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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MaxFabianKirchner

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To find the average price of all 40 items, we need the total price.

Let S be the total price.
The items are in 5 rows and 8 columns.

From the rows:
The sum of prices in each row is 8 times its average. The total price S is the sum of all 5 rows.
S = 8 * (X1 + X2 + ... + X5)

From the columns:
The sum of prices in each column is 5 times its average. The total price S is the sum of all 8 columns.
S = 5 * (Y1 + Y2 + ... + Y8)

Statement (1): X1 + ... + X5 = $850
Using the row formula, we can find the total price: S = 8 * $850 = $6,800.
Since we can find the total price, we can find the average. Statement (1) is SUFFICIENT.

Statement (2): Y1 + ... + Y8 = $1,360
Using the column formula, we can find the total price: S = 5 * $1,360 = $6,800.
Since we can find the total price, we can find the average. Statement (2) is SUFFICIENT.

Each statement alone is sufficient.

The correct answer is (D).

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Not sure. I think the solution can be arrived at calculating a sum of both averages.

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Evaluating option 1:
If we multiply the sum of row averages by 5 (number of rows), we will get the sum of all items on the rack. That divided by 40 will give us the average price of all items on the rack.

1 is SUFFICIENT.

Evaluating option 2:
If we multiply the sum of column averages by 8 (number of columns), we will get the sum of all items on the rack. That divided by 40 will give us the average price of all items on the rack.

2 is also SUFFICIENT.

Hence, answer is D i.e. each statement is sufficient on its own.

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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30 Jun 2025, 08:58

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There are 40 items on a rack arranged in:

5 rows, 8 items each
8 columns, 5 items each

You’re given:

XmX_mXm = average price of items in row mmm (for m = 1 to 5)
YnY_nYn = average price of items in column nnn (for n = 1 to 8)

Let’s call the total average = A
So:
A= Total price of all 40 items/40

Statement (1):
X1+X2+X3+X4+X5=850
Each X is the average of 8 items in that row. So:

Row 1 total = 8*X1
Row 2 total = 8*X2

So the total of all prices:
8(X1+X2+X3+X4+X5)=8*850=6,800
So average:
A=6,800/40=170
Sufficient.
Statement (2):
Y1+Y2+⋯+Y8=1,360
Each Y is the average of 5 items in that column. So total:
5(Y1+Y2+⋯+Y8)=5*1,360=6,800
So again:
A=6,800/40=170
Sufficient.

Bunuel

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Information given:
- 40 items are arranged in a rack: 5 rows x 8 columns
- Average price of items in each row: Xm (for m = 1 to 5)
- Average price of items in each column: Yn (for n = 1 to 8)

Question:
- What is the average price of all 40 items on the rack?

Solution:
- Total price: sum of prices of all items
- Average price: total price / 40
- Therefore, to calculate the average price, we need to know the total price

- Statement 1: X1 + X2 + X3 + X4 + X5 = 850
- Each Xm is a row average for 8 items
- Total price for each row: Row m = 8 * Xm
- Total = 8 * (X1 + X2 + X3 + X4 + X5) = 8 * 850 = 6800
- Average = 6800/40 = 170
- Therefore, statement 1 alone is sufficient

- Statement 2: Y1 + Y2 + Y3 + Y4 + Y5 = 1360
- Each Yn is a column average for 5 items
- Total price for each column: column n = 5 * Yn
- Total = 5 * (Y1 + Y2 + Y3 + Y4 + Y5) = 5 * 1360 = 6800
- Average = 6800/40 = 170
- Therefore, statement 2 alone is sufficient

Answer: D, each statement alone is sufficient

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30 Jun 2025, 09:01

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1. X1+x2+....+x5=850

Since 5 rows, average price is 850/5= 170.. 1 sufficient

Similarly for 2
Y1+y2+....+y8= 1360
8 columns hence per item=1360/8=170 2 also sufficient hence option D

Bunuel

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Correct Answer: 170
Each row have 8 items,
Total Price- 8*850= 6800
Average Price of 40 items (8*5)
6800/40 = 170 Answer.
Similarly we can calculate by 2nd statement as well.
Here we have 5 rows for each column so
Total Price= 5*1360 = 1680
Average price of 40 items (5*8)
6800/40 = 170 H.P.

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We are given the row wise and column wise average notations. We are supposed to find the average of all the observations.

Statement 1 - This statement would be sufficient. The sum total of each row can be found by multiplying its average by 8. Therefore, 8 times the entire sum will provide the sum of all values. Dividing by 40 would provide the required average.

Statement 2 - Similar to Statement 1, the sum total of all columns can be found by multiplying by 5. Dividing by 40 would provide the required average.

Either statement is sufficient.

Option D

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30 Jun 2025, 09:02

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#1
$X_1 + X_2 + ... + X_5 = $850$
avg price of rows is 850/5 ; 170
column info of price not known
insufficient
#2
$Y_1 + Y_2 + ... + Y_8 = $1,360$
avg price of columns is 1360/8 ; 170
row info of price not known
insufficient

from 1 &2
we can say that avg price of rack is 170
option C

Bunuel

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Kanika2409

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We have 40 items arranged in a grid of 5*40
1) sum of means of all rows is 850
Mean of all (850*8)/40
Sufficient
2) sum of means of all columns is 1360
Mean of all (1360*5)/40
Sufficient
Hence D

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The overall average price is calculated as the total sum of all item prices divided by 40. Since each row has the same number of items (8), the overall average is equal to the average of the row averages. Similarly, since each column has the same number of items (5), the overall average is equal to the average of the column averages.

Statement (1): The sum of the row averages is given as
X1+X2+X3+X4+X5=8850
The number of rows is 5.
Overall average =
8850/5=1770
Thus, Statement (1) alone is sufficient to determine the average price.

