Question

Three numbers are in the ratio 2:3 : 4. The sum of their cubes is 33957. Find the
numbers.​with explaination

Answer 1

Answer:

Given :-

• Three numbers are in the ratio of 2 : 3 : 4.
• The sum of their cubes is 33957.

To Find :-

• What are the numbers.

Solution :-

Let,

$\mapsto \sf\bold{First\: number =\: 2x}$

$\mapsto \sf\bold{Second\: number =\: 3x}$

$\mapsto \sf\bold{Third\: number =\: 4x}$

According to the question,

$\implies \sf (2x)^3 + (3x)^3 + (4x)^3 =\: 33957$

$\implies \sf 8x^3 + 27x^3 + 64x^3 =\: 33957$

$\implies \sf 99x^3 =\: 33957$

$\implies \sf x^3 =\: \dfrac{\cancel{33957}}{\cancel{99}}$

$\implies \sf x^3 =\: \dfrac{343}{1}$

$\implies \sf x^3 =\: 343$

$\implies \sf x =\: \sqrt[3]{343}$

$\implies \sf\bold{\purple{x =\: 7}}$

Hence, the required numbers are :

$\mapsto$ First number :

$\longrightarrow \sf 2x$

$\longrightarrow \sf 2(7)$

$\longrightarrow \sf 2 \times 7$

$\longrightarrow \sf\bold{\red{14}}$

$\mapsto$ Second number :

$\longrightarrow \sf 3x$

$\longrightarrow \sf 3(7)$

$\longrightarrow \sf 3 \times 7$

$\longrightarrow \sf\bold{\red{21}}$

$\mapsto$ Third number :

$\longrightarrow \sf 4x$

$\longrightarrow \sf 4(7)$

$\longrightarrow \sf 4 \times 7$

$\longrightarrow \sf\bold{\red{28}}$

Hence,

$\bigstar\: \: \sf\bold{\pink{First\: number =\: 14}}$

$\bigstar\: \: \sf\bold{\pink{Second\: number =\: 21}}$

$\bigstar\: \: \sf\bold{\pink{Third\: number =\: 28}}$

$\therefore$ The numbers are 14, 21 and 28 respectively.

Answer 2

Given:

• Three numbers are in the ratio 2:3 : 4
• sum of their cubes is 33957

To Find:

• The numbers which make the above given statement true

Solution:

Let us assume that,

• The first number is 2x
• The second number is 3x
• The third number is 4x

According to the question:

$\longrightarrow \tt \: (2 {x})^{3} + (3 {x})^{3} + (4 {x})^{3} = 33957$

$\longrightarrow \tt \: 8{x}^{3} + 27{x}^{3} + 64{x}^3 = 33957$

$\longrightarrow \tt \: 99 {x}^{3} = 33957$

$\longrightarrow \tt \: {x}^{3} = \cancel\dfrac{33957}{99}$

$\longrightarrow \tt \: {x}^{3} = 343$

$\longrightarrow \tt \: x = \sqrt[3]{343}$

$\longrightarrow \tt \:{\purple {\underline{\boxed{\frak{x=7}}}\bigstar}}$

Therefore:

${ \longrightarrow} \tt \: 1st \: number = 2x = 2(7) = 14 \\ \\ \longrightarrow \tt \: 2nd \: number = 3x = 3(7) = 21 \\ \\ \longrightarrow \tt \: 3rd \: number = 4x = 4(7) = 28$

∴ The numbers are 14,21,28 respectively