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guddo
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A shoe store purchased 50 pairs of running shoes and sold some of them Updated on: 28 Jan 2024, 07:37
Originally posted by guddo on 24 Jan 2024, 02:35.
Last edited by gmatophobia on 28 Jan 2024, 07:37, edited 1 time in total.
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70% (02:20) correct 30% (02:24) wrong based on 897 sessions

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A shoe store purchased 50 pairs of running shoes and sold some of them at $P per pair. The store sold each of the remaining pairs at $S, where S < P. If the store's average profit per pair of these running shoes was $6, what was the value of S?

(1) The store sold 28 pairs of the running shoes at $P per pair.
(2) P = 108


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A shoe store purchased 50 pairs of running shoes and sold some of them 26 Jan 2024, 23:17
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guddo
A shoe store purchased 50 pairs of running shoes and sold some of them at $P per pair. The store sold each of the remaining pairs at $5, where S < P. If the store's average profit per pair of these running shoes was $6, what was the value of S?

(1) The store sold 28 pairs of the running shoes at $P per pair.
(2) P = 108


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Correct question:

A shoe store purchased 50 pairs of running shoes and sold some of them at $P per pair. The store sold each of the remaining pairs at $S, where S < P. If the store's average profit per pair of these running shoes was $6, what was the value of S?

Avg profit : Total profit/\(50\) = \(6\) or Total profit \(= 300\)

Total profit : Sales Revenue - Total cost

Sales revenue \(= x*p + (50-x)s\)

Total cost : \(50*c\)

(1) The store sold \(28\) pairs of the running shoes at $P per pair.

\(28p +22s -50c = 300\)

One equation three unknowns.

INSUFF.

(2) P = 108

\(108x +(50-x)s -50c =300\)

One equation three unknowns.

INSUFF.

1+2

\(108*28 + 22s-50c=300\)

We don't know \(c\) hence we cannot get the value of \(s\).

INSUFF.

Ans E

Hope it helped.
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A shoe store purchased 50 pairs of running shoes and sold some of them 28 Jan 2024, 08:14
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guddo
A shoe store purchased 50 pairs of running shoes and sold some of them at $P per pair. The store sold each of the remaining pairs at $S, where S < P. If the store's average profit per pair of these running shoes was $6, what was the value of S?

(1) The store sold 28 pairs of the running shoes at $P per pair.
(2) P = 108

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One way is to solve algebraically, another approach is to use a number line. Solving using the number line approach.

Let's assume that the cost price of each pair of shoes is $C. As the store made a profit, we can infer that P > C. On a number line, we can represent P and C as shown below.

Attachment:
Screenshot 2024-01-28 201348.png
Screenshot 2024-01-28 201348.png [ 3.57 KiB | Viewed 16650 times ]

The question also tells us that the value of S is less than P. Hence, we know that S lies to the left of P. We do not have enough information to conclude whether S lies to the left of C, or to the right of C, or at C.

Attachment:
Screenshot 2024-01-28 201650.png
Screenshot 2024-01-28 201650.png [ 14.09 KiB | Viewed 16391 times ]

...the store's average profit per pair of these running shoes was $6 ...

Inference : (C + 6) lies between C and S.

Attachment:
Screenshot 2024-01-28 202816.png
Screenshot 2024-01-28 202816.png [ 17.13 KiB | Viewed 15273 times ]

Statement 1

(1) The store sold 28 pairs of the running shoes at $P per pair.

Inference: 28 pairs are sold at $P, and 14 pairs are sold at $S. We don't know the value of P and C. Hence, we don't know how far P is from C or C+6. Hence, we cannot determine the value of S. The statement alone is not sufficient.

Eliminate A and D.

Statement 2

(2) P = 108

While we know the value of P, the statement provides us with no information on the cost price or the number of pairs sold at this price. Hence, the statement alone is not sufficient.

Eliminate B.

Combined

The statements combined are also not sufficient as we can't still determine the distance between C + 6 and P as the value of C is not known to us.

