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# For positive integers a, b, and c, a < b < c < 100. Which of the follo

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For positive integers a, b, and c, a < b < c < 100. Which of the follo  [#permalink]

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09 Mar 2017, 01:07
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65% (01:50) correct 35% (01:48) wrong based on 222 sessions

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For positive integers a, b, and c, a < b < c < 100. Which of the following has the greatest value?

A. a/100

B. (a+b)/(100+b)

C. (a+c)/(100+c)

D. (a+b+c)/(100+b+c)

E. The answer cannot be determined from the information provided.

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For positive integers a, b, and c, a < b < c < 100. Which of the follo  [#permalink]

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09 Mar 2017, 10:52
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For any proper fraction $$\frac{x}{y}$$ (x<y and both x and y are positive)

$$\frac{x}{y}$$<$$\frac{x+1}{y+1}$$<$$\frac{x+2}{y+2}$$<...<$$\frac{x+n}{y+n}$$

where n is a positive integer. The higher the value of n, the more greater $$\frac{x+n}{y+n}$$ is than the original fraction.

Example
$$\frac{x}{y}$$<$$\frac{x+5}{y+5}$$<$$\frac{x+50}{y+50}$$

Now we need to arrange

$$\frac{a}{100}$$, $$\frac{a+b}{100+b}$$, $$\frac{a+c}{100+c}$$ and $$\frac{a+(b+c)}{100+(b+c)}$$

Since 0<b<c<b+c then the order would be

$$\frac{a}{100}$$<$$\frac{a+b}{100+b}$$<$$\frac{a+c}{100+c}$$<$$\frac{a+(b+c)}{100+(b+c)}$$

D has the greatest value. Hence Answer D
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Re: For positive integers a, b, and c, a < b < c < 100. Which of the follo  [#permalink]

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09 Mar 2017, 04:03
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SHORTCUT:let us assume a, b, c be 25, 50, 75 respectively
A. 25/100 =0.25
B. 75/150 = 0.5
C. 100/175 = 0.57
D. 150/225 = 0.66

Option D
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Re: For positive integers a, b, and c, a < b < c < 100. Which of the follo  [#permalink]

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09 Mar 2017, 06:28
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Agree... D... also it allows all the variables to feature in the option... which is necessary to ascertain the answer in this case

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Re: For positive integers a, b, and c, a < b < c < 100. Which of the follo  [#permalink]

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10 Mar 2017, 10:24
Bunuel wrote:
For positive integers a, b, and c, a < b < c < 100. Which of the following has the greatest value?

A. a/100

B. (a+b)/(100+b)

C. (a+c)/(100+c)

D. (a+b+c)/(100+b+c)

E. The answer cannot be determined from the information provided.

a, b, c = Pos INTs

a < b < c < 100

Let's test numbers, Let try a =1, b =2, & b =3;

so we have

A. a/100 -------------------> 1/100

B. (a+b)/(100+b) ---------> 3/102

C. (a+c)/(100+c) ---------> 4/103

D. (a+b+c)/(100+b+c) ---> 5/105

E. The answer cannot be determined from the information provided.[/quote]

Let's try another integer values: a =97, b =98, c =99

A) 97/100 B) 195/198 C) 196/199 D) 294/297

Since option D has the highest numerator and highest denominator, and the denominator of each option is greater than the numerator by the same value, option D has the greatest value.

Now, assuming at this stage you need to compare each option against the other and you have to deal with big values, here's a shortcut i just found (hope, I'm right ):

For instance, let's check which is bigger, A or B: A) 97/100 vs. B) 195/198 ----> A) 198*97 vs B) 195*100,

just compare the positive differences of the multiplying figures and the option with the lower difference is likely to be bigger.

That is, since 195 - 100 = 97 is less than 199 - 97 = 102, option B is bigger (check!)

For B & C: B) 195/198 vs C) 196/199 ----> B) 199*195 vs. C) 198*196 ----> 199 - 195 = 4 > 198 - 196 = 2, option C is bigger

For C & D: C) 196/199 vs. D) 294/297 ----> C) 297*196 vs. 294*199 ----> 297 - 196 = 201 > 294 - 199 = 195, option D is bigger

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Re: For positive integers a, b, and c, a < b < c < 100. Which of the follo  [#permalink]

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18 Mar 2017, 08:33
picking smart numbers here is great: try 25, 50, 75 and be careful with calculations
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Re: For positive integers a, b, and c, a < b < c < 100. Which of the follo  [#permalink]

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21 May 2017, 13:04
Hi Bunuel,

Thanks for this question!

Although I did get the correct answer ( I used two examples of a, b and c to verify: 1,2,3 and 25,50, 75). And since option D was higher in both the cases, I assumed D would be correct.

However, how can we be so sure that E isn't the correct option? I wanted to understand the question better so that if a similar question comes up during the exam - I can be more certain of why I am eliminating one of the options.

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Joined: 20 May 2017
Posts: 10
Re: For positive integers a, b, and c, a < b < c < 100. Which of the follo  [#permalink]

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21 May 2017, 15:14
poojamathur21 wrote:
Hi Bunuel,

Thanks for this question!

Although I did get the correct answer ( I used two examples of a, b and c to verify: 1,2,3 and 25,50, 75). And since option D was higher in both the cases, I assumed D would be correct.

However, how can we be so sure that E isn't the correct option? I wanted to understand the question better so that if a similar question comes up during the exam - I can be more certain of why I am eliminating one of the options.

I think I know the answer.

1/n < 2/(n+1) < 3/(n+2) etc. with n >1

It could be said in a logic form but I don't know how to put it. Still waiting for Bunuel's explanation.
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Re: For positive integers a, b, and c, a < b < c < 100. Which of the follo  [#permalink]

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21 May 2017, 16:44
MaverickTone wrote:
poojamathur21 wrote:
Hi Bunuel,

Thanks for this question!

