Question

Three rational numbers are in the ratio 2:3:5. If the sum of their cubes is 67.5, the sum of the three numbers (expressed as a decimal) is
A)2.5
B)5.25
C)7.5
D)10​

Answer 1

Answer:

1000000000000000000000

Answer 2

Given : Three rational numbers are in the ratio 2:3:5 & the sum of their cubes is 67.5 .

Need To Find : The sum of three rational numbers (expressed as a decimal ) ?

⠀⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━━━━━━━━━━⠀

❍ Let's say , that the three rational number be 2n , 3n & 5n , respectively.

$\\\bigstar \:\underline {\pmb{\boldsymbol{ According \:\:to\:\: the\:\:Question\: \::\:}}}$

⠀⠀▪︎⠀The cubes of the rational numbers will add up to 67.5 .

$\\:\implies \sf \:\Big\{ \:First_{\:(No.)\:} \:\Big\}^3\:+\:\Big\{ \:Second_{\:(No.)}\:\Big\}^3\:+\:\Big\{ \:Third_{\:(No.)\:} \:\Big\}^3\:=\:67.5\:$

$:\implies \sf \:( \:2n \:)^3\:+\:( \:3n \:)^3\:+\:( \:5n \:)^3\:=\:67.5\:$

$:\implies \sf \:8n^3\:+\:27n^3\:+\:25n^3\:=\:67.5\:$

$:\implies \sf \:160n^3\:=\:67.5\:$

$:\implies \sf \:n^3\:=\:\cancel{\dfrac{67.5}{160}}\:$

$:\implies \sf \:n^3\:=\:0.421875\:$

$:\implies \sf \:n\:=\:\sqrt[3]{\Big( 0.421875\Big)}\:$

$:\implies \pmb {\underline {\boxed {\purple {\:\frak{ \:n\:\:=\:0.75\:}}}}}\:\bigstar \:$

Therefore,

• First Number , 2n = 2 × 0.75 = 1.5 ,
• Second Number , 3n = 3 × 0.75 = 2.25 &
• Third Number , 5n = 5 × 0.75 = 3.75 .

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$\underline {\dag\:\pmb{\frak{\:Sum \:of\:the\:numbers\:\::\:}}}$

$\twoheadrightarrow \sf Sum \:=\: First_{\:(No.)\:} \:+\:Second_{\:(No.)\:} \:+\: Third_{\:(No.)\:} \:\\\\\\\twoheadrightarrow \sf \:Sum \:=\: 1.5 \:+\:2.25 \:+\: 3.75 \:\\\\\\\twoheadrightarrow \sf \:Sum \:=\: 1.5 \:+\:2.25 \:+\: 3.75 \:\\\\\\ \twoheadrightarrow \sf \:Sum \:=\: 3.75 \:+\: 3.75 \:\\\\\\ \dashrightarrow \pmb {\underline {\boxed {\purple {\:\frak{ \:Sum\:\:=\:7.5\:}}}}}\:\bigstar \:$

$\therefore \:\underline {\sf Hence, \:The \:Sum\:of\:the\:numbers\:is\:\pmb{\sf Option \:C)\:7.5\:}\:.}$