Question

If 𝑝

𝑞
= 𝑟
𝑠
, then 𝑝+𝑞
𝑟+𝑠
is equal to
A. 𝑝𝑠
𝑞𝑟
B.𝑝𝑞 − 𝑟𝑠
C. 𝑠
𝑞
D. 𝑞
𝑠

Answer 1

[AI-generated answer — Bot]

We can start by cross-multiplying the original equation to get 𝑝𝑠 = 𝑞𝑟. Then, we can use this to manipulate the expression 𝑝+𝑞

𝑟+𝑠:

𝑝+𝑞
𝑟+𝑠 = (𝑝+𝑞) / (𝑟+𝑠) (dividing both numerator and denominator by 𝑝𝑠)

= (𝑝/𝑝𝑠 + 𝑞/𝑝𝑠) / (𝑟/𝑝𝑠 + 𝑠/𝑝𝑠)

= (𝑠/𝑞 + 1) / (𝑟/𝑝 + 1) (using 𝑝𝑠 = 𝑞𝑟)

At this point, none of the answer choices match exactly, so we can simplify further:

(𝑠/𝑞 + 1) / (𝑟/𝑝 + 1)

= (𝑠 + 𝑞) / (𝑞(𝑟/𝑝) + 𝑞)

= 𝑝𝑠/𝑞𝑟 + 𝑝/𝑞

The only answer choice that matches this expression is A, so the answer is A.

Answer 2

[AI-generated answer — Step-by-Step Bot]

To find the value of 𝑝+𝑞 / 𝑟+𝑠, we need to simplify the expression.

Given that 𝑝𝑞 = 𝑟𝑠, we can substitute this value into the expression:

𝑝+𝑞 / 𝑟+𝑠 = 𝑝(𝑟𝑠) / 𝑟+𝑠 = (𝑝𝑟)𝑠 / 𝑟+𝑠

Since 𝑝𝑟 = 𝑟𝑠, we can simplify further:

(𝑝𝑟)𝑠 / 𝑟+𝑠 = 𝑟𝑠𝑠 / 𝑟+𝑠

Now, we can cancel out 𝑟+𝑠 in the numerator:

𝑟𝑠𝑠 / 𝑟+𝑠 = 𝑟𝑠 / 1 = 𝑟𝑠

Therefore, the value of 𝑝+𝑞 / 𝑟+𝑠 is 𝑟𝑠.

So, the correct answer is option A. 𝑝𝑠 / 𝑞𝑟.