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Five years ago Jim was three times as old as Raoul

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Five years ago Jim was three times as old as Raoul

by BTGmoderatorLU » Sat Mar 31, 2018 2:50 pm
Five years ago Jim was three times as old as Raoul was and Monica was six years older than Raoul was. If all three are still living in five years, which of the following must be true about their ages five years from now?

I. Monica is older than Jim.
II. Raoul is six years younger than Monica.
III. The combined ages of Jim and Raoul are more than Monica's age.

A. I only
B. II only
C. I and II
D. I and III
E. II and III

The OA is E.

Let CURRENT ages of Jim = j, Raoul = r, and Monica = m.

Five years ago Jim = j - 5, Raoul = r - 5, and Monica = m - 5.

Set up equations:

j - 5 = 3(r - 5) . . . . . . . . . . . j + 10 = 3r

m - 5 = r - 5 + 6 . . . . . . . . m = r + 6

Because we have only 2 equations in 3 variables, there are open cases. So we need to be careful to choose numbers that cover multiple cases.

Plug in numbers:

Case 1: r = 7, j = 11, m = 13. After five years, r = 12, j = 16, m = 18.

Case 1: r = 10, j = 20, m = 16. After 5 years, r = 15, j = 25, m = 21.

Checking numeral I as it is most frequent.

From case 1: m > j

From case 2: m < j

To save time, check numeral III, not II. Because if you do II and get correct then you will move to III. But we do III first, you will eliminate one choice in one step.

From case 1: r + j > m.

From case 2: r + j > m.

Therefore, eliminate choice B. Hence option E is the correct answer.

Please, can anyone explain another way to solve this Ps question? Thanks!
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by Jay@ManhattanReview » Tue Apr 03, 2018 1:42 am
BTGmoderatorLU wrote:Five years ago Jim was three times as old as Raoul was and Monica was six years older than Raoul was. If all three are still living in five years, which of the following must be true about their ages five years from now?

I. Monica is older than Jim.
II. Raoul is six years younger than Monica.
III. The combined ages of Jim and Raoul are more than Monica's age.

A. I only
B. II only
C. I and II
D. I and III
E. II and III

The OA is E.
Let the current ages of Jim, Raoul, and Monica are J, R, and M, respectively.

Given five years ago Jim was three times as old as Raoul was, we have

J - 5 = 3(R - 5) => J = 3R - 10 ---(1)

Given Monica was six years older than Raoul was, we have

M - 4 = (R - 5) + 6 => M = R + 6 ---(2)

=> Since M = R + 6, Raoul is six years younger than Monica now and will always remain 6 years younger even after 5 years from now. Statement II is correct.

From (2), we have R = M - 6. By plugging in the value of R in eqn (1), we have J = 3(M - 6) - 10

=> J = 3M - 28 ---(3)

Eqn (3) is useful to analyze Statement I.

Case 1: At lower values of M, we find that Monica is older than Jim.

Example: Say M = 12, then J = 3*12 - 28 = 36 - 28 = 8. M > J.

Case 2: At higher values of M, we find that Monica is NOT older than Jim.

Example: Say M = 40, then J = 3*40 - 28 = 120 - 28 = 92. J > M.

The same scenarios would be there even after 5 years from now.

Thus, Statement I is incorrect. The correct answer must be B or E.

Let's analyze Statement III: The combined ages of Jim and Raoul are more than Monica's age.

The ages 5 years from now of Jim, Raoul, and Monica would be J + 5, R + 5, and M + 5, respectively.

Let's assume that the combined ages of Jim and Raoul are more than Monica's age.

Thus, J + 5 + R + 5 > M + 5 => J + R + 5 > M

Plugging in the value of J from (3) J = 3M - 28

We have (3M - 28) + R + 5> M => 2M + R > 23

Plugging in the value of R from eqn (2), we have

2M + R > 23 => 2M + (M - 6) > 23 => 3M > 29

=> M > 9.67

For Statement III to be correct Monica's age now must be more than 9.67 years. This is correct since 5 years before Raoul was there (alive) and Monica is 6 years older than him, thus, Monica must be more than 11 (= 6 + 5) years now.

Statement III is also correct.

The correct answer: E

Hope this helps!

-Jay
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by Brent@GMATPrepNow » Tue Apr 03, 2018 7:17 am
BTGmoderatorLU wrote:Five years ago Jim was three times as old as Raoul was and Monica was six years older than Raoul was. If all three are still living in five years, which of the following must be true about their ages five years from now?

I. Monica is older than Jim.
II. Raoul is six years younger than Monica.
III. The combined ages of Jim and Raoul are more than Monica's age.

A. I only
B. II only
C. I and II
D. I and III
E. II and III
Let R = Raoul's PRESENT age
So, R - 5 = Raoul's age 5 YEARS AGO

Five years ago .... Monica was six years older than Raoul was.
So, (R - 5) + 6 = Monica's age 5 YEARS AGO
In other words, R + 1 = Monica's age 5 YEARS AGO

Five years ago Jim was three times as old as Raoul was
So, 3(R - 5) = Jim's age 5 YEARS AGO

IMPORTANT: In order for us to know the information about Raoul's age 5 years ago, it must be the case that Raoul's PRESENT age is greater than 5. Otherwise, Raoul wouldn't have been alive 5 years ago

To find the ages 5 years in the FUTURE, we must take these ages for 5 years ago and add 10 years.

