In a store, 40 items are arranged on a rack in 5 rows and 8 columns. The average (arithmetic mean) of the price of items in each row (m) is X_m . The average of the price of items in each column (n) is Y_n . What is the average price of all items on the rack? (1) X_1 + X_2 + ... + X_5 = $850 (2) Y_1 + Y_2 + ... + Y_8 = $1,360

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GMAT Club Olympics 2025 (Day 1): In a store, 40 items are arranged on

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(1) $X_1 + X_2 + ... + X_5 = $850$

let sum of all items on a row equal to $S_m$ , where $m $will be between 1 and 5.
Then
$X_1 = S_1 / 8 >> S_1 = 8X_1 $

$X_2 = S_2 / 8 >> S_2 = 8X_2 $
...
$X_5 = S_5/8 >> S_5 = 8X_5 $

Sum of all elements in the table = $S_1 + S_2 +S_3 + S_4 + S_5 = 8(X_1 + X_2 + X_3 + X_4 + X_5) = 8 * $850 (given from statement 1)$

To find the overall average we can divide the above ( 8 * 850) by total number of elements (40) to get the average
So Statement 1 is sufficient.

(2) $Y_1 + Y_2 + ... + Y_8 = $1,360$

we can follow a similar method as above
let sum of all items in a column equal to $S_m$ , where $m$ will be between 1 and 8.
Then
$Y_1 = S_1 / 5 >> S_1 = 5Y_1 $

$Y_2 = S_2 / 5 >> S_2 = 5Y_2 $
...
$Y_8 = S_8 /5 >> S_8 = 5Y_8 $

Sum of all elements in the table = $S_1 + S_2 +... + S_8 = 5*(Y_1 + Y_2 + .... + Y_8) = 5 * 1360 (given from statement 2)$

To find the overall average we can divide the above ( 8 * 1360 ) by total number of elements (40) to get the average
So Statement 2 is sufficient.

As both 1 and 2 are sufficient by themselves, answer is D

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A) 850 is the sum of all the rows. So dividing it by 5 will give us the average price of a row. and dividing that by 5 will give us average price of all items.
b) Same logic.

Bunuel

In a store, 40 items are arranged on a rack in 5 rows and 8 columns. The average (arithmetic mean) of the price of items in each row (m) is $X_m$ [1 ≤ m ≤5]. The average of the price of items in each column (n) is $Y_n$ [1 ≤ n ≤ 8]. What is the average price of all items on the rack?

(1) $X_1 + X_2 + ... + X_5 = $850$
(2) $Y_1 + Y_2 + ... + Y_8 = $1,360$

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Average = total price of all items/40
(1) X1 + X2 + ... + X5 = 850
This means total price = 8 x 850 = 6800 (as each row will have 8 items on average). we can then use 6800/40 to calc average. Sufficient
(2) Y1 + Y2 +... + Y8 = 1360
This means total price = 5 x 1360 = 6800 (as each column will have 5 items on average). similar as statement 1 - 6800/40 to calculate average. Sufficient
Choose D

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The question mentions a total of 40 items; Tn = 40
Average price of items in each row is same = Xm
Average of the price of items in each column is same = Yn

Considering Statement 1:
This gives sum of the averages of price of items in each row = $850

No. of columns = 8,
850 * 8 = $6800; Sufficient

Considering Statement 2:
This gives sum of the averages of price of items in each column = $1360

No. of rows = 5,
1360 * 5 = $6800; Sufficient

Each statement alone is sufficient.

