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gmatophobia
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Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
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Question Stats:
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32%
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Bob expected to spend a total of $9.00 to buy a given amount of pasta salad at a fixed price per pound. However, the price of the salad was $0.20 more per pound than Bob had expected. Consequently, he spent $9.00 and bought \(\frac{1}{2}\) pound less of the salad. How much did Bob spend per pound for the salad?
A. $1.80
B. $1.85
C. $1.90
D. $1.95
E. $2.00
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A. $1.80
B. $1.85
C. $1.90
D. $1.95
E. $2.00
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Power of Tiny Gains
1.01^(365) = 37.78
0.99^(365) = 0.03
Power of Tiny Gains
1.01^(365) = 37.78
0.99^(365) = 0.03
JeffTargetTestPrep
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Posts: 2,973
gmatophobia
He expected to spend $9 to buy q pounds of salad at p dollars per pound:
Total cost = p × q = $9
However, things turned out differently for him:
Total cost = (p + 0.2) × (q - 0.5) = $9
Let’s solve the system of equations for p.
(p + 0.2) × (q - 0.5) = p × q
pq + 0.2q – 0.5p – 0.1 = pq
0.2q – 0.5p – 0.1 = 0
2q – 5p – 1 = 0
q = (5p + 1)/2
Now we can substitute the above expression for q in the first equation.
p × (5p + 1)/2 = 9
5p^2 + p – 18 = 0
(5p – 9)(p + 2) = 0
If p + 2 = 0, then p = -2, which is an invalid case.
If 5p – 9 = 0, then p = 9/5 = 1.8.
Therefore, he paid 1.8 + 0.2 = $2 per pound for the salad.
Answer: E
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gmatophobia
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gmatophobia
We can use the options and work backward to solve this question. Let's calculate the expected price from the actual price as shown below -
Attachment:
Screenshot 2023-11-18 194320.png [ 54.72 KiB | Viewed 83397 times ]
I would start with Option E, as the numbers seem more friendly.
Expected Quantity (in pounds) = \(\frac{\text{9}}{\text{Expected Price}}\)
Actual Quantity (in pounds) = Expected Quantity - 0.5
The product of Actual Quantity and Actual Price should be equal to $9. As we see Option E matches. We need not test out any other options.
Attachment:
Screenshot 2023-11-18 194412.png [ 66.86 KiB | Viewed 77952 times ]
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Power of Tiny Gains
1.01^(365) = 37.78
0.99^(365) = 0.03
Power of Tiny Gains
1.01^(365) = 37.78
0.99^(365) = 0.03
General Discussion
Posts: 341
avigutman - Can you please share a reasoning based approach to this question.
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Expert reply
Obscurus
Obscurus this is a rare question type for which you either have to solve a quadratic equation (which I don't recommend) or use the answer choices (the better approach in my opinion). Firstly, how to recognize this question type?
When you have the product of two positive numbers (a and b), and you're told that the same product can be reached by adding to one of the numbers and subtracting from the other [(a+x)(b-y), where x and y are positive numbers]. Importantly, we don't know the ratio a:(a+x) nor b:(b-y). If we do know one of those ratios, we can solve using a beautiful reasoning based solution.
Some other official questions in this rare category:
https://gmatclub.com/forum/a-store-curr ... 44780.html
https://gmatclub.com/forum/a-boat-trave ... ml#p778643
https://gmatclub.com/forum/at-his-regul ... 36642.html
The non-integer answer choices make this particular problem more challenging (similar to the boat problem) but we know that the GMAT isn't all that interested in testing our ability to do lots of complicated math in a 2-minute-per-question setting, so it's likely that the answer choice that's easiest to check is the correct answer choice (as is the case in both the boat question and this question).
How I approached this problem: the answer choices represent Bob's cost/pound, and I know he expected to spend 20 cents less than that. Which answer choice is easiest to subtract 20 cents from? Well, (E) looks good to me because it's easy to divide $9 by $2, and also "20 cents less" is exactly 10% below. In other words, he expected to spend 9/10 as much as he did, so he would've been able to buy 10/9 as many pounds if the price were as he expected it to be (this is me using ratios for a beautiful reasoning-based approach).
At $2/pound he bought 4.5 pounds, and 10/9 of 4.5 pounds is exactly 5 pounds - that's the 1/2 pound difference that we were expecting. Done!
Can you imagine having to divide $9 by any of the other answer choices?? It's unlikely that we'd have to do that on the GMAT (unless that was the entire question).
Posts: 1,348
Updated on: 31 May 2024, 17:08
Originally posted by GMATGuruNY on 14 Jan 2024, 04:18.
Last edited by GMATGuruNY on 31 May 2024, 17:08, edited 1 time in total.
Last edited by GMATGuruNY on 31 May 2024, 17:08, edited 1 time in total.
gmatophobia
The actual price per pound and the expected price per pound yield two weights that are 1/2 pound apart.
Implication:
It is extremely likely that -- when divided into the paid amount of 900 cents -- one price-per-pound will yield a weight that is an INTEGER value, while the other yields a weight that is 0.5 MORE than an integer value.
