Question

Four balls are numbered and dropped in a box. The four numbers are in proportion. The sum of
their squares is 50 and the sum of their means is 5. The ratio of the first two numbers is 1:3.
What is the average of the four numbers on the balls?
12

Answer 1

Solution :-

Let us assume that , four balls are numbered as a , b , c and d .

Since they are in proportion ,

So,

→ a : b : : c : d

or,

→ a * d = b * c -------- Eqn.(1) { Product of extreme = Product of means . }

now, given that,

→ a² + b² + c² + d² = 50 -------- Eqn.(2)

→ Sum of means = b + c = 5 -------- Eqn.(3)

→ a : b = 1 : 3 ---------- Eqn.(4)

let us assume that, a is equal to 1 unit and b is equal to 3 unit .

putting value of b in Eqn.(3)

→ 3 + c = 5

→ c = 5 - 3 = 2 unit .

now, putting values of a, b and c in Eqn.(1),

→ a * d = b * c

→ 1 * d = 3 * 2

→ d = 6 unit .

putting all values in Eqn.(2) now,

→ a² + b² + c² + d² = 50

→ (1)² + (3)² + (2)² + (6)² = 50

→ 1 + 9 + 4 + 36 = 50

→ 50 = 50

therefore, we can conclude that,

• a = 1
• b = 3
• c = 2
• d = 6

hence,

→ Required Average = Sum of all terms / 4 = (1 + 3 + 2 + 6) / 4 = 12 / 4 = 3 (Ans.)

∴ The average of the four numbers on the balls is equal to 3 .

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Answer 2

Given: The four numbers are in proportion.

The sum of their squares is 50 and the sum of their means is 5.

The ratio of the first two numbers is 1:3.

To Find : the average of the four numbers on the balls

Solution:

Let say four numbers in proportion are:

a:b::c:d

The ratio of the first two numbers is 1:3.

=> b = 3a

sum of their means is 5

=> b + c = 5

=> c = 5 - b

=>  c = 5 - 3a

a:b:c:d are in proportion

hence

a * d = b * c

=> a * d = 3a * (5 - 3a)

=> d = 3(5 - 3a)

a : b :  c : d

a : 3a : 5 - 3a : 3(5-3a)

average of four numbers = (a + 3a  + 5 - 3a + 15 - 9a) /4

= (20 - 8a)/4

= 5 - 2a

sum of their squares is 50

=>a² + (3a)² + (5-3a)² + (3(5-3a))² = 50

=> a² + 9a² + (5-3a)² + 9(5-3a)² = 50

=> 10a² + 10(5-3a)² = 50

=> a² +  (5-3a)² = 5

=> a² + 25 + 9a² - 30a = 5

=> 10a² - 30a + 20 = 0

=> a² - 3a + 2 = 0

=> (a - 2)(a - 1) = 0

=> a = 2 , a  = 1

5 - 2a

a = 1  => average = 5-2(1) =  3

a = 2  => average =5-2(2) = 1

a = 1  

a : 3a : : 5 - 3a : 3(5-3a)

1  : 3 :: 2 : 6

a = 2

a : 3a : : 5 - 3a : 3(5-3a)

2 : 6 :: -1 : -3

average of the four numbers on the balls  can be  3  or  1

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