∴ the Correct option is Option D

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Bunuel

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In this question we have to find the the average price of the 40 items.
Let us look at each statement

Statement 1:
$X_1 + X_2 + ... + X_5 = $850$

If average of the columns are given then we can find the total price => $850 * 8

From the total we can find the mean of the items => ($850 * 8) / 40

So Statement 1 is Sufficient

Statement 2:
$Y_1 + Y_2 + ... + Y_8 = $1,360$

Similar to statement 1 if average of the columns are given then we can find the total price => $1,360 * 5

From the total we can find the mean of the items => ($1,360 * 5) / 40

So Statement 2 is Sufficient

A simpler method:

In this statement they have specified that the average of a column is $1,360, so to find one value we just have to divide it by the number of columns
=> $1,360 / 8

This logic also works for the first statement:-
-> $850 / 5

Each Statement alone is sufficient

Answer:
(D)

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Now the main part of this would be
The average price of all items = Total sum of all items/40

(1) X1+X2+...+X5=$850
Each Xm is the average price in row m, which has 8 items
Total sum = 8 * (X1+X2+...+X5) = 8*850 = 6800

Average = 6800/40 = 170

(2) Y1+Y2+...+Y8=$1360

Each Xm is the average price in column n, which has 5 items
Total sum = 5 * (Y1+Y2+...+Y8) = 5*1360= 6800

Average = 6800/40 = 170

So, from both independent statements answers are same.

D. Each statement alone is sufficient

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Bunuel

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(1) only tells us the average of each row, however, if you visualize the rows that accounts for the columns. (2) same logic applies. An extra test could be dividing the numbers 850 by 5 and 1360 by 8. Both give the same answer because they cover all 40 items. So D. both self sufficient

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Weighted average method is to be applied for calculating final average.
So in each row 8 items are there, so Average(X)= \frac{Total of 40 items}{40}.
Statement 1: since X1 is average of first row of 8, X2 is average of second row of 8, similarly X3, X4 and X5.
To obtain row total, First Row total = X1x8, Second row total = X2x8, similarly for the rest. Final total is Sum of all Totals above obtained. which is nothing but 8x( X1 + X2 +X3 + X4 +X5), which divided by 40, i.e (X1 + X2 + X3 + X4 + X5)/5. So sufficient.
Statement 2: Same as above, but instead of Row we will go in columns. Hence it is also Sufficient.

So Each statement alone is sufficient. Hence (D).

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Bunuel

A row has 8 items in it, and a column has 5 items in it.

From Statement 1:

R1 items (R11 + R12 + R13 + R14 + R15 + R16 + R17 + R18) = X1 * 8
R2 items (R21 + R22 + R23 + R24 + R25 + R26 + R27 + R28) = X2 * 8
R3 items (R31 + R32 + R33 + R34 + R35 + R36 + R37 + R38) = X3 * 8
R4 items (R41 + R42 + R43 + R44 + R45 + R46 + R47 + R48) = X4 * 8
R5 items (R51 + R52 + R53 + R54 + R55 + R56 + R57 + R58) = X5 * 8

Total = (X1+X2+X3+X4+X5) * 8;

Average Price of all items on the rack = Total / 40 = 850 * 8 / 40 = 170

From Statement 2:
C1 items (C11 + C12 + C13 + C14 + C15) = Y1 * 5
C2 items (C21 + C22 + C23 + C24 + C25) = Y2 * 5
C3 items (C31 + C32 + C33 + C34 + C35) = Y3 * 5
C4 items (C41 + C42 + C43 + C44 + C45) = Y4 * 5
C5 items (C51 + C52 + C53 + C54 + C55) = Y5 * 5
C6 items (C61 + C62 + C63 + C64 + C65) = Y6 * 5
C7 items (C71 + C72 + C73 + C74 + C75) = Y7 * 5
C8 items (C81 + C82 + C83 + C84 + C85) = Y8 * 5

Total = (C1+C2+C3+C4+C5+C6+C7+C8) * 5;

Average Price of all items on the rack = Total / 40 = 1360 * 5 / 40 = 170

Hence, answer is D.

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Bunuel

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Average price of all items on the rack is sum of prize of each item divided by total items that is 40.

Essentially we need some of all items to answer this question.