For different values of C, we will arrive at different values of S. Hence, the statements combined are also not sufficient.

Option E
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A shoe store purchased 50 pairs of running shoes and sold some of them 15 Apr 2025, 04:34
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guddo
A shoe store purchased 50 pairs of running shoes and sold some of them at $P per pair. The store sold each of the remaining pairs at $S, where S < P. If the store's average profit per pair of these running shoes was $6, what was the value of S?

(1) The store sold 28 pairs of the running shoes at $P per pair.
(2) P = 108


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Using the data of both statements, we find

Number of pairs bought = 50
28 were sold at $108 each
22 were sold at $S each
Overall profit per pair = $6

To use the information of overall profit per pair, we need to know the profit per pair for the 28 shoes sold at $108. Since we don't have the cost price of each pair, we do not know the profit per pair for the 28 shoes.
Hence we cannot find the value of S and both statements together are not sufficient.

Answer (E)

Let's see how changes in cost price will change the value of S.

Say Cost Price = $91 per shoe
Total cost price = 91 * 50 = 4550
Total selling price = 4550 + 6*50 = 4850
Selling price of 28 shoes = 108*28 = 3024
Selling price of 22 shoes = (4850 - 3024)/22 = $83 = S

Say Cost Price = $80.
Total cost price = 80 * 50 = 4000
Total selling price = 4000 + 6*50 = 4300
Selling price of 28 shoes = 108*28 = 3024
Selling price of 22 shoes = (4300 - 3024)/22 = $58 = S
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A shoe store purchased 50 pairs of running shoes and sold some of them 08 Nov 2025, 02:52
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Thanks, I too solved it like this but at the end I thought sometimes even if there are 2 variables, 1 equation is enough to get the answer because there's only 1 solution. Can someone help with how we can eliminate that possibility without solving for the various possibilities? I took way too long because I was trying to see if we can get a unique solution from the last eqn
stne


Correct question:

A shoe store purchased 50 pairs of running shoes and sold some of them at $P per pair. The store sold each of the remaining pairs at $S, where S < P. If the store's average profit per pair of these running shoes was $6, what was the value of S?

Avg profit : Total profit/\(50\) = \(6\) or Total profit \(= 300\)

Total profit : Sales Revenue - Total cost

Sales revenue \(= x*p + (50-x)s\)

Total cost : \(50*c\)

(1) The store sold \(28\) pairs of the running shoes at $P per pair.

\(28p +22s -50c = 300\)

One equation three unknowns.

INSUFF.

(2) P = 108

\(108x +(50-x)s -50c =300\)

One equation three unknowns.

INSUFF.

1+2

\(108*28 + 22s-50c=300\)

We don't know \(c\) hence we cannot get the value of \(s\).

INSUFF.

Ans E

Hope it helped.
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A shoe store purchased 50 pairs of running shoes and sold some of them 14 Jul 2026, 01:57
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you cannot use alligation because 6 is profit and "p" and "s" are selling price we need p-c and p-s to do alligation

The key mistake is that alligation applies only when the quantity being averaged is the same. In this question, the average given is average profit per pair, not average selling price per pair. Therefore, the two values that must be averaged are the profits on each type of sale, namely (P - C) and (S - C), where C is the cost price per pair. You cannot directly apply alligation to P and S.

---

From statement (1), the ratio of pairs sold at P and S is 28:22. From statement (2), we know that one selling price is 108, so the corresponding profit is (108 - C), not 108. Applying alligation correctly gives ((108 - C) - 6) / (6 - (S - C)) = 22 / 28. However, both S and C are still unknown, so there is not enough information to determine S uniquely. Hence, statement (2) is not sufficient even when combined with the ratio from statement (1).

---

The takeaway is to always identify what quantity is being averaged before using alligation. If the average is profit, then alligate profits. If the average is selling price, then alligate selling prices. Here, since the average is profit, the unknown cost price C is part of each profit expression. Because C remains unknown, S cannot be determined, which is why choice C is not the correct answer.
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