Question

Divide 56 into four parts which are in ap such that the ratio of product of extremes of the products of means is 5:6 find the mean of the following data

Answer 1

Correct Question

Divide 56 into four parts which are in AP such that the ratio of product of extremes of the products of means is 5:6. Find the AP.

Answer

AP = 8, 12, 16 and 20

Solution

Let the four parts be divided as (a - 3d), (a-d) , (a+d) and (a+3d)

Sum of all these four parts (a-3d), (a-d), (a+d) and (a+3d) is 56

According to the question,

→ (a - 3d) + (a - d) + (a + d) + (a + 3d) = 56

→ a - 3d + a - d + a + d + a + 3d = 56

→ 4a = 56

→ a = 56/4

→ a = 14

Also given that, ratio of product of extremes of the products of means

Means, ( a1 × a4):(a2 × a3) = 5:6

→ (a1 × a4)/(a2 × a3) = 5/6

→ (14 - 3d)(14 + 3d)/(14 - d)(14 + d) = 5/6

Used identity: (a-b)(a+b) = a² - b²

→ (196 - 9d²) / (196 - d²) = 5/6

Cross- multiply them,

→ 6 × 196 - 54d² = 196 × 5 - 5d²

→ 6 × 196 - (5 × 196) = -5d² + 54d²

→ 196(6-5) = 49d²

→ 196 = 49d²

→ d² = 196/49

→ d² = 4

→ d = 2

Now,

Substitute value of a and d in assumed four parts

→ a - 3d = 14 - 3(2) = 8

→ a - d = 14 - 2 = 12

→ a + d = 14 + 2 = 16

→ a + 3d = 14 + 3(2) = 20

Hence, four parts which are in AP are: 8, 12, 16 and 20 .

Answer 2

||✪✪ QUESTION ✪✪||

Divide 56 into four parts which are in ap such that the ratio of product of extremes of the products of means is 5:6 ?

|| ✰✰ ANSWER ✰✰ ||

Let us assume that, the Four parts be (a-3d), (a-d) , (a+d) and (a+3d) .

Given that, Their sum is 56 .

So,

⟿ (a-3d) + (a-d) + (a+d) + (a+3d) = 56

⟿ 4a = 56

Dividing both sides by 4,

⟿ a = 14

So, Required Fours parts are :-

➪ a1 = (14-3d)

➪ a2 = (14-d)

➪ a3 = (14+d)

➪ a4 = (14+3d)

___________________________

Now, given that ratio of product of extremes of the products of means is 5:6 ,

we know that, if a:b::c:d are in ratio proportion , than

➾ a × d = b ×c

⇒ Product of extremes = Product of means

Or,

⇒ a1 x a4 = a2 x a4

____________

Putting values we get,

☞ (14-3d)*(14+3d) : (14-d)*(14+d) = 5 : 6

using (a+b)(a-b) = a² - b²

☞ (196 - 9d²) : (196 - d²) = 5 :6

Or,

☞ (196 - 9d²) / (196 - d²) = 5/6

Cross - Multiply now,

☞ 6(196 - 9d²) = 5(196 - d²)

☞ 6 *196 - 54d² = 5*196 - 5d²

☞ 6*196 - 5*196 = -5d² + 54d²

☞ 196(6 - 5) = 49d²

☞ 49d² = 196

Dividing both sides by 49,

☞ d² = 4

Square root both sides ,

☞ d = 2.

_____________________

∴ Required Fours parts are :-

☛ a1 = (14-3d) = 14 -3*2 = 8

☛ a2 = (14-d) = 14 - 2 = 12

☛ a3 = (14+d) = 14 + 2 = 16

☛ a4 = (14+3d) = 14 + 3*2 = 20.