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If the ratio of the present age of Anna and Paula is 3 : 4, what could

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If the ratio of the present age of Anna and Paula is 3 : 4, what could  [#permalink]

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New post Updated on: 25 Apr 2017, 00:06
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Q. If the ratio of the present age of Anna and Paula is 3 : 4, what could be the ratio of their respective ages after 8 years??

    a. 1 : 2
    b. 3 : 8
    c. 3 : 5
    d. 2 : 3
    e. 4 : 5


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Originally posted by EgmatQuantExpert on 25 Apr 2017, 00:04.
Last edited by EgmatQuantExpert on 25 Apr 2017, 00:06, edited 1 time in total.
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If the ratio of the present age of Anna and Paula is 3 : 4, what could  [#permalink]

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New post 30 Apr 2017, 00:10
4
2

Method 2:



    • Now, the first method is a bit cumbersome and time-consuming too.
    • And the purpose of the question was to teach you a simple concept of ratios/fractions

      o Suppose we have a fraction say \(\frac{N}{D}\), where \(\frac{N}{D}\) is a proper fraction (D>N), for example, \(\frac{1}{2},\frac{2}{3}\) etc, and if we add the same number (say A) to the numerator and denominator, will the new ratio be greater than \(\frac{N}{D}\) or less than?
      o For example, if we have \(\frac{2}{3} = 0.666\)

      o Now say we add 1 to both the numerator and denominator, then the new ratio would be

         \(\frac{(2+1)}{(3+1)} = \frac{3}{4} = 0.75\)
         And we can see clearly that \(\frac{3}{4} > \frac{2}{3}\)
         Hence \(\frac{(N+a)}{(D+a)} > \frac{N}{D}\), when the fraction is a proper fraction.

    • With this knowledge let us now solve the question.
    • Ratio of age of Anna and Paula = 3 : 4 or \(\frac{3}{4} = 0.75\)
    • We have been asked the age after 8 years

      o Now notice we are adding 8 to both the numerator and denominator.
      o Anna: Paula =\(\frac{(3x + 8)}{(4x + 8)}\)
      o Thus, we should immediately conclude that the new ratio(fraction) must be more than 0.75.

    • Thus, we just need to look at the options quickly and find a ratio which is more than 0.75

      A. \(1: 2 = \frac{1}{2} = 0.5\)
      B. \(3: 8 = \frac{3}{8} = 0.375\)
      C. \(3:5 = \frac{3}{5} = 0.6\)
      D. \(2: 3 = \frac{2}{3} = 0.66\)
      E. \(4: 5 = \frac{4}{5} = 0.8\) (Hence Option E is the answer)


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If the ratio of the present age of Anna and Paula is 3 : 4, what could  [#permalink]

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New post Updated on: 30 Apr 2017, 00:01

Originally posted by EgmatQuantExpert on 25 Apr 2017, 00:04.
Last edited by EgmatQuantExpert on 30 Apr 2017, 00:01, edited 1 time in total.
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If the ratio of the present age of Anna and Paula is 3 : 4, what could  [#permalink]

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New post 25 Apr 2017, 00:30
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The ratio of anna's age:paula's age is 3:4
Plugging in numbers,
let anna's age is 24
paula's age is 32

8 years later
Anna will be 32 and paula will be 40 Ratio 4:5(Option E)
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Re: If the ratio of the present age of Anna and Paula is 3 : 4, what could  [#permalink]

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New post 25 Apr 2017, 00:42
ratio of ages after 8 years will be 3x+8:4x+8
put x= 1,2,3,4,5....
you will get
11:12
increase 11 by 3 and 12 by 4 and continue the process.
14:16
17:20
20:24
23:28
26:32
29:36
32:40 = 4:5

hence answer is E
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If the ratio of the present age of Anna and Paula is 3 : 4, what could  [#permalink]

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New post 30 Apr 2017, 00:00

Solution



Let me discuss two ways to solve this question.

Method 1:

    • The ratio of the present age of Anna and Paula has been given as 3: 4
    • Let us assume the age of Anna and Paula to be 3x and 4x respectively.
    • After 8 years, the ratio of ages would be

      o \(\frac{Anna}{Paula} = \frac{(3x+8)}{(4x+8)}\)…………….(i)

    • Now the correct answer has to be one among the 5 options, let us equate equation (i) with all the options one by one.

      A. \(\frac{(3x+8)}{(4x+8)} = \frac{1}{2}\)
        \(6x + 16 = 4x + 8\)
        \(2x = - 8\)
      Since x is negative, this cannot be our answer, as the ages cannot be negative.

