Question

Divide 56 in 4 parts in ap such that the ratio of the product of their extremes to the product of their means is 5:6

Answer 1

Given:

↬ 56 is divided into four parts which form an A.P.

↬ Ratio of the product of extremes of given A.P. to the product of their means is 5:6

To Find:

• All the parts of 56

Things to know before solving question

• (a+b)(a-b)= a² -b²

Solution:

Let the four parts be a-3d, a-d, a+d, a+3d such that they are in A.P.

Now,

Sum of all parts= 56

⇒ a-3d+a-d+a+d+a+3d= 56

⇒ 4a= 56

⇒ a= 14

Now,

According to the question,

$\sf{\dfrac{Product\:of\:extremes\:of\:A.P.}{Product\:of\:means\:of\:A.P.}=\dfrac{5}{6}}$

i.e. $\sf{\dfrac{Product\:of\:first\:and\:fourth\:terms\:of\:A.P.}{Product\:of\:second\:and\:third\:terms\:of\:A.P.}=\dfrac{5}{6}}$

$\longrightarrow\sf{\dfrac{(a-3d)(a+3d)}{(a-d)(a+d)}=\dfrac{5}{6}}$

$\longrightarrow\sf{\dfrac{a^{2}-(3d)^{2}}{a^{2}-d^{2}}=\dfrac{5}{6}}$

$\longrightarrow\sf{\dfrac{a^{2}-9d^{2}}{a^{2}-d^{2}}=\dfrac{5}{6}}$

$\longrightarrow\sf{6(a^{2}-9d^{2})=5(a^{2}-d^{2})}$

$\longrightarrow\sf{6a^{2}-54d^{2}=5a^{2}-5d^{2}}$

$\longrightarrow\sf{6a^{2}-5a^{2}=54d^{2}-5d^{2}}$

$\longrightarrow\sf{a^{2}=49d^{2}}$

$\longrightarrow\sf{49d^{2}=a^{2}}$

On putting value of a in above equation, we get

$\longrightarrow\sf{49d^{2}=14^{2}}$

$\longrightarrow\sf{49d^{2}=196}$

$\longrightarrow\sf{d^{2}=\dfrac{\cancel{196}}{\cancel{49}}}$

$\longrightarrow\sf{d^{2}=2}$

$\longrightarrow\sf{d=\pm2}$

Now,

Case-1 ,when d=2

First part= a-3d= 14-3(2)= 8

Second part= a-d= 14-2= 12

Third part= a+d= 14+2= 16

Fourth part= a+3d= 14+3(2)= 20

Case-2 ,when d= -2

First part= a-3d= 14-3(-2)= 20

Second part= a-d= 14-(-2)= 16

Third part= a+d= 14+(-2)= 12

Fourth part= a+3d= 14+3(-2)= 8

Hence, all four parts are 8, 12, 16 and 20.

Answer 2

Answer:

Step-by-step explanation:

Solution :-

Let the four parts be a - 3d, a - d, a + d and a + 3d.

Then, a - 3d + a - d + a + d + a + 3d = 56

⇒ 4a = 56

⇒ a = 56/4

⇒ a = 14

Hence, four parts are 14 - 3d, 14 - d, 14 + d and 14 + 3d

Now, According to the Question,

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

⇒ 196 - 9d²/196 - d² = 5/6

⇒ 6(196 - 9d²) = 5(196 - d²)

⇒ 6 × 196 - 54d² = 5 × 196 - 5d²

⇒ 6 × 196 - 5 × 196 =54d² - 5d²

⇒ (6 - 5) × 196 = 49d²

⇒ d² = 196/49

⇒ d² = 4

⇒ d² = ± 2

The four parts are {14 - 3(± 2), {14 - (± 2)}

Hence, first possible divisions will be 8, 12, 16 and 20.

And the second possible divisions will be 20, 16, 12 and 8.