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An integer x is chosen at random from the numbers 1 to 50. Find the pr 29 Jun 2021, 02:15
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48% (03:01) correct 52% (03:19) wrong based on 196 sessions

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An integer x is chosen at random from the numbers 1 to 50. Find the probability that \(x + \frac{336}{x} ≤ 50\).

(A) 7/10
(B) 17/25
(C) 19/50
(D) 13/50
(E) 3/10
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A
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An integer x is chosen at random from the numbers 1 to 50. Find the pr 30 Jun 2021, 23:55
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Bunuel
An integer x is chosen at random from the numbers 1 to 50. Find the probability that \(x + \frac{336}{x} ≤ 50\).

(A) 7/10
(B) 17/25
(C) 19/50
(D) 13/50
(E) 3/10
Show more


Two ways

(I) Algebraic

\(x + \frac{336}{x} ≤ 50\)

\(x^2-50x+336\leq 0\)

Factorize to get \((x-8)(x-42)\leq 0\)
Whenever x is between 8 and 42, both inclusive, the answer will be yes.
So 42-8+1=35 values
Total values = 50
P=\(\frac{35}{50}=\frac{7}{10}\)

(II) Analysing the expression

\(x + \frac{336}{x} ≤ 50\)
a) Least value of x: when 336/x is the max possible and \(\frac{336}{x}<50\)
x as 7 gives 336/6 or 56, and 336/7 gives 48........But at x=7, 48+7=55
x as 8 gives 336/8 or 42, and 42+8=50 fits in......
Thus \(x\geq 8\) fits in
a) Max value of x: when 336/x is least and \(x<50\)
x as 50 gives 336/50 or 6.7, and 336/x will increase as x decrease, so x<50-6.7
x as 43 gives 336/43 ....43*7=301, so 336/43>7, and the sum will become >50.
x as 42 gives 336/42 or 8 ....42+8=50.
Thus \(x\leq 42\) fits in

So 42-8+1=35 values
Total values = 50
P=\(\frac{35}{50}=\frac{7}{10}\)


A
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An integer x is chosen at random from the numbers 1 to 50. Find the pr 02 Jul 2021, 00:07
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chetan2u
Bunuel
An integer x is chosen at random from the numbers 1 to 50. Find the probability that \(x + \frac{336}{x} ≤ 50\).

(A) 7/10
(B) 17/25
(C) 19/50
(D) 13/50
(E) 3/10


Two ways

(I) Algebraic

\(x + \frac{336}{x} ≤ 50\)

\(x^2-50x+336\leq 0\)

Factorize to get \((x-8)(x-42)\leq 0\)
Whenever x is between 8 and 42, both inclusive, the answer will be yes.
So 42-8+1=35 values
Total values = 50
P=\(\frac{35}{50}=\frac{7}{10}\)

(II) Analysing the expression

\(x + \frac{336}{x} ≤ 50\)
a) Least value of x: when 336/x is the max possible and \(\frac{336}{x}<50\)
x as 7 gives 336/6 or 56, and 336/7 gives 48........But at x=7, 48+7=55
x as 8 gives 336/8 or 42, and 42+8=50 fits in......
Thus \(x\geq 8\) fits in
a) Max value of x: when 336/x is least and \(x<50\)
x as 50 gives 336/50 or 6.7, and 336/x will increase as x decrease, so x<50-6.7
x as 43 gives 336/43 ....43*7=301, so 336/43>7, and the sum will become >50.
x as 42 gives 336/42 or 8 ....42+8=50.
Thus \(x\leq 42\) fits in

So 42-8+1=35 values
Total values = 50
P=\(\frac{35}{50}=\frac{7}{10}\)


A
Show more

(x−8)(x−42)≤0
from this, how did you realise that x is in between 8 and 42?
wont it be like, x<= 8 and x<=42?
Please clarify my doubt!
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An integer x is chosen at random from the numbers 1 to 50. Find the pr 02 Jul 2021, 01:05
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Solution:

First of all, let us try to find the possible values of \(x\) for which the given equation stands.

We are given: \(x+\frac{336}{x} ≤ 50\)

\(⇒ \frac{(x^2+336)}{x} ≤ 50\)

\(⇒ x^2 + 336 ≤ 50x\)

\(⇒ x^2 - 50x + 336 ≤ 0 \)

We can solve this the way we solve quadratic equation assuming there is an equal to sign instead of ≤ sign.

\(⇒ x^2 - 8x - 42x + 336 ≤ 0 \)

\(⇒ x(x-8) -42(x-8) ≤ 0 \)

\(⇒ (x-8)(x-42) ≤ 0\)

Now we will use the number line technique to get the range of \(x\):

Attachment:
numberline.png
numberline.png [ 2.8 KiB | Viewed 5818 times ]

We can see that the range of x will be in the negative reign i.e., \(8 ≤ x ≤ 42\)

So on these values of \(x\) will satisfy the expression. Now the probability of getting \(x\) in this range among numbers between \(1-50 = \frac{(42-8+1)}{50} = \frac{35}{50} = \frac{7}{10}\)

Hence the right answer is Option A.
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An integer x is chosen at random from the numbers 1 to 50. Find the pr 02 Jul 2021, 02:08
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Solution:

First of all, let us try to find the possible values of \(x\) for which the given equation stands.

