Jim went to the bakery to buy donuts for his office mates. He chose a

Hi,

This is the first time I am posting here because I am losing my mind over this one problem.. i'm really hoping someone can help. Thank you so very much in advance.

This is the problem:

Jim has arrived late for his golf tee time with his friends. When he arrives at the course, he discovers that his friends have already walked 180 yards ahead on the course and that they are continuing to walk forward at a rate of 1.5 yards per second. A bystander with a golf cart takes pity on Jim and drives him to meet his friends at the rate of 4 yards per second. Once Jim meets his friends, he exits the cart and walks with them at their rate for the next 120 yards. What is Jim’s approximate average speed for his total trip?

A )1.8 yards per second
B) 2.0 yards per second
C) 2.4 yards per second
D) 2.7 yards per second
E) 3.0 yards per second


The correct answer is D, and I don't know how to get there.

My approach is:

Calculate total distance and total time traveled, and then say average speed = total distance / total time traveled.

First, for the golf cart piece, I'm saying that Jim is traveling a distance of 180 yards. Given his friends are walking at 1.5 yards per second, Jim's effective speed is 4 yards per second - 1.5 yards per second = 2.5 yards per second. Hence it would take him 180/2.5 = 72 seconds to catch up with his friends. So from the golf piece, distance = 180 yards, time = 72 seconds.

Once he's walking with his friends, he walks at 120 yards at a rate of 1.5 yards per second, so total time is 120/1.5 = 80 seconds.

Combined, then total distance / total time = (180+120)/(72+80) = 300/152. However that is clearly wrong.

If someone could explain why this is not correct I would be deeply grateful. Thanks so much guys.

Hi adm29,

Most of the work (especially the "concept" work) that you did was correct, but here's the error in your calculation:

You correctly deduced that Jim needs 72 seconds to catch-up to his friends, BUT he was traveling 4 yards/second during that time, so he's actually traveled 4(72) = 288 yards (not 180).

He then travels another 120 yards in 80 seconds with his friends.

His total distance is 288+120 = 408 yards
His total time is 72+80 = 152 seconds

Thus, his average speed is:

408 yards = (Av. Sp.)(152 seconds)

408/152 = Av. Sp.
About 2.7 yards/second

Final Answer:

GMAT assassins aren't born, they're made,
Rich

First of all, let me point out that you should make a separate post for every question you put up.

A concept to note is that average speed is the weighted average of two speeds of 4 yards/sec and 1.5 yards/sec where time is the weight.

He travels at 4 yards/sec for 72 secs - perfect
He travels at 1.5 yards/sec for 80 secs - correct again

\(Average Speed = \frac{(Speed1 * Time1 + Speed2 * Time2)}{(Time1 + Time2)}\)

\(Average Speed = \frac{(4*72 + 1.5*80)}{152} = \frac{(4*9 + 1.5*10)}{19} = \frac{51}{19} = 2.7 yards/sec\)

Another way to deal with this problem is using weighted averages:

Two groups of doughnuts are mixed together - n doughnuts costing 15/n per doughnut in one group, and 3 doughtnuts costing 0 in second group.
The average price of the doughnuts is (15/n - 1/6) per doughnut.

You get 3/n = (1/6)/(15/n - 1/6)
On solving, n = 15

So total number of doughtnuts received = 15+3 = 18

Hi Rich,

I am trying to set up a "Test the Answer" equation for this question and I am not getting it correct, I would appreciate your help

Let 12x be the quantity of donuts originally recieved
Let p equal the original price

So we have (12x) * p = $15
Second equation would be (12x +3) * (p-2) = $15

As answer choice A is the correct answer, let us try it
so it means he originally bought 15 donuts

therefore 12x = 15, and p= $1
When i try to substitute it in the second equation, we get a (18) * (1-2) = 15....which would give us a wrong answer

I would be most grateful if you can shed more light on where i am going wrong with this question

Hi Tmoni26,

Part of the difficulty that you're facing is that you decided to refer to the number of donuts as 12X (referring to them as X would have been easier). You can actually avoid all of the Algebra altogether though. Here's how:

We know that a certain number of donuts were bought for $15, which would give us a certain price/dozen.