Statement (2): The sum of the column averages is given as
Y1+Y2+..........+Y8 = 81360
Overall avg. = 81360/8=10170
Thus, Statement (2) alone is sufficient to determine the average price.

Each statement alone provides a unique value for the overall average. In Data Sufficiency, sufficiency is determined by whether a statement allows for a unique answer, regardless of consistency with the other statement. Therefore, each statement alone is sufficient.

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The question states that there items arranged in 5 rows and 8 columns. That means, each row has 8 items and each column has 5 items.

Statement I states that the sum of means of price of items in rows 1 to 5 is $850. So, the sum of the prices of all products can be calculated as 850*8= 6800 (Since, sum of means of each row is 850 and each row has 8 items)
Since the total number of items is 40, the mean of all items can be calculated as, 6800/40= $170

Statement I is sufficient.

Similar calculation can be done for Statement II but instead of multiplying by 8, we multiply the sum of means by 5 because each column has 5 items.

Statement II is also sufficient.

Therefore, each statement alone is sufficient.

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Start by visualizing a table with 5 rows and 8 columns; we will have exactly 40 cells in that table. Since we are given a total of 40 items, we can imagine that each of those items is placed in those 40 cells.

Now that we have an image in our heads, let's see what is given:

Xm, where m is [1,5] > this represents the average price of all items that are placed in the "m" row
Yn, where n is [1,8] > represents the average price of all items that are placed in the "n" column

Let's take row 1 first, it has 8 items

So X1, means the average price of all items in the first row, and consequently for X2, X3, and so on till X5
If we think about only the first row, we can write the sum of all items in the first row as 8*(X1) > basis the concept that average price (X1) = sum of items in row 1/N of items in row 1
Repeating this for all rows we get the row-wise sum of all items as
Row 1 + Row 2 + Row 3 + Row 4 + Row 5 = 8*X1 + 8*X2 + 8*X3 + 8*X4 + 8*X5; which is equal to 8* (X1+X2+X3+X4+X5)

When a similar process for the columns is followed, we get the column-wise sum of all items as 5* (Y1+Y2+Y3+Y4+....+Y8)

Let's look at the statements now:

Statement 1:
X1+X2+...+X5=$850

Basis our result we can use the equation: 8* (X1+X2+X3+X4+X5) = 8*850 = 6,800

This is the sum of all prices of the 40 items we have in the store, so the average price is 6800/40 = $170

Sufficient.

Statement 2:
[ltr][color=#0f0f0f]Y1+Y2+...+Y8=$1,360[/color][/ltr]

Using the second equation we got for columns, we can say the sum of all items = 5* (Y1+Y2+Y3+Y4+....+Y8) = 5*1360 = 6,800

Avergae price of all items is 6800/40 = $170

Sufficient.

Correct answer is D. Both statements are sufficient independently

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No. of items = 40
No. of rows = 5, Avg price = 1, 2, 3, 4, 5
No. of columns = 8, Avg price = 1, 2, 3, ..., 8

Avg price = Total price/40

Statement 1: X1+X2+...+X5 = 850
Each row has 8 items
Avg price = (8*850)/40

Sufficient

Statement 2: Y1+Y2+...+Y8 = 1360
Each column has 5 items
Avg price = 5*1360/40

Sufficient

Ans. D

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X Represent the average of each row and we have 5 rows but the total element in each of the row is 8 which means average of 1 row would be sum of 8 numbers / 8 (total number) and this average would be calculate same way for all 5 rows so
we know average of rows is 850 and we know total number in each row is 8 so the total sum of all the rows would be 850*8 i.e. 6800 and total number of items are 8*5 which is 40

so average is 6800/40

Option 1 help us answer this using and similarly in option 2 we can leverage the no of items in each column so the average * no of items in each column = sum of all columns

1360*5= 6800
6800/40 = 170

so both are sufficient

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We know the answer can be 8 x (X1 +... +X5)/40 OR 5 x (Y1 +... + Y8)/40

(1) only: we can calculate the answer as above
(2) only: we can calculate the answer as above

Answer: D

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so there are 5 rows and 8 columns.

One shortcut is that, anyways, there are 40 items, and we can divide 850/5 = 170, which is sufficient, and 1360/8 = 170, which is also sufficient. The answer is D: each statement alone is sufficient.

Bunuel

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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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1) Total X sum is given so we have 8 items in each rows-so total =8X/40 =we can solve
2) total Y sum is given so we have 5 items in each column-so total =5Y/40 =we can solve hence answer is D

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Answer
(D) Each statement ALONE is sufficient.

Explanation

To find the average price of all 40 items, we need the total price of all 40 items. We can find this total in two ways: by adding up the sum of each row, or by adding up the sum of each column.

Total Price (using rows) = (Sum of row 1) + (Sum of row 2) + ... + (Sum of row 5)
Total Price (using columns) = (Sum of col 1) + (Sum of col 2) + ... + (Sum of col 8)

Analyzing the Statements:

Statement (1): Gives the sum of the average price of each ROW.
- The total price of one row is Average * 8 (since there are 8 items in a row).
- The total price of all items is (X1 + X2 + ... + X5) * 8.
- Since we are given X1 + ... + X5 = $850, we can calculate the total price ($850 * 8).
- This is SUFFICIENT.
Statement (2): Gives the sum of the average price of each COLUMN.
- The total price of one column is Average * 5 (since there are 5 items in a column).
- The total price of all items is (Y1 + Y2 + ... + Y8) * 5.
- Since we are given Y1 + ... + Y8 = $1,360, we can calculate the total price ($1,360 * 5).
- This is also SUFFICIENT.

Since each statement alone allows us to find the total price, the answer is D.

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