Question

Of the four numbers, the first is twice the second, the second is one-third of the third, and the third is 5 times the fourth. The average of the numbers is 24.75. The largest of these numbers is

• A. $9$
• B. $25$
• C. $30$
• D. $45$

Answer 1

The correct option is D $45$

Let the four numbers be $x_1,x_2,x_3,x_4$

Given that first number is twice the second number.

Therefore, $x_1=2x_2$ ------(1)

Given that second number is one-third of the third.

Therefore, $x_2=\frac{1}{3}{x}_3$ -------(2)

Given that third number is 5 times the fourth.

Therefore, $x_3=5x_4$ ------(3)

Given that average of the four numbers is $24.75$

Therefore, $24.75=\frac{x_1+x_2+x_3+x_4}{4}$

$⟹24.75\times 4=x_1+x_2+x_3+x_4$

$⟹x_1+x_2+x_3+x_4=99$

$⟹2x_2+x_2+x_3+x_4=99$ (from (1))

$⟹3x_2+x_3+x_4=99$

$⟹3\left(\frac{1}{3}{x}_3\right)+x_3+x_4=99$ (from (2))

$⟹2x_3+x_4=99$

$⟹2\left(5x_4\right)+x_4=99$

$⟹11x_4=99$

$⟹x_4=9$

$⟹x_3=5x_4=5\left(9\right)=45$

$⟹x_2=\frac{1}{3}{x}_3=\frac{1}{3}\left(45\right)=15$

$⟹x_1=2x_2=2\left(15\right)=30$

Therefore, the largest number is $x_3=45$