So, (R - 5) + 10 = Raoul's age 5 YEARS IN THE FUTURE
R + 1 + 10 = Monica's age 5 YEARS IN THE FUTURE
3(R - 5) + 10 = Jim's age 5 YEARS IN THE FUTURE

SIMPLIFY to get:
R + 5 = Raoul's age 5 YEARS IN THE FUTURE
R + 11 = Monica's age 5 YEARS IN THE FUTURE
3R - 5 = Jim's age 5 YEARS IN THE FUTURE

Now, let's examine the statements:

I. Monica is older than Jim.
MUST it be the case that R + 11 is greater than 3R - 5?
No.
If R = 10, then R + 11 = 21 and 3R - 5 = 25
So, if R = 10, Monica is NOT older than Jim (5 years from now)
So, statement 1 need not be true.

We can ELIMINATE answer choices A, C and D

IMPORTANT: Notice that the remaining answer choices (B and E) both say that statement II is correct.
So, we need not check statement II, since it MUST be correct.

III. The combined ages of Jim and Raoul are more than Monica's age.

Is it true that (3R - 5) + (R + 5) > (R + 11)?
Let's simplify to get: 4R > R + 11
Subtract R from both sides to get: 3R > 11
Divide both sides by 3 to get: R > 11/3
MUST this be TRUE?
Yes. It must be true, because we earlier concluded that it must be the case that R is greater than 5
So statement III must be true.

Answer: E
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Jim, Raoul and Monica

by GMATGuruNY » Tue Apr 03, 2018 7:49 am
BTGmoderatorLU wrote:Five years ago Jim was three times as old as Raoul was and Monica was six years older than Raoul was. If all three are still living in five years, which of the following must be true about their ages five years from now?

I. Monica is older than Jim.
II. Raoul is six years younger than Monica.
III. The combined ages of Jim and Raoul are more than Monica's age.

A. I only
B. II only
C. I and II
D. I and III
E. II and III
Test the SMALLEST POSSIBLE CASE.

Five years ago Jim was three times as old as Raoul was and Monica was six years older than Raoul was.
Let R=1, implying that J = 3R = 3*1 = 3 and that M = R+6 = 1+6 = 7.
In this case, their ages 5 years from now -- in other words, 10 years beyond 5 years ago -- are as follows:
R = 1+10 = 11.
J = 3+10 = 8.
M = 7+10 = 17.
Here, statements I, II and III are all true.

Test an EXTREME CASE:

Five years ago Jim was three times as old as Raoul was and Monica was six years older than Raoul was.
Let R=30, implying that J = 3R = 3*30 = 90 and that M = R+6 = 30+6 = 36.
In this case, their ages 5 years from now -- in other words, 10 years beyond 5 years ago -- are as follows:
R = 30+10 = 40.
J = 90+10 = 100.
M = 36+10 = 46.
Here, only statements II and III are true.
Eliminate any answer choice that includes statement I.
Eliminate A, C and D.

Since statements II and III are true not only when R is very small but also when R is very large, these statements must always be true.

The correct answer is E.
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by Jeff@TargetTestPrep » Thu Apr 05, 2018 4:18 pm
BTGmoderatorLU wrote:Five years ago Jim was three times as old as Raoul was and Monica was six years older than Raoul was. If all three are still living in five years, which of the following must be true about their ages five years from now?

I. Monica is older than Jim.
II. Raoul is six years younger than Monica.
III. The combined ages of Jim and Raoul are more than Monica's age.

A. I only
B. II only
C. I and II
D. I and III
E. II and III
We can let Jim's age today = J, Raoul's age today = R, and Monica's age today = M.

Let's set up their ages 5 years ago: Jim was (J - 5), Raoul was (R - 5), and Monica was (M - 5).

Since, five years ago, Jim was three times as old as Raoul was, we have:

J - 5 = 3(R - 5)

J - 5 = 3R - 15

J = 3R - 10

Since, five years ago, Monica was six years older than Raoul was, we have:

M - 5 = (R - 5) + 6

M - 5 = R + 1

M = R + 6

Notice that Raoul is the youngest of the three people, and Raoul must be more than 5 years old since only then can we talk about their ages 5 years ago.

Let's now test each Roman numeral (keep in mind that in 5 years, Jim's, Raoul's and Monica's ages will be J + 5, R + 5 and M + 5, respectively):

I. Monica is older than Jim.

Is M + 5 > J + 5 ?

Is (R + 6) + 5 > (3R - 10) + 5 ?

Is R + 11 > 3R - 5 ?

Is 16 > 2R ?

Is 8 > R ?

Is R < 8?

We know that R > 5; however, we can't determine whether R < 8. Thus, we cannot determine whether Monica is older than Jim.

II. Raoul is six years younger than Monica.

Since M = R + 6, in 5 years Raoul will still be 6 years younger than Monica. Roman numeral II is true.

III. The combined ages of Jim and Raoul are more than Monica's age.

We already see that in 5 years, Monica's age will be R + 11, Jim's age will be 3R - 5, and Raoul's age will be R + 5. We can create the following inequality:

Is (3R - 5) + (R + 5) > R + 11 ?

Is 4R > R + 11 ?

Is 3R > 11 ?

Is R > 11/3 ?

We know that R > 5, so R > 11/3. Thus Roman numeral III is true.

Answer: E

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Head of GMAT Instruction
[email protected]

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