Question

If x and y are inversely proportional, find the values of ${X_1},{X_2},{Y_1},{Y_2}$ in the table given below:

$x$	$8$	${X_1}$	$16$	${X_2}$	$80$
$y$	${Y_1}$	$4$	$5$	$2$	${Y_2}$

Answer 1

Hint: Inverse proportionality is the relation between two quantities in which if one quantity increases, then the other quantity decreases, and if one quantity decreases, then the other quantity increases. Here we will use this relation to find the proportionality constant between $x$ and $y$.

Complete step-by-step solution:
The given condition is: $x$ is inversely proportional to $y$.
According to inverse proportionality relation,
\[x\propto \dfrac{1}{y}\]
Here, $\propto $ is the proportionality sign.
Removing the proportionality sign and replacing it with a constant $k$,
$x = k \times \dfrac{1}{y}$
Here, $k$ is the proportionality constant.
Taking $y$ to the left side of the equation,
$xy = k$
Now we can say that,

$x$	$8$	${X_1}$	$16$	${X_2}$	$80$	$k$
$y$	${Y_1}$	$4$	$5$	$2$	${Y_2}$

Consider, $x = 8,y = {Y_1}$ and $x = 16,y = 5$

$x$	$8$	$16$	$k$
$y$	${Y_1}$	$5$

According to $xy = k$,
$8 \times {Y_1} = k = 16 \times 5$
Equating the required two equations,
$\therefore 8 \times {Y_1} = 16 \times 5$
Taking all the numbers on one side and variables on other,
${Y_1} = \dfrac{{16 \times 5}}{8}$
Dividing the numbers,
${Y_1} = 2 \times 5$
Multiplying the numbers,
$\therefore {Y_1} = 10$
Consider, $x = {X_1},y = 4$ and $x = 16,y = 5$

$x$	${X_1}$	$16$	$k$
$y$	$4$	$5$

According to $xy = k$,
${X_1} \times 4 = k = 16 \times 5$
Equating the required two equations,
$\therefore {X_1} \times 4 = 16 \times 5$
Taking all the numbers on one side and variables on other,
${X_1} = \dfrac{{16 \times 5}}{4}$
Dividing the numbers,
${X_1} = 4 \times 5$
Multiplying the numbers,
$\therefore {X_1} = 20$
Consider, $x = {X_2},y = 2$ and $x = 16,y = 5$

$x$	$16$	${X_2}$	$k$
$y$	$5$	$2$

According to $xy = k$,
${X_2} \times 2 = k = 16 \times 5$
Equating the required two equations,
$\therefore {X_2} \times 2 = 16 \times 5$
Taking all the numbers on one side and variables on other,
${X_2} = \dfrac{{16 \times 5}}{2}$
Dividing the numbers,
${X_2} = 8 \times 5$
Multiplying the numbers,
$\therefore {X_2} = 40$
Consider, $x = 80,y = {Y_2}$ and $x = 16,y = 5$

$x$	$16$	$80$	$k$
$y$	$5$	${Y_2}$

According to $xy = k$,
${Y_2} \times 80 = k = 16 \times 5$
Equating the required two equations,
$\therefore {Y_2} \times 80 = 16 \times 5$
Taking all the numbers on one side and variables on other,
${Y_2} = \dfrac{{16 \times 5}}{{80}}$
Dividing the numbers,
${Y_2} = \dfrac{5}{5}$
Multiplying the numbers,
$\therefore {Y_2} = 1$
Therefore, ${X_1} = 20,{X_2} = 40,{Y_1} = 10,{Y_2} = 1$

Note: Two quantities can be compared using direct or indirect relation between them. In direct proportionality, if one quantity increases then the other one also increases, and if one quantity decreases, then the other one also decreases. Direct proportionality between $x$ and $y$ is given by $x\propto y$ and after removing the proportionality sign and introducing a proportionality constant $k$, the relation becomes $xy = k$.