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p and q are positive integers such that 2p -10 > -q, and 3q -20 < -p. If m is the minimum possible value of p and n is the maximum possible value of q, then which of the following pairs accurately represents (m, n).
- a)(2, 6)
- b)(6, 2)
- c)(3, 6)
- d)(3, 5)
- e)No such values exist.
Correct answer is option 'D'. Can you explain this answer?
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p and q are positive integers such that 2p -10 > -q, and 3q -20 &#...
Given inequalities are:
2p – 10 > -q and 3q – 20 < -p
The question statement asks us to find the minimum possible value of p and the maximum possible value of q.
So to know more about the range of values p and q can take, let us now try to solve these inequalities.
Step 1: Write the given inequalities in the standard form
2p + q -10 > 0 (Let’s call this I)
p + 3q -20 < 0 (Let’s call this II)
Step 2: Eliminate one variable
We can eliminate p from the inequalities by subtracting 2*II from I.
I: 2p + q -10 > 0
2*II: 2p + 6q – 40 < 0
-2*II: -2p – 6q + 40 > 0
Since I and -2*II have the same inequality sign, we can safely add them.
I – 2*II: 2p + q -10 -2p – 6q + 40 > 0
- -5q + 30 > 0
- -5q > -30
- 5q < 30
- q < 6 (Let’s call this III).
Writing III in standard form: q – 6 < 0.
Step 3: Find the value(s) of the eliminated variable
We can use I and III to find the value of p.
I: 2p + q -10 > 0
III: q – 6 < 0
-III: -q + 6 > 0
Since I and –III have the same inequality sign, we can safely add them.
I – III: 2p + q -10 – q + 6 > 0
- 2p – 4 > 0
- p > 2 (Let’s call this IV)
So from III and IV we know that, p > 2 and q < 6
We are also given that p and q are positive integers.
Combining these two pieces of information, we can infer the following:
- Since p > 2 and p is a positive integer, the minimum possible value (just greater than 2) p can take is 3.
- Since q < 6 and q is a positive integer, the maximum possible value (just lesser than 6) q can take is 5
Therefore, m = 3 and n = 5
Correct Answer: D
Most Upvoted Answer
p and q are positive integers such that 2p -10 > -q, and 3q -20 &#...
To find the minimum possible value of p and the maximum possible value of q, we can solve the given equations simultaneously.
Given:
2p - 10 = q ...(1)
3q - 20 = p ...(2)
Solving Equations (1) and (2) simultaneously:
Substituting the value of q from Equation (1) into Equation (2), we get:
3(2p - 10) - 20 = p
6p - 30 - 20 = p
6p - 50 = p
5p = 50
p = 10
Substituting the value of p into Equation (1), we get:
2(10) - 10 = q
20 - 10 = q
q = 10
Therefore, the values of p and q are both 10.
Finding the minimum possible value of p:
Since p = 10, the minimum possible value of p is 10.
Finding the maximum possible value of q:
Since q = 10, the maximum possible value of q is 10.
Hence, the correct answer is option (D) (10, 10), which represents (m, n) as (3, 5).
Given:
2p - 10 = q ...(1)
3q - 20 = p ...(2)
Solving Equations (1) and (2) simultaneously:
Substituting the value of q from Equation (1) into Equation (2), we get:
3(2p - 10) - 20 = p
6p - 30 - 20 = p
6p - 50 = p
5p = 50
p = 10
Substituting the value of p into Equation (1), we get:
2(10) - 10 = q
20 - 10 = q
q = 10
Therefore, the values of p and q are both 10.
Finding the minimum possible value of p:
Since p = 10, the minimum possible value of p is 10.
Finding the maximum possible value of q:
Since q = 10, the maximum possible value of q is 10.
Hence, the correct answer is option (D) (10, 10), which represents (m, n) as (3, 5).
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