Kimberly sent gifts to her friends. Each gift box was either rectangular or cylindrical and each contained exactly one gift: either a fragrant soap or a pack of spicy roasted almonds. If half of the boxes she sent were cylindrical and a third of the rectangular boxes contained soap, then how many cylindrical boxes contained soap?
>(1) Of the gifts Kimberly sent, the total number of cylindrical boxes was 20% greater than the total number of gifts of almonds.
>(2) Kimberly sent 16 rectangular boxes containing almonds.

For Stat.(1+2) combined, put $$\color{blue}{1.2z}$$ in the Cyl. / Total box and $$\color{blue}{z}$$ in the Total / Almonds box. Put {color:dark-blue}16{/color} in the Rect. / Almonds box.
| | SOAP | ALMONDS | TOTAL |
|-------|----------------------|-----------------------------------|--------------------------|
| Rectangular | $$\frac{x}{3}$$ | $$\frac{2x}{3}=\color{blue}{16}$$ | $$x$$ |
| Cylindrical | $$\color{red}{?}$$ | | $$x=\color{blue}{1.2z}$$ |
| Total | | $$\color{blue}{z}$$ | $$2x$$ |
Now what happens when you put them together in the table?

For Stat. (2), add "{color:dark-blue}Kimberly sent 16 rectangular boxes containing almonds{/color}" to the table. Put {color:dark-blue}16{/color} in the Rect. / Almonds box.
| | SOAP | ALMONDS | TOTAL |
|-------|----------------------|-----------------------------------|--------|
| Rectangular | $$\frac{x}{3}$$ | $$\frac{2x}{3}=\color{blue}{16}$$ | $$x$$ |
| Cylindrical | $$\color{red}{?}$$ | | $$x$$ |
| Total | | | $$2x$$ |
Completing the Rect. row and then the Total column, you can reveal the value of $$x$$. But even then you cannot "reach" the question mark ({color:red}?{/color}) by completing columns and rows. Therefore, **Stat.(2) → IS → CE**.

For Stat.(1+2) combined, put $$\color{blue}{1.2z}$$ in the Cyl. / Total box and $$\color{blue}{z}$$ in the Total / Almonds box. Put {color:dark-blue}16{/color} in the Rect. / Almonds box.
| | SOAP | ALMONDS | TOTAL |
|-------|----------------------|-----------------------------------|--------------------------|
| Rectangular | $$\frac{x}{3}$$ | $$\frac{2x}{3}=\color{blue}{16}$$ | $$x$$ |
| Cylindrical | $$\color{red}{?}$$ | | $$x=\color{blue}{1.2z}$$ |
| Total | | $$\color{blue}{z}$$ | $$2x$$ |
Now that you have a numerical value in the Rect. / Almonds box, you can reveal the values of $$x$$ and $$z$$. Then simple subtraction will reveal the number of cylindrical boxes containing soap. Fortunately, this is a __DS__ question, so you don't actually have to do these calculations; just know that you can. Therefore, **Stat.(1+2) → S → C**.

For Stat. (2), add "{color:dark-blue}Kimberly sent 16 rectangular boxes containing almonds{/color}" to the table. Put {color:dark-blue}16{/color} in the Rect. / Almonds box.
| | SOAP | ALMONDS | TOTAL |
|-------|----------------------|-----------------------------------|--------|
| Rectangular | $$\frac{x}{3}$$ | $$\frac{2x}{3}=\color{blue}{16}$$ | $$x$$ |
| Cylindrical | $$\color{red}{?}$$ | | $$x$$ |
| Total | | | $$2x$$ |
Completing the Rect. row and then the Total column, you can reveal the value of $$x$$. But even then you cannot "reach" the question mark ({color:red}?{/color}) by completing columns and rows. Therefore, **Stat.(2) → IS → ACE**.

For Stat. (1), add "{color:dark-blue}the total number of cylindrical boxes was 20% greater than the total number of gifts of almonds{/color}" to the table. Put $$\color{blue}{1.2z}$$ in the Cyl. / Total box and $$\color{blue}{z}$$ in the Total / Almonds box.
| | SOAP | ALMONDS | TOTAL |
|-------|----------------------|---------------------|-----------|
| Rectangular | $$\frac{x}{3}$$ | $$\frac{2x}{3}$$ | $$x$$ |
| Cylindrical | $$\color{red}{?}$$ | | $$x=\color{blue}{1.2z}$$ |
| Total | | $$\color{blue}{z}$$ | $$2x$$ |
Even though you can "reach" the question mark ({color:red}?{/color}) by completing columns and rows, there is an infinite number of values for $$x$$ and $$z$$ that satisfy the equations in the table. Therefore, **Stat.(1) → IS → BCE**.

Incorrect.
[[snippet]]
Now look at the statements.

