Tag: composition of ratios

Questions Related to composition of ratios

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, $x:z=15:8$

If $\cfrac{a}{2}=\cfrac{b}{3}=\cfrac{c}{4}$, then $a:b:c=$

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

  1. $2:3:4$

  2. $4:3:2$

  3. $3:2:4$

  4. None of these

Show answer
Correct Option: A
Explanation:

Given, $\displaystyle \frac{a}{2} = \frac{b}{3} = \frac{c}{4}$


Lets take $\displaystyle \frac{a}{2} = \frac{b}{3} = \frac{c}{4} = k$


So, $\dfrac{a}{2}  = k$

$a = 2k$

$\dfrac{b}{3}  = k$

$b = 3k$

$\dfrac{c}{4} = k$

$c = 4k$

i.e., $a : b: c = 2k : 3k : 4k$

$a : b; c = 2 : 3 : 4$  

If $a:b=3:4$, then $4a:3b=$

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

  1. $4:3$

  2. $3:4$

  3. $1:1$

  4. None of these

Show answer
Correct Option: C
Explanation:

Given $a:b=3:4$

or, $\dfrac{a}{b}=\dfrac{3}{4}$
or, $4a={3b}$.....(1).
Now,
$\dfrac{4a}{3b}$
$=\dfrac{3b}{3b}$ [ Using (1)]
$=\dfrac{1}{1}$.
So $4a:3b=1:1$.

What is the compounded ratio of $x : y, y : z$ and $z : w$

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

  1. $y : w$

  2. $x : w$

  3. $y : z$

  4. $x : z$

Show answer
Correct Option: B
Explanation:

By the defination of compound ratio these ratio can be expressed as
$\dfrac {x}{y} \times \dfrac {y}{z}\times \dfrac {z}{w} = \dfrac {x}{w}$
Hence $x : w$

If $a:b=5:7$ and $b:c=6:11$, then $a:b:c=$

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

  1. $35:49:66$

  2. $30:42:77$

  3. $30:42:55$

  4. None of these

Show answer
Correct Option: B
Explanation:

$\dfrac{a}{b} = \dfrac{5}{7} $       $\dfrac{b}{c} = \dfrac{6}{11}$


$\left.\begin{matrix} a = \dfrac{5}{7} b \end{matrix}\right|\begin{matrix} c = \dfrac{11 b}{6} \end{matrix}$

$a : b : c$

$\dfrac{5}{7} b : b : \dfrac{11b}{6}$

$\dfrac{5}{7} : 1 : \dfrac{11}{6}$

L.C.M of $7, 6$ is $42$

$\dfrac{5}{7} \times 42 : 42 \times 1 : \dfrac{11}{6} \times 42$

$5 \times 6 : 42 : 11 \times 7$

$30 : 42 : 77$

The compounded ratio of (2 : 3), (6 : 11), and (11 : 2) is

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

  1. 1:2

  2. 2:1

  3. 11:24

  4. 36:121

Show answer
Correct Option: B
Explanation:

Required compounded ratio $\displaystyle=\frac{2}{3}\times\frac{6}{11}\times\frac{11}{2}=2:1$