Four balls ae numbered and dropped in a box. The four
numbers are in proportion. The sum of the squares is 50
and the sum of the mean is 5. The ratio of the first two is
1:3. What is the average of the four numbers on the balls?
1) 12
3) 6
4) 4
5) 11
2)3
Answers
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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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
Average = 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
From the given options , 3 is the correct answer.
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