Bob's Flat tire

[jean-baptiste]

"Bob bikes to school every day at a steady rate of x miles per hour. On a particular day, Bob had a flat tire exactly halfway to school. He immediately started walking to school at a steady pace of y miles per hour. He arrived at school exactly t hours after leaving his home. How many miles is it from the school to Bob's home?

A: (x + y) / t
B :2(x + t) / xy
C : 2xyt / (x + y)
D : 2(x + y + t) / xy
E : x(y + t) + y(x + t)
"

I Got this problem wrong. I looked at the answer explanation, and I found it way too complicated, algebraically speaking.

I was told by a Gmat professor that plugging numbers was an efficient way of dealing with such obscure problems.

However, no matter how long I reflected, I was not able to come with a decent plan.

Is it possible to solve this problem by plugging numbers, this to say by adopting an easier way than a rough algebraic way ?

Thank you for your help,
JB

[RonPurewal]

sure -- although it's much easier if you don't pick numbers for the variables themselves.
here's the lesson/takeaway: You don't necessarily have to pick values for the variables. You can pick numbers for whatever is easiest to start with.

in this case, the easiest thing to start with is exactly the thing that's NOT a variable: the distance to school.

let's say it's 20 miles to school, so that "halfway to school" is 10 miles. (long commute!)

then... let's say that the biking is at 5 mi/h (x = 5). so, 2 hours on the bike.
let's say that the walking is at 2 mi/h (y = 2). so, 5 hours on foot.
... so 2 + 5 = 7 hours = t.

the answer we want is the total distance, which is 20 miles.

(A) (2 + 5)/7 = 1 = not 20
(B) 2(5 + 7)/(5 x 2) = 12/5 = not 20
(C) 2(5)(2)(7)/(5 x 2) = 20
(D) 2(5 + 2 + 7)/(5 x 2) = 14/5 = not 20
(E) 5(2 + 7) + 2(5 + 7) = waayyyy more than 20

done!

[RonPurewal]

by the way, you can do other things, too.

* one thing you can realize is that adding times to speeds doesn't make sense. so...
... you can't add or subtract x and t,
... you can't add or subtract y and t.
(note: there's no issue with multiplying or dividing disparate units; in fact, that's how all unit conversions are accomplished.)
this humble observation is actually enough to kill every choice except (a) and (c).

[RonPurewal]

* you can also notice that "x" and "y" are interchangeable in this problem -- both are speeds that prevail for exactly the same part of the distance, so switching them doesn't matter.
i.e., if you switch the "first" and "second" halves of the trip, then nothing should change. (this would NOT be true if the walking and biking were for different numbers of miles.)
therefore, the correct answer has to be a formula in which switching x and y produces the same formula.

here, this only eliminates choice (b) -- there's a remarkable degree of symmetry in the rest of the answer choices -- but, in other problems, similar observations could get you a lot more mileage. (ha! "mileage"! i'm funny.)

[jean-baptiste]

Thank you,

Your help was awesome. Now I get it :)

[RonPurewal]

Thank you,

Your help was awesome. Now I get it :)

no problem!

[griffin.811]

Hi Team, can someone help me figure out where I'm going wrong on this one?

I chose 10 for the total distance, so half the distance to school is 5mi. I then chose 5mph for the biking rate yielding a time of 1hour.

Next, I chose a walking rate of 2mph yielding a time of 5/2.

Here's where Im a little confused. When I try to complete the final row of my RTD chart, I get a rate of 7mph multiplied by a time of 7/2 which does not equal 10mph.

My initial guess is that one of the RTD groups doesnt have an additive relationship, but they all seem to.

Also, any general advice for trickier RTD problems? This one wasn't too bad because I could solve algebraically, but some others are quite difficult. I've been spending a good deal of time trying to hammer these out, but without much success. Unfortunately, I find myself trying to do every rate problem as much as I can to "memorize" how to attack each type, but I'm pretty sure this will get me in trouble come test day if I can't have a more flexible thought process.

Thanks!

[RonPurewal]

When it comes to figuring out whether RTD quantities are related in certain ways, that's 100% good old-fashioned God-given common sense.

This problem consists of a journey with 2 parts. If you don't know whether certain quantities combine in some way, just think about a real two-part trip.

• Can I add times?
Well... let's say it takes me 5.5 hours to drive from San Jose to Barstow and 2.5 hours from Barstow to Las Vegas.
Does it take me 8 hours to drive from San Jose (through Barstow) to Las Vegas?
Yep.
Ok, I can add times.

• Can I add speeds?
Let's say I drive an average of 70 mph from San Jose to Barstow, and an average of 80 mph from Barstow to Las Vegas.
Does the figure "150 mph" have any real meaning whatsoever in this situation?
Nope. It's total hocus pocus. Complete nonsense. A random number that means nothing at all.
I guess I can't add speeds.

You get the point.

[RonPurewal]

Here's where Im a little confused. When I try to complete the final row of my RTD chart, I get a rate of 7mph multiplied by a time of 7/2 which does not equal 10mph.

That. That pink thing. That's what we call "lack of focus". Or, "loss of focus".