We can PLUG IN THE ANSWERS, which represent the actual price per pound.
When divided into 900 cents, options B, C and D -- 185 cents, 190 cents, and 195 cents -- will yield neither an integer value nor 0.5 more than an integer value.
Option A implies an expected price of 160 cents per pound, 20 cents less than the actual price per pound (180 cents).
When divided into 900 cents, 160 will yield neither an integer value nor 0.5 more than an integer value.
The correct answer is almost certain to be E.
E: Actual price = 200 cents per pound, expected price = 180 cents per pound
Weight purchased at the actual price \(= \frac{900-cents}{200-cents-per-pound} = 4.5\) pounds
Weight that would have been purchased at the expeced price \(= \frac{900-cents}{180-cents-per-pound} = 5\) pounds
Success!
The actual weight is 0.5 pound less than the expected weight.
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gmatophobia
Given: Bob expected to spend a total of \($9.00\) to buy a given amount of pasta salad at a fixed price per pound. However, the price of the salad was \($0.20\) more per pound than Bob had expected. Consequently, he spent \($9.00\) and bought \(\frac{1}{2}\) pound less of the salad.
Asked: How much did Bob spend per pound for the salad?
Let us assume that Bob spent $x per pound for the salad
Bob expected to purchase = $9/(x-.2) pounds
$9 = x ($9/(x-.2) - .5) = 9x/(x-.2) - .5x
9(x-.2) = 9x - .5x(x-.2)
-1.8 = -.5x^2 + .1x
5x^2 - x - 18 = 0
5x^2 - 10x + 9x - 18 = 0
5x(x-2) + 9(x-2) = 0
(5x+9)(x-2) = 0
x = 2 since x = -9/5 is not feasible
IMO E
Given: Bob expected to spend a total of \($9.00\) to buy a given amount of pasta salad at a fixed price per pound. However, the price of the salad was \($0.20\) more per pound than Bob had expected. Consequently, he spent \($9.00\) and bought \(\frac{1}{2}\) pound less of the salad.
Asked: How much did Bob spend per pound for the salad?
Let us assume that Bob spent $x per pound for the salad
Bob expected to purchase = $9/(x-.2) pounds
$9 = x ($9/(x-.2) - .5) = 9x/(x-.2) - .5x
9(x-.2) = 9x - .5x(x-.2)
-1.8 = -.5x^2 + .1x
5x^2 - x - 18 = 0
5x^2 - 10x + 9x - 18 = 0
5x(x-2) + 9(x-2) = 0
(5x+9)(x-2) = 0
x = 2 since x = -9/5 is not feasible
IMO E
avigutman
Thank you very much for this super explanation! I followed the same rationale however once I got 4,5 pounds and 5 pounds respectively, I did not recognize that the 0,5 difference was the 1/2 pounds less... I am still struggling with it as I am looking at it as "4,5 is not the half of 5". Apologies as I must be asking sth pretty obvious but I am unable to see the error in my rationale.
Thanks again
ruis
"1/2 pound less" does not mean half as much. So, it's not dividing by 2; it's subtracting 1/2.
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avigutman
Highly elegant and intricate explanation.
Are there people who a] are not naturally good at 'math' and b] have learnt to think and execute this stuff within the time constraints and under the exam pressure by way of GMAT preparation? If yes, please reply to this and tell me how you developed this skill. I have been preparing for the test for a long time and still can never do this in any mock or actual attempt.
Posts: 145
Answer is wrong. should be 5x^2 +x - 18
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Expert reply
Honeybadger269
Yes, this is a great explanation by avigutman!
Honeybadger269 These quadratics do come up sometimes, often with this kind of wording, where a price goes up or down and the quantity goes in the other direction. It's usually easiest to backsolve from the answer choices, since you'd have to guess and check anyways to do the factoring.
Easier said than done, but thorough preparation helps a lot, so we don't have to recreate the wheel on the actual test: once we've done several of these problems, it helps us quickly recognize this wording and make it a habit to use the answer choices to our advantage. Remember to first test the answer choice that is easiest to test; this boat one is a great example:
https://gmatclub.com/forum/a-boat-trave ... ml#p778643
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Posts: 1
Please help me:
Would it also be possible to say:
p=price
n=weight
p*(n-0.5) = (p-0.2)*n
pn-0.5p = pn-0.2n
0.5p = 0.2n
5/2p = n
Then try out "p=2" --> 5/2*2=5 and 5/2*1.8=4.5 which is correct
I just dont understand why exactly that is right, can anyone explain please?
Would it also be possible to say:
p=price
n=weight
p*(n-0.5) = (p-0.2)*n
pn-0.5p = pn-0.2n
0.5p = 0.2n
5/2p = n
Then try out "p=2" --> 5/2*2=5 and 5/2*1.8=4.5 which is correct
I just dont understand why exactly that is right, can anyone explain please?
stephaniepongracz
You should clearly state what the variables represent.