For Stat. (1), add "{color:dark-blue}the total number of cylindrical boxes was 20% greater than the total number of gifts of almonds{/color}" to the table. Put $$\color{blue}{1.2z}$$ in the Cyl. / Total box and $$\color{blue}{z}$$ in the Total / Almonds box.
| | SOAP | ALMONDS | TOTAL |
|-------|----------------------|---------------------|--------------------------|
| Rectangular | $$\frac{x}{3}$$ | $$\frac{2x}{3}$$ | $$x$$ |
| Cylindrical | $$\color{red}{?}$$ | | $$x=\color{blue}{1.2z}$$ |
| Total | | $$\color{blue}{z}$$ | $$2x$$ |
Even though you can "reach" the question mark ({color:red}?{/color}) by completing columns and rows, there is an infinite number of values for $$x$$ and $$z$$ that satisfy the equations in the table. Therefore, **Stat.(1) → IS → BCE**.

For Stat. (2), add "{color:dark-blue}Kimberly sent 16 rectangular boxes containing almonds{/color}" to the table. Put {color:dark-blue}16{/color} in the Rect. / Almonds box.
| | SOAP | ALMONDS | TOTAL |
|-------|----------------------|-----------------------------------|--------|
| Rectangular | $$\frac{x}{3}$$ | $$\frac{2x}{3}=\color{blue}{16}$$ | $$x$$ |
| Cylindrical | $$\color{red}{?}$$ | | $$x$$ |
| Total | | | $$2x$$ |
Completing the Rect. row and then the Total column, you can reveal the value of $$x$$. But even then you cannot "reach" the question mark ({color:red}?{/color}) by completing columns and rows. Therefore, **Stat.(2) → IS → CE**.

Incorrect.
[[snippet]]
Now look at the statements.

For Stat. (1), add "{color:dark-blue}the total number of cylindrical boxes was 20% greater than the total number of gifts of almonds{/color}" to the table. Put $$\color{blue}{1.2z}$$ in the Cyl. / Total box and $$\color{blue}{z}$$ in the Total / Almonds box.
| | SOAP | ALMONDS | TOTAL |
|-------|----------------------|---------------------|--------------------------|
| Rectangular | $$\frac{x}{3}$$ | $$\frac{2x}{3}$$ | $$x$$ |
| Cylindrical | $$\color{red}{?}$$ | | $$x=\color{blue}{1.2z}$$ |
| Total | | $$\color{blue}{z}$$ | $$2x$$ |
Even though you can "reach" the question mark ({color:red}?{/color}) by completing columns and rows, there is an infinite number of values for $$x$$ and $$z$$ that satisfy the equations in the table. Therefore, **Stat.(1) → IS → BCE**.

Correct.
[[snippet]]
Now look at the two statements.

For Stat. (1), add "{color:dark-blue}the total number of cylindrical boxes was 20% greater than the total number of gifts of almonds{/color}" to the table. Put $$\color{blue}{1.2z}$$ in the Cyl. / Total box and $$\color{blue}{z}$$ in the Total / Almonds box.
| | SOAP | ALMONDS | TOTAL |
|-------|----------------------|---------------------|--------------------------|
| Rectangular | $$\frac{x}{3}$$ | $$\frac{2x}{3}$$ | $$x$$ |
| Cylindrical | $$\color{red}{?}$$ | | $$x=\color{blue}{1.2z}$$ |
| Total | | $$\color{blue}{z}$$ | $$2x$$ |
Even though you can "reach" the question mark ({color:red}?{/color}) by completing columns and rows, there is an infinite number of values for $$x$$ and $$z$$ that satisfy the equations in the table. Therefore, **Stat.(1) → IS → BCE**.

For Stat. (2), add "{color:dark-blue}Kimberly sent 16 rectangular boxes containing almonds{/color}" to the table. Put {color:dark-blue}16{/color} in the Rect. / Almonds box.
| | SOAP | ALMONDS | TOTAL |
|-------|----------------------|-----------------------------------|--------|
| Rectangular | $$\frac{x}{3}$$ | $$\frac{2x}{3}=\color{blue}{16}$$ | $$x$$ |
| Cylindrical | $$\color{red}{?}$$ | | $$x$$ |
| Total | | | $$2x$$ |
Completing the Rect. row and then the Total column, you can reveal the value of $$x$$. But even then you cannot "reach" the question mark ({color:red}?{/color}) by completing columns and rows. Therefore, **Stat.(2) → IS → CE**.

Incorrect.
[[snippet]]
Now look at the two statements.

Incorrect.
[[snippet]]
Now look at the statements.

Statement (1) ALONE is sufficient, but Statement (2) alone is not sufficient to answer the question asked.

Statement (2) ALONE is sufficient, but Statement (1) alone is not sufficient to answer the question asked.

BOTH Statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.

EACH statement ALONE is sufficient to answer the question asked.

Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

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