After getting another 3 donuts for free, the price/dozen for all the donuts drops $2 EXACTLY. The fact that the difference is an integer is interesting - this makes me think that the original number of donuts (X) and the larger number of donuts (X+3) both "relate" nicely to the number 12. $2 is also relatively small compared to $15, so the correct answer will probably be one of the smaller answers.

Starting with Answer A, we'd have....

Final number of donuts = 18 (this is a nice number - it's exactly 1.5 dozen donuts)
Initial number of donuts = 15 (this is also relatively nice - it's exactly 1.25 dozen donuts)

18 donuts for $15 = 1.5 dozen for $15 = $10 per dozen
15 donuts for $15 = 1 donut for $1 = $12 per dozen

The difference here is $2, which is a MATCH for what we were told.

Final Answer:

GMAT assassins aren't born, they're made,
Rich

Hi Rich,
Thanks for the explanation.

When I first attempted the question, I did not use 12X, I used X but I got stumped with the "per dozen" aspect

My first set of equations were
So we have (x) * p = $15
Second equation would be (x +3) * (p-2) = $15
But the second equation got me confused because I was like, how would we be getting a negative value for (p-2)
Arrrghhh!

Hi Tmoni26,

If you were dead-set on using those equations, then you would have to pay careful attention to what the variables REALLY mean and adjust your math accordingly....

In your setup, if you're going to use P and (P-2) to refer to the prices, then the variables are...

P = price per DOZEN donuts
X = number of DOZEN donuts

So if you have 12 donuts, then X = 1
if you have 24 donuts, then X = 2
if you have 18 donuts, then X = 1.5
etc.

This seems like an overly complicated way to do the work, but it could have gotten you to the correct answer.

GMAT assassins aren't born, they're made,
Rich

Let Jim bought X dozen at $15
So, Old Price = 15/X per dozen

New Quantity = X + 1/4 dozen (3 nos. = 1/4 dozen)
New Price = 15/(X + 1/4)

Old Price - New Price = 2

15/X - 15/(X + 1/4) = 2
Solving you get X = 5/4 dozen (eliminating the -ve solution)
So new quantity = (5/4 + 1/4) = 6/4 dozen or 18 nos.

How do I get from the first line to the second line? Stupid math question but I don't get it...

I solved like this:

x + 3 * (15/x - 1/6) = 15
x * (15/x - 1/6) = 12
15x/x - 1/6 * x = 12
15 - 1/6 * x = 12
3 = 1/6 *x

x = 18

Let y = No of Donuts & X = Price for each donut

Equation -I : X * Y = 15

Equation-II: (x - 1/6) * (Y+3) = 15

Solving for Y

Y = 3 * (6X -1)

let X = 1 => Y = 15 we get X * Y = 15 i.e. Our first Equation

He left the bakery with X + 3 i.e. 18 Donuts

\((x+3)(\frac{15}{x} - \frac{1}{6}) = 15\)
is different from
\(x(\frac{15}{x} - \frac{1}{6}) + 3 = 15\)

In the first equation, 3 is also multiplied by the entire \((\frac{15}{x} - \frac{1}{6})\). So you cannot take it to the right like that. Following your lead, this should be solved like this:

\((x+3)(\frac{15}{x} - \frac{1}{6}) = 15\)

\(x(\frac{15}{x} - \frac{1}{6}) + 3*(\frac{15}{x} - \frac{1}{6}) = 15\)

\(15 - \frac{x}{6} + \frac{45}{x} - \frac{1}{2} = 15\)

Now multiply by 6x to get

\(90x - x^2 + 270 - 3x = 90x\)

\(x^2 + 3x - 270 = 0\) (multiplied by -1)

oh, wow, tough one, took me some time to get to the answer:

ok, so suppose we have n - number of doughnuts and x - price paid for each one. nx = 15$.
now, because the doughnuts were not in good shape, he received n+3 doughnuts.
we are told that with the new quantity, the price for 12 would be -2$. this is where the trick part is...
1 doughnut cost 15/n. now, 15/n * 12 = 180/n. this is the price for 12 doughnuts, and from this, we deduct 2.

