Question

A discount series of 10% ,20%,40%is equivalent to a single discount of

Answer 1

Let the initial price of the article be $x.$

After a discount of $10\%,$ the new price of the article will be,

$\longrightarrow x_1=\left[1-\dfrac{10}{100}\right]x$

$\longrightarrow x_1=\dfrac{90}{100}\,x\quad\quad\dots(1)$

After a discount of $20\%,$ the new price of the article will be,

$\longrightarrow x_2=\left[1-\dfrac{20}{100}\right]x_1$

$\longrightarrow x_2=\dfrac{80}{100}\,x_1$

From (1),

$\longrightarrow x_2=\dfrac{80}{100}\times\dfrac{90}{100}\,x$

$\longrightarrow x_2=\dfrac{72}{100}\,x\quad\quad\dots(2)$

After a discount of $30\%,$ the new price of the article will be,

$\longrightarrow x_3=\left[1-\dfrac{40}{100}\right]x_2$

$\longrightarrow x_3=\dfrac{60}{100}\,x_2$

From (2),

$\longrightarrow x_3=\dfrac{60}{100}\times\dfrac{72}{100}\,x$

$\longrightarrow x_3=\dfrac{43.2}{100}\,x\quad\quad\dots(3)$

So the single discount of this reduction is,

$\longrightarrow d=\dfrac{x-x_3}{x}\times100$

From (3),

$\longrightarrow d=\dfrac{x-\dfrac{43.2}{100}\,x}{x}\times100$

$\longrightarrow d=\dfrac{\left[1-\dfrac{43.2}{100}\right]x}{x}\times100$

$\longrightarrow d=\left[1-\dfrac{43.2}{100}\right]\times100$

$\longrightarrow d=\dfrac{100-43.2}{100}\times100$

$\longrightarrow\underline{\underline{d=56.8\%}}$

So the discount series is equivalent to a single discount of $\bf{56.8\%}.$