Tag: composition of ratios

Questions Related to composition of ratios

The ratio compound of $2:3$ and sub-duplicate ratio of $4:9$ is __________.

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

  1. $16:81$

  2. $4:9$

  3. $2:1$

  4. $12:81$

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Correct Option: B
Explanation:

we have to find the ratio compound of $2:3$ and sub-duplicate ratio of $4:9$ 

The duplicate ratio of $a:b$ is also called compound ratio of $a:b$ and is equal to $a^{2}:b^{2}$
 Similarly, sub-duplicate ratio of  $a:b$ is $\sqrt{a}:\sqrt{b}$  
Therefore compound ratio of $2:3=4:9$ 
Sub-duplicate ratio of $4:9=\sqrt{4}:\sqrt{9}=2:3$
 Ratio compound =$\left ( \dfrac{4/9}{2/3} \right )^{2}=4:9$

What is the reciprocal ratio of $21 : 31$?

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

  1. $42 : 62$

  2. $62 : 42$

  3. $35 : 37$

  4. $31 : 21$

Show answer
Correct Option: D
Explanation:

The reciprocal ratio of $a : b$ is $b : a$
$\therefore$ reciprocal ratio of $21 : 31$ is $31 : 21$

The value of $x : y$ is _____, if $(4x + 7y) : (5x - y)$ is the duplicate ratio of $5 : 1$

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

  1. $32 : 121$

  2. $25 : 1$

  3. $1 : 5$

  4. $1 : 25$

Show answer
Correct Option: A
Explanation:

$(4x + 7y) : (5x - y)$ is the duplicate ratio of $5 : 1$.
Also, the duplicate ratio of $5 : 1$ is $25 : 1$.
$\therefore \dfrac {4x + 7y}{5x - y} = \dfrac {25}{1}\Rightarrow 4x + 7y = 125x - 25y$
$\therefore 125x - 4x = 7y + 25y$
$\therefore 121x = 32y$
$\therefore \dfrac {x}{y} = \dfrac {32}{121}$
$\therefore x : y = 32 : 121$

If $(x + y) : (x - y)$ is equal to the duplicate ratio of $3 : 1$, then $x : y = $ _____

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

  1. $1 : 3$

  2. $4 : 5$

  3. $5 : 4$

  4. $3 : 1$

Show answer
Correct Option: C
Explanation:

The duplicate ratio of $a : b$ is $a^{2}:b^{2}$
$\therefore$ The duplicate ratio of $3 : 1$ is $9 : 1$.
$\therefore \dfrac {x + y}{x - y} = \dfrac {9}{1}$
$\therefore x + y = 9x - 9y$
$\therefore y + 9y = 9x - x$
$\therefore 10y = 8x$
$\therefore \dfrac {x}{y} = \dfrac {10}{8} = \dfrac {5}{4}$
$\therefore x : y = 5 : 4$.

The value of $x$ is ____ if $(3x + 1) : (5x - 4)$ is the duplicate ratio of $5 : 6$

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

  1. $2$

  2. $4$

  3. $8$

  4. $6$

Show answer
Correct Option: C
Explanation:

$(3x + 1) : (5x - 4)$ is the duplicate ratio of $5 : 6$.
Also, the duplicate ratio of $5 : 6$ is $5^{2} : 6^{2} = 25 : 36$
$\therefore \dfrac {3x + 1}{5x - 4} = \dfrac {25}{36}$
$\therefore 108 x + 36 = 125x - 100$
$\therefore 125x - 108x = 36 + 100$
$\therefore 17x = 136$
$\therefore x = \dfrac {136}{17} = 8$

The duplicate ratio of $\dfrac {1}{6} : \dfrac {1}{5}$ is ____

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

  1. $36 : 25$

  2. $\dfrac {1}{5} : \dfrac {1}{6}$

  3. $25 : 36$

  4. $30 : 15$

Show answer
Correct Option: C
Explanation:

The duplicate ratio of $a : b$ is $a^{2} : b^{2}$
$\therefore$ The duplicate ratio of $\dfrac {1}{6} : \dfrac {1}{5}$ is $\left (\dfrac {1}{6}\right )^{2} : \left (\dfrac {1}{5}\right )^{2} = \left (\dfrac {1}{36}\right ) : \left (\dfrac {1}{25}\right )$
$= \dfrac {\dfrac {1}{36}}{\dfrac {1}{25}} = \dfrac {25}{36}$.

