Bob expected to spend a total of $9.00 to buy a given amount of pasta : Problem Solving (PS)

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Re: Bob expected to spend a total of $9.00 to buy a given amount of pasta [#permalink]

16 Jul 2023, 00:11

avigutman - Can you please share a reasoning based approach to this question.

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Re: Bob expected to spend a total of $9.00 to buy a given amount of pasta [#permalink]

18 Nov 2023, 07:20

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gmatophobia wrote:

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 $

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 13969 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 13272 times ]

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Re: Bob expected to spend a total of $9.00 to buy a given amount of pasta [#permalink]

14 Jan 2024, 05:18

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gmatophobia wrote:

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 $

The actual price per pound and the expected price per pound yield two weights 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.

Show Spoiler

E

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Re: Bob expected to spend a total of $9.00 to buy a given amount of pasta [#permalink]

14 Jan 2024, 07:03

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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

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Re: Bob expected to spend a total of $9.00 to buy a given amount of pasta [#permalink]

21 Jan 2024, 13:18

avigutman wrote:

Obscurus wrote:

avigutman - Can you please share a reasoning based approach to this question.

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).

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

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Re: Bob expected to spend a total of $9.00 to buy a given amount of pasta [#permalink]

21 Jan 2024, 13:51

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ruis wrote:

avigutman wrote:

Obscurus wrote:

avigutman - Can you please share a reasoning based approach to this question.

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).

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

"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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Re: Bob expected to spend a total of $9.00 to buy a given amount of pasta [#permalink]

15 Feb 2024, 04:02

avigutman wrote:

Obscurus wrote:

avigutman - Can you please share a reasoning based approach to this question.

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+[i]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).

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.

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Re: Bob expected to spend a total of $9.00 to buy a given amount of pasta [#permalink]

19 Mar 2024, 18:36

Answer is wrong. should be 5x^2 +x - 18

Kinshook wrote:

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

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Bob expected to spend a total of $9.00 to buy a given amount of pasta [#permalink]

31 Mar 2024, 13:57

Expert Reply

Honeybadger269 wrote:

avigutman wrote:

Obscurus wrote:

avigutman - Can you please share a reasoning based approach to this question.

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+[i]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).

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.

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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Bob expected to spend a total of $9.00 to buy a given amount of pasta [#permalink]

10 Apr 2024, 02:38

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?

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Re: Bob expected to spend a total of $9.00 to buy a given amount of pasta [#permalink]

10 Apr 2024, 03:15

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stephaniepongracz wrote:

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

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?

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.

gmatclubot

Re: Bob expected to spend a total of $9.00 to buy a given amount of pasta [#permalink]

10 Apr 2024, 03:15

GMAT Problem Solving (PS) Questions

Bob expected to spend a total of $9.00 to buy a given amount of pasta