Tag: types of ratios

Questions Related to types of ratios

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

If $(5x + 3) : (3x + 1)$ is the triplicate ratio of $4 : 3$, then $x =$ _____

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

  1. $57$

  2. $17$

  3. $\dfrac {17}{57}$

  4. $4$

Show answer
Correct Option: C
Explanation:

The triplicate ratio of $4 : 3$ is $4^{3} : 3^{3}$
$\therefore \dfrac {5x + 3}{3x + 1} = \dfrac {64}{27}$
$\Rightarrow 135x + 81 = 192x + 64$
$\Rightarrow 192x - 135x = 81 - 64$
$\Rightarrow 57x = 17$
$\therefore x = \dfrac {17}{57}$

If $x : y = 2 : 5, y : z = 15 : 8$ and $z : w = 3 : 2$, then find the triplicate ratio of $x : w$

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

  1. $\sqrt [3]{9} : \sqrt [3]{8}$

  2. $3 : 4$

  3. $729 : 512$

  4. $81 : 64$

Show answer
Correct Option: C
Explanation:

$x : y = 2 : 5, y : z = 15 : 8, z : w = 3 : 2$
$\therefore \dfrac {x}{y}\times \dfrac {y}{z}\times \dfrac {z}{w} = \dfrac {2}{5} \times \dfrac {15}{8}\times \dfrac {3}{2} = \dfrac {9}{8}$
$\therefore x : w = 9 : 8$
$\therefore$ The triplicate ratio of $9 : 8$ is $9^{3} : 8^{3} = 729 : 512$

The subduplicate ratio of $9 : 1$ is $(x + y) : (x - y)$. Then $x : y =$ _____

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

  1. $2 : 1$

  2. $1 : 2$

  3. $4 : 3$

  4. $6 : 4$

Show answer
Correct Option: A
Explanation:

The subduplicate ratio of $9 : 1$ is $\sqrt {9} : \sqrt {1} = 3 : 1$
$\therefore \dfrac {x + y}{x - y} = \dfrac {3}{1} \Rightarrow x + y = 3x - 3y$
$\Rightarrow 2x = 4y \Rightarrow \dfrac {x}{y} = \dfrac {4}{2} = \dfrac {2}{1}$

The value of $x$ is ____ if $(x - 4) : (x + 2)$ is the triplicate ratio of $1 : 2$

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

  1. $\dfrac {34}{7}$

  2. $\dfrac {7}{34}$

  3. $\dfrac {30}{7}$

  4. $\dfrac {7}{30}$

Show answer
Correct Option: A
Explanation:

The triplicate ratio of $1 : 2$ is $1^{3} : 2^{3} = 1 : 8$
$\therefore \dfrac {x - 4}{x + 2} = \dfrac {1}{8}\Rightarrow 8x - 32 = x + 2$
$\Rightarrow 8x - x = 2 + 32$
$\Rightarrow 7x = 34$
$\Rightarrow x = \dfrac {34}{7}$

The triplicate ratio of $(x + y)^{\frac {2}{3}} : (x - y)^{\frac {2}{3}}$ is _____

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

  1. $(x + y)^{2} : (x - y)^{2}$

  2. $(x + y) : (x - y)$

  3. $(x + y)^{3} : (x - y)^{3}$

  4. $(x + y)^{6} : (x - y)^{6}$

Show answer
Correct Option: A
Explanation:

The triplicate ratio of $a : b$ is $a^{3} : b^{3}$
$\therefore$ The triplicate ratio of $(x + y)^{\frac {2}{3}} : (x - y)^{\frac {2}{3}}$ is $[(x + y)^{\frac {2}{3}}]^{3} : [(x - y)^{\frac {2}{3}}]^{3}$
$= (x + y)^{2} : (x - y)^{2}$

The subduplicate ratio of $25x^{2} : 196y^{2}$ is _____

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

  1. $25x : 196 y$

  2. $x : y$

  3. $5x : 14y$

  4. $196 y^{2} : 25x^{2}$

Show answer
Correct Option: C
Explanation:

The subduplicate ratio of $a : b$ is $\sqrt {a} : \sqrt {b}$
$\therefore$ The subduplicate ratio of $25x^{2} : 196y^{2}$ is $\sqrt {25x^{2}} : \sqrt {196y^{2}}$
$= 5x : 14y$.

The duplicate ratio of $\sqrt {16} : \sqrt {64}$ is ____

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

  1. $1 : 2$

  2. $1 : 4$

  3. $4 : 16$

  4. $2 : 1$

Show answer
Correct Option: B
Explanation:

The given ratio can be simplified as $\sqrt {16} : \sqrt {64} = 4 : 8 = 1 : 2$
Now, the duplicate ratio of $a : b$ is $a^{2} : b^{2}$ and so that of $1 : 2$ is $1^{2} : 2^{2} = 1 : 4$.