A certain toy store sold 20 toys yesterday, each of which was either

Question

A certain toy store sold 20 toys yesterday, each of which was either a $40 toy or a $20 toy. How many $20 toys did the toy store sell?

1) The average price of the toys sold yesterday was $35.
(2) The total price of the 20 toys sold yesterday was between $650 and $750.

gmatophobia · Answer

Given

Number of toys sold = 20

Each toy was either $20 or $40

Statement 1

The average price of the toys sold yesterday was $35.

We can use the concept of weighted average and find the number of toys

20 ---------- 30 ---- 35 ---- 40

No. of $20 toys / No. of $40 toys is = 1/3

As we know the ratio and the total number of toys are give we can find the number of $20 toys that the store sold.

The information is sufficient and we can eliminate B, C and E.

Statement 2

Let the number of $20 toys sold = x & number of $40 toys sold is y

Given 650 < 20x + 40y < 750

We need not solve this, as its very clear the statement is not sufficient.

This is because, even if we get one value of x and y that fits the criteria , we can replace one $40 toy with two $20 toys without affecting the total.

Hence we cannot conclusively tell the number of $20 toys the toy store sold.

Option A

ArjunKumar25 · Answer

Hello Gmatphobia, How is B not sufficient? if we solve then 65<2x+4y<75. We only have 1 set of values of x and y satisfying it.

Bunuel · Answer

650 < 20x + 40y < 750
650 < 20x + 40(20 - x) < 750
65 < 2x + 80 - 4x < 75
65 < -2x + 80 < 75
-15 < -2x < -5
5 < 2x < 15
2.5 < x < 7.5

x can be 3, 4, 5, 6, or 7.

pratiksha1998 · Answer

Hi,

Not able to understand how statement 1 is sufficient. Can someone please elaborate?

egmat · Answer

Hi! Let me show you why Statement 1 is sufficient.

Key Setup: 
Let x = number of  $20  toys
Then ( 20  - x) = number of  $40  toys

Statement 1:  The average price of the toys sold yesterday was  $35 .

Key insight: When you have only two types of items and know their average price, you can find the exact mix!

Here's why:

Average price = Total price ÷ Number of toys
 $35  = Total price ÷  20 
Therefore, Total price =  $35  ×  20  =  $700

Now, we also know:
Total price = (Number of  $20  toys ×  $20 ) + (Number of  $40  toys ×  $40 )
 $700  = x ×  $20  + ( 20  - x) ×  $40 
 $700  =  20 x +  800  -  40 x
 $700  =  800  -  20 x
 20 x =  100 
x =  5

Therefore, the store sold exactly 5 toys at $20. 
 
 Quick verification: 
 5  toys at  $20  =  $100 
 15  toys at  $40  =  $600 
Total =  $700  ✓
Average =  $700  ÷  20  =  $35  ✓

DS Principle: A statement is sufficient when it leads to exactly ONE answer. Statement 1 gives us a unique equation that has only one solution.

Answer: A

A certain toy store sold 20 toys yesterday, each of which was either

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GMAT Data Sufficiency (DS) Questions

A certain toy store sold 20 toys yesterday, each of which was either

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