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## OGQR 2020: Question No. 80

If x, y, and k are positive and x is less than y, then (x + k) / (y + k) is

### Solution

#### Given

In this question, we are given

• The numbers x, y, and k are positive.
• Also, x is less than y.

#### To Find

We need to determine

• The possible range of (x+ k) / (y+ k).

#### Approach & Working

We know that

• x < y

Or, xk < yk [Since k is a positive number, multiplication does not change the sign of inequality.]

Or, xk + xy < yk + xy [Adding xy on both sides]

Or, x (y + k) < y (x + k) [Taking x and y common on each side]

Or,  x /y< (x+ k) / (y+ k). [Since y and (y + k) are positive numbers, division does not change the sign of inequality.]

Hence, the correct answer is option B.

## Takeaways:

1. Multiplying/dividing both sides of an inequality by a positive number does not change the sign of the inequality.
1. Example: If x < y, then 2x < 2y.
2. Multiplying/dividing both sides of an inequality by a negative number reverses the sign of the inequality.
1. Example: If x < y, then -2x > -2y.

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