Question

Ram and Sham together have 55 marbles .Both of them have lost 5 marbles each and
the product of the no. of marbles is 164 .Find how many marbles they have to start with?

Answer 1

ram 55-5=50

sham 55-50

ram and sham now have 50 marbles

164 +55=219

Answer 2

Answer:

At the start, Ram had 46 marbles and Sham had 9 marbles.

Step-by-step explanation:

Given,Ram and Sham together have 55 marbles.

So total number of marbles are 55.

But they both lost 5 marbles each.

Therefore they lost $(5 \times 2) = 10$marbles.

We are also know the product of the no. of marbles is 164.

Let now Ram has x number of marbles and sham has y number of marbles.

According to question,

$x + y = 55 - 10 = 45....(1)$

and $xy = 164....(2)$

We know,

${(x - y)}^{2} = {(x + y)}^{2} - 4xy = {45}^{2} - 4 \times 164 = 1369$

So,$x - y = \sqrt{1369} = 37...(3)$

We are adding equation (1) and (3),

$x + y + x - y = 45 + 37$

$2x = 82$

$x = \frac{82}{2} = 41$

From equation (1),

$41 + y = 45$

$y = 4$

Therefore now Ram has 41 marbles and sham has 4 marbles.

Hence at the first,Ram had (41+5)=46 marbles and Sham had (4+5)=9 marbles.

Here we applied rule of Algebra.

Some important rule of Algebra,

a + b)² = a² + 2ab + b²

(a − b)² = a² − 2ab − b²

(a + b)³ = a³ + 3a²b + 3ab² + b³

(a - b)³ = a³ - 3a²b + 3ab² - b³

a³ + b³ = (a + b)³ − 3ab(a + b)

a³ - b³ = (a -b)³ + 3ab(a - b)

e.t.c