrac { x+\sqrt { 5 }  }{ x-\sqrt { 5 }  } +\cfrac { x+\sqrt { 3 }  }{ x-\sqrt { 3 }  } $

properties of proportion ratio and proportions ratio and proportion maths

  1. $\sqrt {5}$

  2. $\sqrt {3}$

  3. $\sqrt {15}$

  4. $2$

Show answer
Correct Option: D
Explanation:

$x=\cfrac { 2\sqrt { 5 }  }{ \sqrt { 3 } +\sqrt { 5 }  } \Rightarrow \cfrac { x }{ \sqrt { 3 }  } =\cfrac { 2\sqrt { 5 }  }{ \sqrt { 3 } +\sqrt { 5 }  } $
and $\cfrac { x }{ \sqrt { 5 }  } =\cfrac { 2\sqrt { 3 }  }{ \sqrt { 3 } +\sqrt { 5 }  } $
Applying components and dividendo, we get
$\cfrac { x+\sqrt { 5 }  }{ x-\sqrt { 5 }  } =-\left( 7+2\sqrt { 15 }  \right) $
and
$\cfrac { x+\sqrt { 3 }  }{ x-\sqrt { 3 }  } =9+2\sqrt { 15 } $
$\Rightarrow \cfrac { x+\sqrt { 5 }  }{ x-\sqrt { 5 }  } +\cfrac { x+\sqrt { 3 }  }{ x-\sqrt { 3 }  } =2$

Which of the following ratios is equal to $13:4$ in its simplest form?

properties of proportion ratio and proportions ratio and proportion maths

  1. $18:8$

  2. $105:36$

  3. $91:28$

  4. $144:250$

Show answer
Correct Option: C
Explanation:

A.  $18 : 8 = \dfrac{18}{8}$

      Cancelling both numerator and denominator by $2$, the ratio becomes $9 : 4$
      Hence this option is wrong.

B. $105 : 36 = \dfrac{105}{36}$
    Cancelling both numerator and denominator by $3$, the ratio becomes $35 : 12$
     Hence this option is wrong.

C.  $91 : 28 = \dfrac{91}{28}$
      Cancelling both numerator and denominator by $7$, the ratio becomes $13 : 4$
      Therefore this is correct option. 

D.  $144 : 250 = \dfrac{144}{250}$
      Cancelling both numerator and denominator by $2$, the ratio becomes $72 : 125$
      Hence this option is also wrong. 

If $a : b = 3 : 5$  then $ a - b : a + b =$

properties of proportion ratio and proportions ratio and proportion maths

  1. $\displaystyle \frac{-1}{4}$

  2. $\displaystyle \frac{1}{4}$

  3. $-4$

  4. $4$

Show answer
Correct Option: A
Explanation:

Let $a = 3x$ and $b = 5x$.
$\therefore \displaystyle \frac{a-b}{a+b}=\frac{3x-5x}{3x+5x}=\frac{-2x}{8x}=-\frac{1}{4}$

If $\left( {{p^2} + {q^2}} \right)/\left( {{r^2} + {s^2}} \right) = \left( {pq} \right)/\left( {rs} \right)$, then what is the value of $\left( {p - q} \right)/\left( {p + q} \right)$ in terms of $r$ and $s$?

properties of proportion ratio and proportions ratio and proportion maths

  1. $\left( {r + s} \right)/\left( {r - s} \right)$

  2. $\left( {r - s} \right)/\left( {r + s} \right)$

  3. $\left( {r + s} \right)/\left( {r s} \right)$

  4. $\left( {r s} \right)/\left( {r - s} \right)$

Show answer
Correct Option: A

If $ \displaystyle \frac {1}{x} : \frac {1}{y} : \frac {1}{z} = 2:3:5, $ then $x:y:z =?$

properties of proportion ratio and proportions ratio and proportion maths

  1. $2:3:5$

  2. $15:10:6$

  3. $5:3:2$

  4. $6:10:15$

Show answer
Correct Option: B
Explanation:

$\dfrac { 1 }{ x } :\dfrac { 1 }{ y } :\dfrac { 1 }{ z } =\quad 2:3:5\ \dfrac { yz:xz:xy }{ xyz } =\quad 2:3:5\ yz:xz:xy\quad =\quad 2xyz:3xyz:5xyz\ 1:1:1=\quad 2x:3y:5z\ x:y:z=\quad \dfrac { 1 }{ 2 } :\dfrac { 1 }{ 3 } :\dfrac { 1 }{ 5 } =\dfrac { 15:10:6 }{ 30 } \ So,\quad x:y:z\quad is\quad 15:10:6$

The reciprocal of $\dfrac {-5}{13}$ is _____

composition of ratios types of ratios ratio and proportions ratio and proportion maths

  1. $\dfrac {5}{13}$

  2. $\dfrac {-13}{5}$

  3. $\dfrac {13}{5}$

  4. $\dfrac {-5}{13}$

Show answer
Correct Option: B
Explanation:
The reciprocal (also known as the multiplicative inverse) is the number we have to multiply to get an answer equal to the multiplicative number with recipocal of it is 1.
Then $\frac{-5}{13}\times \frac{-13}{5}=1$.
So recipocal of $\frac{-5}{13}$ is $\frac{-13}{5}$.
So answer is (B) $\frac{-13}{5}$.

