Question

A number 36 is divided into four parts that are in AP such that the ratio of the product of the first and fourth part two the product of the second and the third part is 9:10 the lowest number of the AP is__

Options: 10/12/8/6

Answer 1

Answer :

Lowest part = 6

Solution :

Let the four parts in AP be ,

a - 3d , a - d , a + d , a + 3d

Thus ,

=> (a - 3d) + (a - d) + (a + d) + (a + 3d) = 36

=> 4a = 36

=> a = 36/4

=> a = 9

According to the question , the ratio of the product of the first and fourth part two the product of the second and the third part is 9:10 .

Thus ,

=> (a - 3d)(a + 3d) : (a - d)(a + d) = 9 : 10

=> (a - 3d)(a + 3d) / (a - d)(a + d) = 9 / 10

=> [a² - (3d)²] / [a² - d²] = 9/10

=> (a² - 9d²) / (a² - d²) = 9/10

=> 10(a² - 9d²) = 9(a² - d²)

=> 10a² - 90d² = 9a² - 9d²

=> 10a² - 9a² = 90d² - 9d²

=> a² = 81d²

=> d² = a²/81

=> d² = 9²/81

=> d² = 81/81

=> d² = 1

=> d = √1

=> d = ±1

• f a = 9 and d = 1 , then

1st part = a - 3d = 9 - 3•1 = 9 - 3 = 6

2nd part = a - d = 9 - 1 = 8

3rd part = a + d = 9 + 1 = 10

4th part = a + 3d = 9 + 3•1 = 9 + 3 = 12

• If a = 9 and d = -1 , then

1st part = a - 3d = 9 - 3•(-1) = 9 + 3 = 12

2nd part = a - d = 9 - (-1) = 9 + 1 = 10

3rd part = a + d = 9 + (-1) = 9 - 1 = 8

4th part = a + 3d = 9 + 3•(-1) = 9 - 3 = 6

Hence ,

The four parts are : 6 , 8 , 10 , 12

The smallest part = 6

Answer 2

The lowest number of the AP is 6.

• Since the number is divided into four parts so

        $4a=36\\a=9$

• The four consecutive numbers of the AP are in the form of $(a-3d)(a-d)(a+d)(a+3d)$
• It is given that the ratio of the product of the first and fourth term to the product of the second and the third part is 9:10, so writing this in the equation $\frac{(a-3d)(a+3d)}{(a-d)(a+d)}=\frac{9}{10}$
• Substitute a=9 into the above equation and solve for d.

        $\frac{(9-3d)(9+3d)}{(9-d)(9+d)}=\frac{9}{10}\\\frac{(81-9d^2)}{(81-d^2)}=\frac{9}{10}\\\\810-90d^2=729-9d^2\\81=81d^2\\d=1$

• Substitute a=9 and d=1 into $(a-3d)(a-d)(a+d)(a+3d)$ and find the four terms of the AP.

      $(9-3\times 1)(9-1)(9+1)(9+3\times 1)=6,\,8,\,10,\,{\rm and}\,12$

• The lowest term of the AP is 6.

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