      B. \(\frac{(3x+8)}{(4x+8)} = \frac{3}{8}\)
      \(24x + 64 = 12x + 24\)
      x = negative
      Since x is negative, this cannot be our answer, as the ages cannot be negative.

      C. \(\frac{(3x+8)}{(4x+8)} = \frac{3}{5}\)
      \(15x + 40 = 12x + 24\)
      x = negative
      Since x is negative, this cannot be our answer, as the ages cannot be negative.

      D. \(\frac{(3x+8)}{(4x+8)} = \frac{2}{3}\)
      \(9x + 24 = 8x + 16\)
      x = negative
      Since x is negative, this cannnot be our answer, as the ages cannot be negative.

      E. \(\frac{(3x+8)}{(4x+8)} = \frac{4}{5}\)
      \(15x + 40 = 16x + 32\)
      \(x = 8\)
      Since x is positive, this can be our answer, as the ages cannot be negative.

    • As we can see, in only one case, we are getting x as positive, hence the correct answer is Option E.


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Re: If the ratio of the present age of Anna and Paula is 3 : 4, what could  [#permalink]

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New post 16 Sep 2018, 09:21
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EgmatQuantExpert wrote:
Q. If the ratio of the present age of Anna and Paula is 3 : 4, what could be the ratio of their respective ages after 8 years??

    a. 1 : 2
    b. 3 : 8
    c. 3 : 5
    d. 2 : 3
    e. 4 : 5



Here's a nice property of fractions: If a, b and k are positive, then (a + k)/(b + k) approaches 1 as k gets bigger.
For example, the fraction (2+11)/(3+11) is closer to 1 than 2/3 is.
Likewise, the fraction (1+7)/(2+7) is closer to 1 than 1/2 is.

Let A = Anna's present age
Let P = Paula's present age
So, A/P = 3/4

In EIGHT YEARS, we can conclude that:
Let A + 8= Anna's future age
Let P + 8 = Paula's future age
So, in EIGHT YEARS, the ratio of their ages = (A + 8)/(P + 8)

By the above rule, we know that (A + 8)/(P + 8) is closer to 1 than is A/P is.
In other words, (A + 8)/(P + 8) is closer to 1 than 3/4 is.

Check the answer choices.....
Only answer choice E (aka 4/5) is closer to 1 than 3/4 is.

Answer: E

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If the ratio of the present age of Anna and Paula is 3 : 4, what could  [#permalink]

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New post 07 Oct 2018, 21:27
let us use fractions properties. When we add the same number to numerator and denominator the fraction increases.
In our case we do so.
So, we have 3/4 and look answer choices. Only E, in which 4/5>3/4 can be correct

E
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Re: If the ratio of the present age of Anna and Paula is 3 : 4, what could  [#permalink]

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New post 10 Apr 2019, 17:55
EgmatQuantExpert wrote:
Q. If the ratio of the present age of Anna and Paula is 3 : 4, what could be the ratio of their respective ages after 8 years??

    a. 1 : 2
    b. 3 : 8
    c. 3 : 5
    d. 2 : 3
    e. 4 : 5


Some possible ratios of their current ages are:

A/P = 3/4, 6/8, 9/12, 12/15, 15/20, 18/24, 21/28, 24/32…

So after 8 years, these ratios, would be:

11/12, 14/16 = 7/8, 17/20, 20/23, 23/28, 26/32 = 13/16, 29/36, 32/40 = 4/5, …
We see that if Anna is 24 and Paula is 32, the ratio of their ages 8 years later is 32/40 = 4/5, or 4 : 5.

Answer: E
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Re: If the ratio of the present age of Anna and Paula is 3 : 4, what could  [#permalink]

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New post 19 Apr 2019, 09:00
ScottTargetTestPrep wrote:
EgmatQuantExpert wrote:
Q. If the ratio of the present age of Anna and Paula is 3 : 4, what could be the ratio of their respective ages after 8 years??

    a. 1 : 2
    b. 3 : 8
    c. 3 : 5
    d. 2 : 3
    e. 4 : 5


Some possible ratios of their current ages are:

A/P = 3/4, 6/8, 9/12, 12/15, 15/20, 18/24, 21/28, 24/32…

So after 8 years, these ratios, would be:

11/12, 14/16 = 7/8, 17/20, 20/23, 23/28, 26/32 = 13/16, 29/36, 32/40 = 4/5, …
We see that if Anna is 24 and Paula is 32, the ratio of their ages 8 years later is 32/40 = 4/5, or 4 : 5.