 

The subtriplicate ratio of $a : b$ is ____

composition of ratios types of ratios ratio and proportions ratio and proportion maths

  1. $a^{2} : b^{2}$

  2. $a^{3} : b^{3}$

  3. $\sqrt {a} : \sqrt {b}$

  4. $\sqrt [3]{a} : \sqrt [3]{b}$

Show answer
Correct Option: D
Explanation:

The subtriplicate ratio of $a : b$ is $\sqrt [3]{a} : \sqrt [3]{b} = (a)^{\frac {1}{3}} : (b)^{\frac {1}{3}}$

If $\dfrac {y}{x-z}=\dfrac{y+x}{z}=\dfrac{x}{y}$ then find $x:y:z$

composition of ratios types of ratios ratio and proportions ratio and proportion maths

  1. $1:2:3$

  2. $3:2:1$

  3. $4:2:3$

  4. $2:4:7$

Show answer
Correct Option: C
Explanation:


$ \dfrac{y}{x-z}=\dfrac{y+x}{z}=\dfrac{x}{y} $

 

Now,

$ \dfrac{y}{x-z}=\dfrac{x}{y} $

$ {{y}^{2}}={{x}^{2}}-xz\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ......(1) $

 

And

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

$ {{y}^{2}}+xy=xz\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ......(2) $

$ {{x}^{2}}-xz+xy=xz $

$ x-z+y=z $

$ 2z=x+y\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ......(3) $

 

$ And $

$ \dfrac{y}{x-z}=\dfrac{y+x}{z} $

$ yz=xy-yz+{{x}^{2}}-xz $

$ 2yz=xy+{{x}^{2}}-xz $

$ 2yz=x\left( y+x \right)-xz $                    [From equation (3)]

$ 2yz=2xz-xz $

$ 2yz=xz $

$ 2y=x $

$ \dfrac{x}{y}=\dfrac{2}{1}\,\,\,\,\,\,\,\,......\,\,\left( 4 \right) $


Substituting this value in equation (3), we get

$ 2z=2y+y $

$ 2z=3y $

$ \dfrac{y}{z}=\dfrac{2}{3}\,\,\,\,\,......\,\,\left( 5 \right) $


By equation (4) and (5), we get

$ x:y:z=4:2:3 .$


Hence, this is the answer.

If $\left( {p - q} \right)\,:\left( {q - x} \right)\,$ be the duplicate ratio of $p:q$, then : $\dfrac{1}{p} + \dfrac{1}{q} = \dfrac{1}{x}$

composition of ratios types of ratios ratio and proportions ratio and proportion maths

  1. True

  2. False

Show answer
Correct Option: A
Explanation:
$\left(p-x\right):\left(q-x\right)$ is the duplicate ratio of $p:q$

we know,
 if $a^2 : b^2$ is the duplicate  ratio of $a : b$
         now a/c to question,
$(p -x) : (q - x)$ is the duplicate ratio of $p : q$ 
so, from above rule,
$(p -x ) : (q - x ) = p^2 : q^2$


So,$\dfrac{{p}^{2}}{{q}^{2}}=\dfrac{p-x}{q-x}$

$\Rightarrow\,\dfrac{q-x}{{q}^{2}}=\dfrac{p-x}{{p}^{2}}$

$\Rightarrow\,\dfrac{q}{{q}^{2}}-\dfrac{x}{{q}^{2}}=\dfrac{p}{{p}^{2}}-\dfrac{x}{{p}^{2}}$

$\Rightarrow\,\dfrac{1}{q}-\dfrac{x}{{q}^{2}}=\dfrac{1}{p}-\dfrac{x}{{p}^{2}}$

$\Rightarrow\,\dfrac{1}{q}-\dfrac{1}{p}=\dfrac{x}{{q}^{2}}-\dfrac{x}{{p}^{2}}$

$\Rightarrow\,\dfrac{p-q}{pq}=\dfrac{x\left({p}^{2}-{q}^{2}\right)}{{p}^{2}{q}^{2}}$

$\Rightarrow\,p-q=\dfrac{x\left(p-q\right)\left(p+q\right)}{pq}$

$\Rightarrow\,1=\dfrac{x\left(p+q\right)}{pq}$

$\Rightarrow\,\dfrac{1}{x}=\dfrac{\left(p+q\right)}{pq}$

$\Rightarrow\,\dfrac{1}{x}=\dfrac{1}{q}+\dfrac{1}{p}$

$\therefore\,\dfrac{1}{p}+\dfrac{1}{q}=\dfrac{1}{x}$

Hence the given statement is true.

If $2x=3y$ and $4y=5z$, then $x:z=$

composition of ratios types of ratios ratio and proportions ratio and proportion maths

  1. $4:3$

  2. $8:15$

  3. $3:4$

  4. $15:8$

Show answer
Correct Option: D
Explanation:

Given,

$2x=3y$

or, $\dfrac{x}{y}=\dfrac{3}{2}$.....(1).

Again 

$4y=5z$

or, $\dfrac{y}{z}=\dfrac{5}{4}$.....(2).

Now multiplying (4) and (5) we get,

$\dfrac xy \times \dfrac yz=\dfrac 32 \times \dfrac 54$

$\dfrac{x}{z}=\dfrac{15}{8}$

or, $x:z=15:8$