180/n -2. let's rearrange everything: (180-2n)/n. now looks better.

ok, so if this is the new price for 12 doughnuts, then the price for 1 doughnut would be (180-2n)/n * 1/12 or (90-n)/6n.

now we are getting closer:
he received n+3 at the price (90-n)/6n. and the total is 15.
we can make a new equation:
(n+3)(90-n)/6n = 15 | multiply both sides by 6n
(n+3)(90-n) = 90n
90n-n^2+270-3n = 90n | subtract 90n from both sides, then multiply by -1.
n^2+3n-270=0.
factor the equation: (n-15)(n+18)=0.
we have n=15 and n=-18. Since we are talking about real things, n can't be -18. thus, n must be 15.
now, n is the initial number of doughnuts he bought, but then he received 3 more. so the total number should be 15+3 = 18.

Let total doughnuts at first be x and price per doughnut be y
Then xy=15
but the clerk added 3 more doughnuts and bcoz of that the cost reduced by $2 per dozen or $1/6 per piece
So, new eqn will be
(x+3)*(y-1/6)=15
on solving the two equations we get y=1 and y=-5/6
since it is price it cant be negative so y=1 and thus x=15
but questions asks how many doughnuts he has at the end
so at end no. of doughnuts=x+3=15+3=18

Given:
1. Jim went to the bakery to buy donuts for his office mates. He chose a quantity of similar donuts, for which he was charged a total of $15.
2. As the donuts were being boxed, Jim noticed that a few of them were slightly ragged-looking so he complained to the clerk. The clerk immediately apologized and then gave Jim 3 extra donuts for free to make up for the damaged goods.
3. As Jim left the shop, he realized that due to the addition of the 3 free donuts, the effective price of the donuts was reduced by $2 per dozen.

Asked: How many donuts did Jim receive in the end?

Let the price per donut be $x

Quantity of donuts purchased = 15/x
After receiving 3 extra donuts, Jim had = 15/x + 3 donuts

Effective price / donut reduced by = $2/dozen = $2/12 = $1/6

\(x - \frac{1}{6}= 15 / (\frac{15}{x} +3)\)
\((x - \frac{1}{6})(\frac{15}{x} + 3) = 15\)
\(15 - \frac{1}{6}* \frac{15}{x} +3x -\frac{1}{2} = 15\)
-2.5/x + 3x - .5 =0
6x - 5/x -1 =0
6x^2 -x -5=0
6x^2 - 6x + 5x -5 =0
(6x+1)(x-1)=0
x = 1

Quantity of donuts received in the end = 15/x +3 = 15 +3 = 18

IMO A

Hello Karishma,

Can you explain the calculation. I was not able to understand how did you arrive at

The average price of the doughnuts is (15/n - 1/6) per doughnut.

I get that 15/n is the price of each donut. Can can you let me know why did you substract 1/6 to get the net average ?

Suppose he bought 'n' doughnuts at $p per doughnut.

So, pn = 15

Actually Jim bought n + 3 doughnuts.
Effective price per doughnut = p - 2/12 = p - 1/6
(n + 3)(p - 1/6) = 15

np - n/6 + 3p - 1/2 = 15
15 - n/6 + 3p - 1/2 = 15
-n/6 + 3p = 1/2
-n + 18p = 3
n = 18p - 3

So, p*(18p - 3) = 15
18p^2 -3p - 15 = 0
6p^2 - p - 5 = 0
6p^2 - 6p + 5p - 5 = 0
6p(p - 1) + 5(p - 1) = 0
So, p = 1

So, n = 15

So, Jim received 18 (n+3) doughnuts at the end.