Why would you need a "final row"?

Take a look at the problem again. The only quantities named in the problem are:
• total distance from home to school;
• speed on two wheels;
• speed on foot;
• total time.

You already have the first three quantities. They're 10, 5, and 2. So, all you need is total time.

It's pretty clear that total time is the sum of the two smaller times. Those, you also have (1 hour and 5/2 hours).

That's everything.
Everything.

There are no more things.

If you are even thinking about "completing a final row" at this point, it means that you've forgotten what you're actually trying to do.

Oops.

[RonPurewal]

And of course you can't generalize from situation to situation. If two situations are meaningfully different, then the relevant mathematical relationships will be different, too.

E.g., if I start from Las Vegas, my friend starts from San Jose, and we both drive to Bakersfield (= approx halfway point) and meet up there, then ...
... I can add speeds this time. (If he's driving 70 miles/hour and I'm driving 80 miles/hour, then, yes, we are really covering 150 miles together in each hour of driving.)
... I cannot add times. (We're doing things at the same time, so "total time" is not a thing here.)

[RonPurewal]

Also, any general advice for trickier RTD problems?

"Tricky" is nonsense.

In the context of word problems, the honest meaning of "tricky" is...
... I didn't organize this problem well enough, or
... I didn't write enough stuff down.

"Tricky" is a dangerous word because it's a dodge. By calling a problem "tricky", you're placing the onus on the writers of the problem, rather than on you.
I.e., as soon as you label a problem "tricky", you are MUCH less likely to say, "OK, ____ was a shortcoming in my approach, and I can fix it by doing ____ next time."

If you think there are "tricks", there are two negative consequences:

1/
Because you think the root of the problem is "They tricked me" (rather than "My approach was lacking"), you're less likely to shore up existing shortcomings. Or even to acknowledge that they are shortcomings in the first place. (A "trick", by definition, is something that will foil approaches that DON'T have fundamental shortcomings.)

2/
You'll think that you need to learn gimmicky approaches. Like, anti-trick tricks. More tricks, to defend against their tricks.
Needless to say, this is not a good situation. It will devolve into an arms race of gimmicks. Which you will lose, because you're actually the only one racing.

There are no "tricks".

Are there annoying problems, with lots of information, that require greater-than-normal attention to organization and writing stuff down?
Yes.

Gimmicks?
No.

GMAT problems are honest. That's why I like them. They don't have booby traps.


--


(NB: If you were genuinely using "tricky" ONLY to mean "I found this difficult", then this little lecture doesn't apply to you. But that's unlikely.)

[RonPurewal]

I find myself trying to do every rate problem as much as I can to "memorize" how to attack each type, but I'm pretty sure this will get me in trouble come test day if I can't have a more flexible thought process.

^^ This.

You've reached a milestone here.

Specifically, you've reached the point of acknowledging that you cannot memorize a "formula" for how to set up word problems. (Amazingly, some people try to do this for years before finally acknowledging that it doesn't work and/or makes easy things really hard. Hard-working in all the wrong ways.)

So, you're halfway out of the woods. On the other hand, unfortunately, you're still halfway into the woods.
E.g., you just added speeds up there, for a two-part journey. When I gave you the example of 70mph + 80mph = 150mph between SJ and Las Vegas, you almost certainly thought, "No, I can't do that. Even if I thought about doing that, well, 150mph is obviously nonsense."
Now, you need to invoke the same common sense when you're solving these problems.

Classrooms steal common sense (and self-driven organizational skills). Your job is to steal those skills back from the evil classroom monster.

[RonPurewal]

And, finally, this is a response to something in the original post. Yes, the original post is a year old, but this is one of those things that can never be repeated too much.

I was told by a Gmat professor that plugging numbers was an efficient way of dealing with such obscure problems.

The purple thing is a big problem.

The point of learning new strategies——not just for the GMAT, but really for anything in life——is to become more self-sufficient.

It shouldn't be "Ron said that strategy X works on problems that look like this."

It SHOULD be:
• "I know strategy X." (Maybe I learned it from Ron; maybe I learned it in a psychedelic dream. Maybe I even think I invented it myself. Doesn't really matter.)
• "I think I'll throw strategy X at this problem and see how things turn out."
• IF IT WORKS: "Let me think about why strategy X is well adapted to this problem."
• IF IT DOESN'T: "Let me think about what makes strategy X inapplicable here."

Etc.

In fact, if you know more strategies, you should be able to appeal less to "authority" / GMAT tutors / answer keys. Because you'll have a Plan B, and a Plan C, and so forth.

[griffin.811]

Thanks so much Ron. This must have taken a good deal of time, so hopefully it can help others that view this page as well. The part where you used examples to discuss that rates are not additive was particularly helpful, and I can't believe how foolish my initial attempt was. Also I completely agree with having multiple approaches, it's something I spend a decent amount of time working on during review. As you stated, however, I need to do a better job of finding out why a particular method was better suited for a problem than maybe another would be. There are a ton of good takeaways here, so thanks again!

[RonPurewal]

You're welcome.