A store sells only two types of shirts - branded and non-branded

Let branded shirts =X and non-branded shirts =Y. Given, X+Y=30 and we are asked to find X.

1. The store sold more than 20 branded shirts on that day. --->>> Clearly Not Sufficient.
2. On that day, the total sales from shirts were between $1604 and $1674. --->>> this leads to the below equation.

1604 < 60X+20Y < 1674
1604 < 60X+20(30-X) < 1674 ---- Substituting Y=30-X
1604 < 60X+600-20X < 1674
1004 < 40X < 1074
25.1 < X < 26.85

As, shirt has to be an integer X will be 26 and Y=30-26=4.
Validation: Confirm the total sales of shirts by putting the values 26*60+20*4=1560+80=1640 which is between 1604 and 1674.

Hence, statement 2 is sufficient.

Answer (B).

The correct answer is Option B

Here is the solution:

From the question we can gather that Branded (B) and Non-Branded (B) sum up to 30 units. So B+NB=30 . Another important thing we get from the question is that both B and NB are positive integers.
Now lets consider the options:
1) NB>20 - This is of no help as there are multiple cases that fulfil this. Hence option A & D are eliminated.
2) Mathematically, the statement will look something like - 1604< 60B+20NB < 1674 or 1604< 20(3B+NB) < 1674
So 20(3B+NB) can only 1620 or 1640 or 1660. Now we have two equations and two variables. On solving, we further realise that 20(3B+NB) can only be 1640, such that B and NB are positive integers.

Hence, option B.

A store sells only two types of shirts - branded and non-branded. All the branded shirts are priced at $60 per unit and all the non-branded shirts are priced at $20 per unit. On a certain day, the store sold a total of 30 shirts. What is the number of branded shirt that the store sold on that day?

1. The store sold more than 20 branded shirts on that day.

2. On that day, the total sales from shirts were between $1604 and $1674.

Let the number of branded shirts sold be x. Since 30 shirts were sold in total, number of Non branded shirts sold was 30-x. The question asks to find: x

Total sales from sales of shirts = 60(x) + 20(30-x) = 40 x + 600 = 40 (x+15). Since x is an integer greater than 0, the total sales should be a multiple of 40.

Step 1 - Evaluate statement (1) independently:- Store sold more than 20 branded shirts => x >20. Clearly, there is no unique value of x. Statement (1) is insufficient.

Step 2 - Evaluate statement (2) independently:- total sales from shirts were between $1604 and $1674. From our pre-thinking step, we know that total sales should be a multiple of 40. Clearly, there is only one multiple of 40 between between $1604 and $1674 i.e. $1640. Hence x = 1640/40 - 15 = 26. Hence, statement (2) is sufficient.

Step 3 - Evaluate statements (1) and (2) combines:- There is no need of this step as statement (2) alone is sufficient.

Hence answer is choice B

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Given: All the branded shirts are priced at $60 per unit and all the non-branded shirts are priced at $20 per unit. On a certain day, the store sold a total of 30 shirts.
Let B = # of branded shirts sold
Let N = # of non-branded shirts sold
If a TOTAL of 30 shirts were sold, we can write: N + B = 30

Target question: What is the value of B?

Statement 1: The store sold more than 20 branded shirts on that day.
All we know so far is that N + B = 30
So, it's possible that B = 21, B = 22, B = 23, etc
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: On that day, the total sales from shirts were between $1604 and $1674.
Branded shirts cost $60 each, and all the non-branded shirts cost $20 each
We can write: 20N + 60B = TOTAL sales
This means: 20N + 60B = some value between $1604 and $1674

Let's see if we can also use the fact that N + B = 30 to help us answer the target question.
We can rewrite 60B as 20B + 40B and see what happens.
We get: 20N + 20B + 40B = some value between $1604 and $1674
Factor first two terms to get: 20(N + B)+ 40B = some value between $1604 and $1674
Replace N + B with 30 to get: 20(30)+ 40B = some value between $1604 and $1674
Evaluate: 600 + 40B = some value between $1604 and $1674
Subtract 600 from both sides to get: 40B = some value between $1004 and $1074

IMPORTANT: B is a POSITIVE INTEGER.
If B = 25, then 40B = 40(25) = 1000, which is NOT between $1004 and $1074
If B = 26, then 40B = 40(26) = 1040, which IS between $1004 and $1074
If B = 27, then 40B = 40(27) = 1080, which is NOT between $1004 and $1074

So, there's only 1 possible value that satisfies statement 2.
The answer to the target question is B = 26
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer: B

Cheers,
Brent

\(30\,{\text{units}}\,\,\,\left\{ \begin{gathered}\\
\,B\,\,{\text{branded}}\,{\text{,}}\,\,{\text{\$ 60}}\,\,{\text{each}} \hfill \\\\
\,N\,\,{\text{non - branded}}\,{\text{,}}\,\,{\text{\$ 20}}\,\,{\text{each}} \hfill \\ \\
\end{gathered} \right.\)

\(? = B\)


\(\left( 1 \right)\,\,B > 20\,\,\left\{ \begin{gathered}\\
\,{\text{Take}}\,\,\left( {B,N} \right) = \left( {21,9} \right)\,\,\,\, \Rightarrow \,\,\,\,? = 21 \hfill \\\\
\,{\text{Take}}\,\,\left( {B,N} \right) = \left( {22,8} \right)\,\,\,\, \Rightarrow \,\,\,\,? = 22 \hfill \\ \\
\end{gathered} \right.\,\,\,\,\, \Rightarrow \,\,\,\,\,{\text{INSUFF}}.\)


\(\left( 2 \right)\,\,1604 < 60B + 20\left( {30 - B} \right) < 1674\,\,\,\,\,\,\left[ \$ \right]\)

\(1604 < 40B + 600 < 1674\)

\(25 \cdot 40 + 4 = 1004 < 40B < 1074 = 26 \cdot 40 + 34\)

\(25 + \frac{4}{40} < B < 26 + \frac{34}{40}\,\,\,\,\, \Rightarrow \,\,\,\,\,? = B = 26\,\,\,\,\, \Rightarrow \,\,\,\,\,{\text{SUFF}}.\)


This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.

B Branded, N Not branded
B+N=30 --- (a)

Total Sales must be: 60B+20N= 40B+20(B+N)= 40B+20*30=40B+600 --- (b)

1. B>20
B can be 21, 22,...
No eqn. that puts any restriction on value of B
Not sufficient

2. 1604< Total Sales < 1674
From b, 1604 < 40B+600 < 1674
It is possible for only one value of B, i.e. 26
Hence, sufficient.

Option: B

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