If $x : y = 4 : 9$ and $y : z = 3 : 8$, then the duplicate ratio of $x : z$ is ____

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

  1. $16 : 64$

  2. $9 : 64$

  3. $16 : 81$

  4. $1 : 36$

Show answer
Correct Option: D
Explanation:

$x : y = 4 : 9$
$y : z = 3 : 8$
$\therefore \dfrac {x}{y} \times \dfrac {y}{z} = \dfrac {4}{9}\times \dfrac {3}{8}$
$\therefore \dfrac {x}{z} = \dfrac {1}{6}\Rightarrow x : z = 1 : 6$
$\therefore$ The duplicate ratio of $x : z$ is $(1)^{2} : (6)^{2} = 1 : 36$.

If $a : b = 2 : 3$ and $b : c = 9 : 8$ then the duplicate ratio of $a : c$ is ____

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

  1. $9 : 16$

  2. $4 : 9$

  3. $81 : 64$

  4. $4 : 64$

Show answer
Correct Option: A
Explanation:

$a : b = 2 : 3$
$b : c = 9 : 8$
$\therefore \dfrac {a}{b} \times \dfrac {b}{c} = \dfrac {2}{3} \times \dfrac {9}{8}$
$\therefore \dfrac {a}{c} = \dfrac {3}{4} \Rightarrow a : c = 3 : 4$.
$\therefore$ The duplicate ratio of $3 : 4$ is $3^{2} : 4^{2} = 9 : 16$

One year ago the ratio between Laxman's and Gopal's salary was $3:4$. The ratio of their individual salaries between last year's and this year's salaries are $4:5$ and $2:3$ respectively. At present the total of their salary is $Rs. 4160$. At present, the salary of Laxman, is _______.

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

  1. $Rs. 1040$

  2. $Rs. 1600$

  3. $Rs. 2560$

  4. $Rs. 3120$

Show answer
Correct Option: B
Explanation:

Let the salaries of Laxman and Gopal one year before be ${L} _{1} \; & \; {G} _{1}$ respectively and now be ${L} _{2} \; & \;  {G} _{2}$ respectively.

Therefore, as given:-
$\cfrac{{L} _{1}}{{G} _{1}} = \cfrac{3}{4} \; \longrightarrow {eq}^{n} (i)$
$\cfrac{{L} _{1}}{{L} _{2}} = \cfrac{4}{5} \; \longrightarrow {eq}^{n} (ii)$
$\cfrac{{G} _{1}}{{G} _{2}} = \cfrac{2}{3} \; \longrightarrow {eq}^{n} (iii)$
${L} _{2} + {G} _{2} = 4160 \; \longrightarrow {eq}^{n} (iv)$
From ${eq}^{n} \; (ii) \; & \; (iii)$, we get
${L} _{1} = \cfrac{4}{5} {L} _{2} \; & \; {G} _{1} = \cfrac{2}{3} {G} _{2}$
On putting the value of ${L} _{1} \; & \; {G} _{1} \; in \; {eq}^{n} (i)$, we get
$\cfrac{\cfrac{4}{5} {L} _{2}}{\cfrac{2}{3} {G} _{2}} = \cfrac{3}{4}$
$\Rightarrow \cfrac{12 {L} _{2}}{10 {G} _{2}} = \cfrac{3}{4}$
$\Rightarrow \cfrac{{L} _{2}}{{G} _{2}} = \cfrac{5}{8}$
$\Rightarrow {G} _{2} = \cfrac{8}{5} {L} _{2} \; \longrightarrow {eq}^{n} {v}$
On solving ${eq}^{n} (iv) \; & \; (v)$, we get
${L} _{2} + \cfrac{8}{5} {L} _{2} = 4160$
$\Rightarrow \cfrac{13}{5} {L} _{2} = 4160$
$\Rightarrow {L} _{2} = 4160 \times \cfrac{5}{13} = 1600$
Hence, the salary of Laxman, at present, is Rs.1600

If $(4x + 3) : (9x + 10)$ is the triplicate ratio of $3 : 4$, then the value of x is ___

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

  1. $6$

  2. $12$

  3. $5$

  4. $4$

Show answer
Correct Option: A
Explanation:

The triplicate ratio of $3 : 4$ is $3^{3} : 4^{3} = 27 : 64$
$\therefore \dfrac {4x + 3}{9x + 10} = \dfrac {27}{64}$
$\Rightarrow 256x + 192 = 243x + 270$
$\Rightarrow 256x - 243x = 270 - 192$
$\Rightarrow 13x = 78$
$\Rightarrow x = \dfrac {78}{13} = 6$
$\Rightarrow x = 6$