We are given: \(x+\frac{336}{x} ≤ 50\)

\(⇒ \frac{(x^2+336)}{x} ≤ 50\)

\(⇒ x^2 + 336 ≤ 50x\)

\(⇒ x^2 - 50x + 336 ≤ 0 \)

We can solve this the way we solve quadratic equation assuming there is an equal to sign instead of ≤ sign.

\(⇒ x^2 - 8x - 42x + 336 ≤ 0 \)

\(⇒ x(x-8) -42(x-8) ≤ 0 \)

\(⇒ (x-8)(x-42) ≤ 0\)

Now we will use the number line technique to get the range of \(x\):

Attachment:
numberline.png

We can see that the range of x will be in the negative reign i.e., \(8 ≤ x ≤ 42\)

So on these values of \(x\) will satisfy the expression. Now the probability of getting \(x\) in this range among numbers between \(1-50 = \frac{(42-8+1)}{50} = \frac{35}{50} = \frac{7}{10}\)

Hence the right answer is Option A.
Show more

(x−8)(x−42)≤0
from this, how did you realise that x is in between 8 and 42?
wont it be like, x<= 8 and x<=42?
Please clarify my doubt!

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An integer x is chosen at random from the numbers 1 to 50. Find the pr 02 Jul 2021, 03:04
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Chitra657
chetan2u
Bunuel
An integer x is chosen at random from the numbers 1 to 50. Find the probability that \(x + \frac{336}{x} ≤ 50\).

(A) 7/10
(B) 17/25
(C) 19/50
(D) 13/50
(E) 3/10


Two ways

(I) Algebraic

\(x + \frac{336}{x} ≤ 50\)

\(x^2-50x+336\leq 0\)

Factorize to get \((x-8)(x-42)\leq 0\)
Whenever x is between 8 and 42, both inclusive, the answer will be yes.
So 42-8+1=35 values
Total values = 50
P=\(\frac{35}{50}=\frac{7}{10}\)

(II) Analysing the expression

\(x + \frac{336}{x} ≤ 50\)
a) Least value of x: when 336/x is the max possible and \(\frac{336}{x}<50\)
x as 7 gives 336/6 or 56, and 336/7 gives 48........But at x=7, 48+7=55
x as 8 gives 336/8 or 42, and 42+8=50 fits in......
Thus \(x\geq 8\) fits in
a) Max value of x: when 336/x is least and \(x<50\)
x as 50 gives 336/50 or 6.7, and 336/x will increase as x decrease, so x<50-6.7
x as 43 gives 336/43 ....43*7=301, so 336/43>7, and the sum will become >50.
x as 42 gives 336/42 or 8 ....42+8=50.
Thus \(x\leq 42\) fits in

So 42-8+1=35 values
Total values = 50
P=\(\frac{35}{50}=\frac{7}{10}\)


A

(x−8)(x−42)≤0
from this, how did you realise that x is in between 8 and 42?
wont it be like, x<= 8 and x<=42?
Please clarify my doubt!
Show more

For (x−8)(x−42)≤0, both x-8 and x-42 should have opposite sign.
When x<8, both are negative and when x>42, both are positive.

Thus when x is between 8 and 42, both will have opposite sign and their product will be negative
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An integer x is chosen at random from the numbers 1 to 50. Find the pr 12 Feb 2024, 22:10
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Bunuel
An integer x is chosen at random from the numbers 1 to 50. Find the probability that \(x + \frac{336}{x} ≤ 50\).

(A) 7/10
(B) 17/25
(C) 19/50
(D) 13/50
(E) 3/10
Show more
­\(x^2 - 50x + 336 <=0\) or, (x-42)(x-8) <= 0 so, 8<=x<=42
possible values are 42-8+1=35
total values are 50
Therefore, 35/50=7/10. Option (A) is correct.
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An integer x is chosen at random from the numbers 1 to 50. Find the pr 03 Aug 2025, 05:49
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An integer x is chosen at random from the numbers 1 to 50. Find the probability that \(x + \frac{336}{x} ≤ 50\).

336 = 2*2*2*2*2*3*7
x + 336/x <= 50
x^2 - 50x + 336 <=0
x^2 - 42x - 8x + 336 <=0
(x-8)(x-42) <=0
8<=x<=42;
Favorable cases = 42-8+1 = 35
Total cases = 50-1+1= 50

An integer x is chosen at random from the numbers 1 to 50. Find the probability that \(x + \frac{336}{x} ≤ 50\) = 35/50 = 7/10

IMO A
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