Statement 1 gives us Average of all rows from which we can get sum of items in each row and finally sum of all items.
Statement 2 gives us Average of all columns from which we can get sum of items in each columns and finally sum of all items,

So Correct option is D we can answer the question with Statement 1 alone or Statement 2 alone

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01 Jul 2025, 10:25

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My approach is slightly varying that the traditional one:
such as
8 columns * 5 rows = 40 equal vacant positions
total items = 40

Assuming that each item is distributed proportionally for each position and question didn't mention that the arrangement is irregular

Average prices of the items in rack = (Total sum of prices)/Total number

A) (X1+X2+.......+X5) = $850
Total number of rows = 5
Average prices of all items = 850/5 = $170
Statement 1 alone is sufficient

B) (Y1+......+Y8) = $1360
Total number of columns = 8
Average prices of all items = 1360/8 = $170
Statement 2 alone is sufficient

Therefore, correct option D) Both statement alone are sufficient

However, if the items were not uniformly distributed then : {(Total number of Column or Row) * (Total sum of Row or column)}/Total number of items
A = 8*(850)/40 = $170
B = 5 * (1360)/40 = $170

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02 Jul 2025, 02:13

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There are 5 rows and 8 columns, for a total of 5 * 8 = 40 items.
Xm is the average price of items in row m (where m ranges from 1 to 5).
Yn is the average price of items in column n (where n ranges from 1 to 8).
to find the average price of all 40 items.

Statement (1): X1 + X2 + ... + X5 = $850.
Statement (2): Y1 + Y2 + ... + Y8 = $1,360.

Statement (1):
- The sum of the average prices of the 5 rows is $850.
- The sum of the prices of all items in a row is 8 times the average price of that row (since there are 8 items in each row).
  8 * (X1 + X2 + ... + X5) = 8 * $850.(8 * $850) / 40.This allows the calculation of the average price of all items can be calculated. statement (1) is sufficient.
Statement (2):
- The sum of the average prices of the 8 columns is $1,360.
- The sum of the prices of all items in a column is 5 times the average price of that column (since there are 5 items in each column).
  5 * $1,360. (5 * $1,360) / 40.
- This allows the calculation of the average price of all items. statement (2) is sufficient.

the answer is D.

Bunuel

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twinkle2311

Given : 40 items in a 5 X 8 rack.
We need the average price of all items.

Statement 1
X1 + X2 + ... + X5 = 850
Each Xm is the average price of 8 items in row m, so the total price in row m is 8 × Xm.
So, total price of all 5 rows =
8 × (X1 + X2 + ... + X5) = 8 × 850 = 6800
There are 40 items in total, so average = 6800 ÷ 40 = 170
Statement 1 is sufficient.

Statement 2
Y1 + Y2 + ... + Y8 = 1360
Each Yn is the average price of 5 items in column n, so total price in column n = 5 × Yn
Total price = 5 × (Y1 + Y2 + ... + Y8) = 5 × 1360 = 6800
Average = 6800 ÷ 40 = 170
Statement 2 is sufficient.

Answer: (D) Each statement alone is sufficient.

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bb Bunuel
Didn't get Kudos for this. Didn't use any AI. Please help on this.

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Some of your recent responses were flagged as AI-generated. As per the competition rules, solutions created using AI tools like ChatGPT are not eligible for kudos. Please make sure all submissions are your own work.

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Bunuel, Sorry to nag you on this but I didn't use AI on this. Sharing a ss of AI detector that I used just now.

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Bunuel, Sorry to nag you on this but I didn't use AI on this. Sharing a ss of AI detector that I used just now.

Giving you the benefit of the doubt. If you're saying it wasn’t AI, then I’ll award kudos to you.

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Bunuel, Really appreciate it!

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Bunuel, Sorry to nag you on this but I didn't use AI on this. Sharing a ss of AI detector that I used just now.

Giving you the benefit of the doubt. If you're saying it wasn’t AI, then I’ll award kudos to you.

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