Although I did get the correct answer ( I used two examples of a, b and c to verify: 1,2,3 and 25,50, 75). And since option D was higher in both the cases, I assumed D would be correct.

However, how can we be so sure that E isn't the correct option? I wanted to understand the question better so that if a similar question comes up during the exam - I can be more certain of why I am eliminating one of the options.

I think I know the answer.

1/n < 2/(n+1) < 3/(n+2) etc. with n >1

It could be said in a logic form but I don't know how to put it. Still waiting for Bunuel's explanation.

Yup, you're right. I think I got it. Bunuel, please confirm.

If we have a fraction say a / b, any number (x>0) when added to both the numerator and denominator makes the fraction bigger, and when subtracted makes the fraction smaller.

=> a-x/b-x < a/b < a+x/b+x

Posted from my mobile device
Math Expert
Joined: 02 Sep 2009
Posts: 52108
Re: For positive integers a, b, and c, a < b < c < 100. Which of the follo  [#permalink]

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21 May 2017, 22:07
poojamathur21 wrote:
MaverickTone wrote:
poojamathur21 wrote:
Hi Bunuel,

Thanks for this question!

Although I did get the correct answer ( I used two examples of a, b and c to verify: 1,2,3 and 25,50, 75). And since option D was higher in both the cases, I assumed D would be correct.

However, how can we be so sure that E isn't the correct option? I wanted to understand the question better so that if a similar question comes up during the exam - I can be more certain of why I am eliminating one of the options.

I think I know the answer.

1/n < 2/(n+1) < 3/(n+2) etc. with n >1

It could be said in a logic form but I don't know how to put it. Still waiting for Bunuel's explanation.

Yup, you're right. I think I got it. Bunuel, please confirm.

If we have a fraction say a / b, any number (x>0) when added to both the numerator and denominator makes the fraction bigger, and when subtracted makes the fraction smaller.

=> a-x/b-x < a/b < a+x/b+x

Posted from my mobile device

Here is a POST by Magoosh which discusses this issue in detail.

Hope it helps.
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Re: For positive integers a, b, and c, a < b < c < 100. Which of the follo  [#permalink]

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21 May 2017, 22:13
1
poojamathur21 wrote:
MaverickTone wrote:
poojamathur21 wrote:
Hi Bunuel,

Thanks for this question!

Although I did get the correct answer ( I used two examples of a, b and c to verify: 1,2,3 and 25,50, 75). And since option D was higher in both the cases, I assumed D would be correct.

However, how can we be so sure that E isn't the correct option? I wanted to understand the question better so that if a similar question comes up during the exam - I can be more certain of why I am eliminating one of the options.

I think I know the answer.

1/n < 2/(n+1) < 3/(n+2) etc. with n >1

It could be said in a logic form but I don't know how to put it. Still waiting for Bunuel's explanation.

Yup, you're right. I think I got it. Bunuel, please confirm.

If we have a fraction say a / b, any number (x>0) when added to both the numerator and denominator makes the fraction bigger, and when subtracted makes the fraction smaller.

=> a-x/b-x < a/b < a+x/b+x

Posted from my mobile device

It depends on whether the fraction a/b is a proper fraction (a<b) or its an improper fraction (a>b)

If a/b is a proper fraction, then adding any positive number x (x>0) to both numerator and denominator Increases the value of the fraction
OR a/b < (a+x)/(b+x)

But if c/d is say an improper fraction (c>d), then adding any positive number x (x>0) to both numerator and denominator Decreases the value of the fraction.
OR c/d > (c+x)/(d+x)

So lets consider a proper fraction a/100 (a<100). If we add a positive number b to both numerator and denominator, its value will increase
Thus a/100 < (a+b)/(100+b)
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Re: For positive integers a, b, and c, a < b < c < 100. Which of the follo  [#permalink]

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21 May 2017, 23:52
Bunuel wrote:
For positive integers a, b, and c, a < b < c < 100. Which of the following has the greatest value?

A. a/100

B. (a+b)/(100+b)

C. (a+c)/(100+c)

D. (a+b+c)/(100+b+c)

E. The answer cannot be determined from the information provided.

since b > a
so, a/100 < (a+b)/(100+b)

since c > b
so, (a+b)/(100+b) < (a+c)/(100+c)

since b+c > c
so, (a+c)/(100+c) < (a+b+c)/(100+b+c)

So , a/100 < (a+b)/(100+b) < (a+c)/(100+c) < (a+b+c)/(100+b+c)

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Re: For positive integers a, b, and c, a < b < c < 100. Which of the follo  [#permalink]

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22 May 2017, 02:05
Bunuel wrote:
For positive integers a, b, and c, a < b < c < 100. Which of the following has the greatest value?

A. a/100

B. (a+b)/(100+b)

C. (a+c)/(100+c)

D. (a+b+c)/(100+b+c)

E. The answer cannot be determined from the information provided.

Logical Thinking

I took 3 numbers < 100 for a,b and c such as 97,98 and 99.

I use a numerator and denominator rule (mentioned in Manhattan guide )to solve this question without calculation. To be frank I couldn't think of solid solution.

Rule : Adding the same number to both the numerator and denominator brings the fraction close to 1, regardless of the fraction value.

since, a<b<c< 100. The fraction < 1.

Now scanning through the answer options, A to D are combination of a, b and c.

Using the above rule I know that D is closed to 1. Hence D option is largest among A to D. Therefor marked D.

To understand this approach please refer to Manhattan guide

Ans: D
Re: For positive integers a, b, and c, a < b < c < 100. Which of the follo &nbs [#permalink] 22 May 2017, 02:05
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