Problem Solving

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

I am not able to understand the question line itself. Can someone explain?

What's going on in these age questions is that, while the absolute differences in ages don't change over the years, the relative proportions of the ages do.

For example, if Joe is 10 years old, and Mary is 5 years old, Joe is 5 years older than Mary, and twice as old.

Fifteen years later, Joe would be 25 years old and Mary would be 20 years old. Joe would still be 5 years older, but he would only be 1.25 times as old as Mary at that point.

So this question gives us a relative proportion of ages, five years ago, Jim's age was 3 times Raoul's age.

It also gives us an absolute difference in ages, five years ago, Monica's age was 6 years greater than Raoul's age.

As time goes on, the relative proportions of their ages will change, but the absolute differences will not.

Now let's look at the statements.

I. For Monica to be older than Jim in five years, she needs to be older than Jim is now, and she had to be older five years ago. What we know is that Monica is six years older than Raoul. We also know that five years ago, Jim was three times as old as Raoul. So, five years ago Raoul could have been 4 years old, Jim could have been 12 years old and Monica could have been 10 years old. So it's not necessarily the case that Monica was older than Jim and will be in five years.

II. If Monica was 6 years older than Raoul five years ago, she will still be five years from now. So this is a sure thing.

III. We know that Raoul's age is 6 less than Monica's. So for their combined ages to be greater than hers, Jim age has to be over 6 years. The time period being discussed includes five years before the present and five years after the present. So five years from now Jim will be at least 10 years old. So this one has to be true too.

So the correct answer is E.

nitinmenon89 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 EXTREMES.

Case 1: 5 years ago R=1
Since Jim was times as old as Raoul was, J = 3*1 = 3.
Since Monica was 6 years older than Raoul, M = 1+6 = 7.

Ages 5 years from now:
R = 1+10 = 11.
J = 3+10 = 13.
M = 7+10 = 17.

In this case, I, II and III are all true.

Case 2: 5 years ago R=20
Since Jim was times as old as Raoul was, J = 3*20= 60.
Since Monica was 6 years older than Raoul, M = 20+6 = 26.

Ages 5 years from now:
R = 20+10 = 30.
J = 60+10 = 70.
M = 26+10 = 36.

In this case, only II and III are true.

Since only II and III are true in both cases, the correct answer is E.

Hi,

Can you please explain the solution if I do like

Let present age of Raoul R


5 years before

Raoul's age : R-5

Jim's age : 3 * (R-5)

Monica's age : (R-5) + 6


As Jim's age can't be -ve minimum value of R is 6 but then I am getting statement 1 and 2 as true and hence my answer is option C.

Please tell me where I am making mistake in my approach.

Hi,

Can you please explain the solution if I do like

Let present age of Raoul R


5 years before

Raoul's age : R-5

Jim's age : 3 * (R-5)

Monica's age : (R-5) + 6


As Jim's age can't be -ve minimum value of R is 6 but then I am getting statement 1 and 2 as true and hence my answer is option C.

Please tell me where I am making mistake in my approach.

vishalwin wrote:Hi,

Can you please explain the solution if I do like

Let present age of Raoul R


5 years before

Raoul's age : R-5

Jim's age : 3 * (R-5)

Monica's age : (R-5) + 6


As Jim's age can't be -ve minimum value of R is 6 but then I am getting statement 1 and 2 as true and hence my answer is option C.

Please tell me where I am making mistake in my approach.

It could be that Raoul's age is at some minimum such that Jim's age is less than Monica's, but the question regards which of the statements must be true. Raoul's age, R, could be a high number such that 3(R - 5) > (R - 5) + 6.

For instance, R could be 20, in which case 3(15) > (15) + 6.

Hi Marty,

I didn't get your point.


Are you saying "combined ages of Jim and Raoul are more than Monica's age."

Raoul's age : R-5

Jim's age : 3 * (R-5)

Monica's age : (R-5) + 6

put R=20

Then

Raoul's age : R-5 = 20 - 5 = 15

Jim's age : 3 * (R-5) = 3*15 = 45

Monica's age : (R-5) + 6 = 21

Here, 3(R - 5) > (R - 5) + 6 is true


and, combined ages of Jim and Raoul are more than Monica's age

15 + 21 < 45


but when R= 6

Raoul's age : R-5 = 6 - 5 = 1

Jim's age : 3 * (R-5) = 3*5 = 15

Monica's age : (R-5) + 6 = 7

Here, 3(R - 5) > (R - 5) + 6 is true


and, combined ages of Jim and Raoul are more than Monica's age

1 + 7 < 15


so R=20 is similar when R =20 as per your point.

What say?

vishalwin wrote:Hi,

Can you please explain the solution if I do like

Let present age of Raoul R


5 years before

Raoul's age : R-5

Jim's age : 3 * (R-5)

Monica's age : (R-5) + 6


As Jim's age can't be -ve minimum value of R is 6 but then I am getting statement 1 and 2 as true and hence my answer is option C.

Please tell me where I am making mistake in my approach.

Your algebraic setup looks good. It's likely that you've made a careless mistake somewhere. Mitch addressed the scenario where R = 6 above. If Raoul is 6 now, then five years ago, the ages of each would be

FIVE years ago
Raoul: 6-5 = 1
Jim: 3 * (6-5) = 3*1 = 3
Monica: (6-5) + 6 = 1 + 6 = 7


Which means their ages NOW are as follows:
Raoul: 6
Jim: 8
Monica: 12

And their ages in five years:
Raoul: 11
Jim: 13
Monica: 17

So all three statements are true in this case.

But if make Raoul is significantly older, then statement 1 need not be true. Say R = 30

FIVE years ago
Raoul: 30-5 = 25
Jim: 3 * (30-5) = 25 = 75
Monica: (30-5) + 6 = 31


Which means their ages NOW are as follows:
Raoul: 30
Jim: 80
Monica: 36

And their ages in five years:
Raoul: 35
Jim: 85
Monica: 41

Now only II and III are true.

vishalwin wrote:Hi Marty,

I didn't get your point.

Are you saying "combined ages of Jim and Raoul are more than Monica's age." What say?

The point is that the statements refer to their ages 5 years from now, at which point combined, (R + 5) + [3(R - 5)] + 10 will be greater than (R + 6) + 5.

Since Raoul was alive 5 years ago, the minimum value of R is 5.

Five years ago:
Raoul's age R = R
Jim's age J = 3R
and Monica's age M = R + 6

Five years from now:
Raoul's age R = R + 10
Jim's age J = 3R + 10
and Monica's age M = R + 16

I. Monica is older than Jim.
M > J
R + 16 > 3R + 10
6 > 2R
R < 3
Statement (1) will hold true as along as Raoul's age is less than 3 years.
There is no info on Raoul's age, so statement (1) can or cannot be true.

II. Raoul is six years younger than Monica
It was already given that "Monica was six years older than Raoul" and it won't change if we add years because it's a relative phenomenon."
So, it's always true that Raoul is six years younger than Monica.

III. The combined ages of Jim and Raoul are more than Monica's age.
J + R > M
(3R + 10) + (R+10) > (R + 16)
(4R + 20) > (R + 16)
LHS will always greater than RHS

Answer E

BOOM - There it IZZZZ!!!!