Tag: ratio and proportions

The reciprocal ratio of $a : b$ is $b : a$
$\therefore$ reciprocal ratio of $121x^{3} : 25x^{3}$ is $25x^{3} : 121x^{3}$

The reciprocal ratio of $a : b$ is $b : a$
$\therefore$ reciprocal ratio of $81 : 121$ is $121 : 81$.

The reciprocal ratio of $a : b$ is $b : a$
$\therefore$ reciprocal ratio of $8 : 15$ is $15 : 8$

The reciprocal ratio of $a : b$ is $b : a$
$\therefore$ reciprocal ratio of $(x - 5) : (x - 7)$ is $(x - 7) : (x - 5)$

The reciprocal ratio of $a : b$ is $b : a$
$\therefore$ reciprocal ratio of $(x + 12) : (x + 21)$ is $(x + 21) : (x + 12)$

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

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

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

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

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