If p is the old price, and n is the weight of the salad Bob could have bought at that price, then we'd get: pn = (p + 0.2)(n - 0.5) = 9
If p is the new price, and n is the weight of the salad Bob bought at that price, then we'd get: pn = (p - 0.2)(n + 0.5) = 9.
You could have used only one variable p to denote the new price per pound to get: 9/p = 9/(p - 0.2) - 0.5, which gives n = 2.
Hope this helps.
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gmatophobia
One of the question types in which I would use variables and equations to get to the quadratic and then plug in values. Just using reasoning from options straight away might lead to a lot of time wasted with high possibility of calculation errors.
Whatever you are asked to find, you should prefer to use that as a variable. So price he paid = p and weight he bought = w
pw = 9
Also, as per his initial expectation,
\((p - \frac{1}{5})(w + \frac{1}{2}) = 9\)
\(pw + \frac{p}{2} - \frac{w}{5} - \frac{1}{10} = 9\)
\(2w - 5p = 1\) (Eliminate w since we need the value of p)
\(\frac{18}{p} - 5p = 1\)
Now plug in. I know that 18/p will likely be an integer so I will try 2 first. I get -1. Incorrect.
Next try 1.8 because it will divide 18 evenly. I get 1. Valid.
Answer (A)
Avoiding calculations with variables in denominator is discussed here:
https://anaprep.com/avoiding-calculatio ... nominator/
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You can also solve it in an easier way involving single variable.
Assuming x is pound
9 = (9/x + 0.2)(x - 0.5)
simplify to
45 = 2x^2 - x
Thus x - 5
9/5 + 0.2 = 2
Assuming x is pound
9 = (9/x + 0.2)(x - 0.5)
simplify to
45 = 2x^2 - x
Thus x - 5
9/5 + 0.2 = 2
gmatophobia
Hey folks, this right here is one of the most mentally challenging and time consuming and error prone problems, but dont worry, to every problem i can find a sweet simple solution which would expect one to use creative thinking, cause GMAT rewards creative thinking, not "structured approach", especially in the given time frame.
So well Here we go,
Given:
The total cost is fixed at $9.00.
Bob bought 1⁄2 pound less because the price per pound increased by $0.20.
Let p be price per pound
So he expected to get 9/p lb of pasta
But the actual amount he got was 9/p+0.2
He bought 1⁄2 lb less, so the difference between expected qty of pasta and actual quantity of pasta is 1⁄2
Which is
9/p – 9/p+0.2 = 1⁄2
Now let's check for what values of p would this equation hold, use options in AEBDC
p = 1.8
9/1.8 – 9/2 = 1⁄2
90/18-9/2 = 5-4.5 = 0.5
Hence p = 1.8
Now the question asks how much did he spend per pound of salad, which is 1.8+0.2 = $2
I dont think there's any solution out there as simple, effective and error free as this, this doesnt test your skills, rather comes to you if you visualize and the key part to visualize is the question, in this question its important we visualize the difference between expected qty of pasta and actual quantity of pasta, which would be our key towards unlocking the answer.
How do we visualize? Write down the given information and dumb it down.
violá, c'est ça!
let's see,
What is Given:
now don't solve the equation, to save time, just put options (a,b,c,d,e) quickly and check.
you will see, option a. x=1.8 suits the best.
but but but , question is asking , how much bob actually paid (x+.20) not expected the price (x)
so answer will be (1.8+.2)= 2 . option e.
What is Given:
- Bob expected the price per pound to be x.
- Bob actually paid x+0.20x per pound.
- Bob spent $9 total in both cases:
- Expected pounds: 9/x
- Actual pounds: 9\{x + 0.20}
- Bob bought 0.5 pounds less than expected due to the price increase:
9/x - 9\{x + 0.20} = 0.5
now don't solve the equation, to save time, just put options (a,b,c,d,e) quickly and check.
you will see, option a. x=1.8 suits the best.
but but but , question is asking , how much bob actually paid (x+.20) not expected the price (x)
so answer will be (1.8+.2)= 2 . option e.
gmatophobia
Answer is E , not A. RIGHT ?:think:
KarishmaB
Used the quadratic formula which took a long time -- really need to master when to go down this path, i made the same mistake in exam and took ages in one question
-- same as everyone got to quadratic
5p2+p−18=0
We’ll use the quadratic formula get original price:
Calculate the discriminant:
b2−4ac=12−4×5×(−18)=1+360=361
Compute the square root of the discriminant:
361=19
Now, find the roots:
p=−(1)+19/2×5=18/10=1.8
p=-(1)- 19/2×5=-20/10=-2. -- cant be negative
so original price 1.8 +0.2 2$
-- same as everyone got to quadratic
5p2+p−18=0
We’ll use the quadratic formula get original price:
Calculate the discriminant:
b2−4ac=12−4×5×(−18)=1+360=361
Compute the square root of the discriminant:
361=19
Now, find the roots:
p=−(1)+19/2×5=18/10=1.8
p=-(1)- 19/2×5=-20/10=-2. -- cant be negative
so original price 1.8 +0.2 2$
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