Answer: E


Hi Scott,

Just wanted to clarify - you've picked an example here of Anna being 24 and Paula being 32. I.e. 24/32 = 3/4 so these numbers fit the initial ratio. When I did this question, I picked a different set of numbers (Anna is 12, Paula is 16) and then went 12+8/16+8 = 20/24 = 5/6. Why did my numbers not work? Alternatively, why did you pick the numbers 24 and 32? 12/16 still reduces down to 3/4 after all...

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Re: If the ratio of the present age of Anna and Paula is 3 : 4, what could  [#permalink]

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New post 23 Apr 2019, 16:06
YB101 wrote:
ScottTargetTestPrep wrote:
EgmatQuantExpert wrote:
Q. If the ratio of the present age of Anna and Paula is 3 : 4, what could be the ratio of their respective ages after 8 years??

    a. 1 : 2
    b. 3 : 8
    c. 3 : 5
    d. 2 : 3
    e. 4 : 5


Some possible ratios of their current ages are:

A/P = 3/4, 6/8, 9/12, 12/15, 15/20, 18/24, 21/28, 24/32…

So after 8 years, these ratios, would be:

11/12, 14/16 = 7/8, 17/20, 20/23, 23/28, 26/32 = 13/16, 29/36, 32/40 = 4/5, …
We see that if Anna is 24 and Paula is 32, the ratio of their ages 8 years later is 32/40 = 4/5, or 4 : 5.

Answer: E


Hi Scott,

Just wanted to clarify - you've picked an example here of Anna being 24 and Paula being 32. I.e. 24/32 = 3/4 so these numbers fit the initial ratio. When I did this question, I picked a different set of numbers (Anna is 12, Paula is 16) and then went 12+8/16+8 = 20/24 = 5/6. Why did my numbers not work? Alternatively, why did you pick the numbers 24 and 32? 12/16 still reduces down to 3/4 after all...

Thanks


First of all, your numbers work too; you determined that the ratio of ages could be 5/6. It’s just that 5/6 is not among the answer choices.

About your second question on why the numbers 24 and 32; what I’ve done is actually listing all numbers with a ratio of 3:4 or at least, I’ve started listing such numbers (there are infinitely many of them). In the next step, I added 8 to the numerator and the denominator to get the possible values for the ratio after 8 years. I’ve continued this process until I got one of the values among the answer choices and that’s why I ended the list when I reached 24 and 32.
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If the ratio of the present age of Anna and Paula is 3 : 4, what could  [#permalink]

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New post 25 Apr 2019, 17:27
EgmatQuantExpert wrote:

Method 2:



    • Now, the first method is a bit cumbersome and time-consuming too.
    • And the purpose of the question was to teach you a simple concept of ratios/fractions

      o Suppose we have a fraction say \(\frac{N}{D}\), where \(\frac{N}{D}\) is a proper fraction (D>N), for example, \(\frac{1}{2},\frac{2}{3}\) etc, and if we add the same number (say A) to the numerator and denominator, will the new ratio be greater than \(\frac{N}{D}\) or less than?
      o For example, if we have \(\frac{2}{3} = 0.666\)

      o Now say we add 1 to both the numerator and denominator, then the new ratio would be

         \(\frac{(2+1)}{(3+1)} = \frac{3}{4} = 0.75\)
         And we can see clearly that \(\frac{3}{4} > \frac{2}{3}\)
         Hence \(\frac{(N+a)}{(D+a)} > \frac{N}{D}\), when the fraction is a proper fraction.

    • With this knowledge let us now solve the question.
    • Ratio of age of Anna and Paula = 3 : 4 or \(\frac{3}{4} = 0.75\)
    • We have been asked the age after 8 years

      o Now notice we are adding 8 to both the numerator and denominator.
      o Anna: Paula =\(\frac{(3x + 8)}{(4x + 8)}\)
      o Thus, we should immediately conclude that the new ratio(fraction) must be more than 0.75.

    • Thus, we just need to look at the options quickly and find a ratio which is more than 0.75

      A. \(1: 2 = \frac{1}{2} = 0.5\)
      B. \(3: 8 = \frac{3}{8} = 0.375\)
      C. \(3:5 = \frac{3}{5} = 0.6\)
      D. \(2: 3 = \frac{2}{3} = 0.66\)
      E. \(4: 5 = \frac{4}{5} = 0.8\) (Hence Option E is the answer)


Thanks,
Saquib
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Simple, elegant, excellent.

Although I knew such property existed, I got the question wrong. I assumed (terribly) that because an odd + even must equal odd, then the new ratio of their ages would also be odd:odd. Of course, I had the foresight in forgetting that their ages could be \(\frac{3(even)+8}{4(even)+8}\) which would collapse my whole theory.
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If the ratio of the present age of Anna and Paula is 3 : 4, what could   [#permalink] 25 Apr 2019, 17:27
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