Ratios & Proportions: Combining Ratios with Different Multipliers - Equate the Common Member
There are 3 times as many hot dogs as pizzas and 5 times as many brownies as hot dogs in the Junk Food "R" Us restaurant. The restaurant sells only whole products (you cannot buy half a pizza, for example). How many brownies are there in the restaurant?
>(1) There are at least 10 pizzas in the restaurant.
>(2) There are 32 hot dogs at most in the restaurant.
According to Stat. (1+2), when constraints are placed on two of the items, namely pizza and hot dogs, it is possible to narrow down the possibilities to one that satisfies both constraints.
The only ratio that involves at least 10 pizzas and at most 32 hot dogs is 10:30:150. Thus, there must be 150 brownies. **Stat.(1+2) → S → C**.
According to Stat. (1+2), when constraints are placed on two of the items, namely pizza and hot dogs, it is possible to narrow down the possibilities to one that satisfies both constraints.
The only ratio that involves at least 10 pizzas and at most 32 hot dogs is 10:30:150. Thus, there must be 150 brownies. **Stat.(1+2) → S → C**.
Incorrect.
[[snippet]]
According to Stat. (2), from the question you know two ratios—hot dogs to pizzas and brownies to hot dogs:
| | Hot dogs | : | Pizzas | : | Brownies |
|-----------|---|---|---|----|----|
| 1st Ratio | $$\hspace{0.3in} 3$$ | : | $$\hspace{0.1in} 1$$ | | |
| 2nd Ratio | $$\hspace{0.3in} 1$$ | | | : | $$\hspace{0.2in} 5$$ |
Arrange the two ratios so that the common entity (hot dogs) is in the middle:
| | Pizzas | : | Hot dogs | : | Brownies |
|-----------|---|---|----|----|----|
| 1st Ratio | $$\hspace{0.2in} 1$$ | : | $$\hspace{0.3in} 3$$ | | |
| 2nd Ratio | | | $$\hspace{0.3in} 1$$ | : | $$\hspace{0.2in} 5$$ |
The least common multiple of 1 and 3 is their product, 3. Thus, you
have to expand the second ratio by 3:
| | Pizzas | : | Hot dogs | : | Brownies |
|-----------|---|---|----|----|----|
| 1st Ratio | $$\hspace{0.2in} 1$$ | : | $$\hspace{0.3in} 3$$ | | |
| 2nd Ratio | | | $$\hspace{0.3in} 3$$ | : | $$\hspace{0.2in} 15$$ |
Thus, the ratio of Pizzas to Hot dog to Brownies is 1:3:15.
It does not help
to know that there are 32 hot dogs at most, since it is still possible
that the numbers of pizzas, hot dogs, and brownies are 1, 3, and 15, or
maybe 2, 6, 30, or maybe even 3, 9, and 45. **Stat.(2) → IS → ACE**.
Incorrect.
[[snippet]]
According to Stat. (1), you can find the ratio of pizzas to hot dogs to brownies, but then
what? "At least 10 pizzas" allows for endless trios of real numbers for
the three entities. **Stat.(1) → IS → BCE**.
Incorrect.
[[snippet]]
According to Stat. (1), you can find the ratio of pizzas to hot dogs to brownies, but then
what? "At least 10 pizzas" allows for endless trios of real numbers for
the three entities. **Stat.(1) → IS → BCE**.
According to Stat. (2), from the question you know two ratios—hot dogs to pizzas and brownies to hot dogs:
| | Hot dogs | : | Pizzas | : | Brownies |
|-----------|---|---|---|----|----|
| 1st Ratio | $$\hspace{0.3in} 3$$ | : | $$\hspace{0.1in} 1$$ | | |
| 2nd Ratio | $$\hspace{0.3in} 1$$ | | | : | $$\hspace{0.2in} 5$$ |
Arrange the two ratios so that the common entity (hot dogs) is in the middle:
| | Pizzas | : | Hot dogs | : | Brownies |
|-----------|---|---|----|----|----|
| 1st Ratio | $$\hspace{0.2in} 1$$ | : | $$\hspace{0.3in} 3$$ | | |
| 2nd Ratio | | | $$\hspace{0.3in} 1$$ | : | $$\hspace{0.2in} 5$$ |
The least common multiple of 1 and 3 is their product, 3. Thus, you
have to expand the second ratio by 3:
| | Pizzas | : | Hot dogs | : | Brownies |
|-----------|---|---|----|----|----|
| 1st Ratio | $$\hspace{0.2in} 1$$ | : | $$\hspace{0.3in} 3$$ | | |
| 2nd Ratio | | | $$\hspace{0.3in} 3$$ | : | $$\hspace{0.2in} 15$$ |
Thus, the ratio of Pizzas to Hot dog to Brownies is 1:3:15.
It does not help
to know that there are 32 hot dogs at most, since it is still possible
that the numbers of pizzas, hot dogs, and brownies are 1, 3, and 15, or
maybe 2, 6, 30, or maybe even 3, 9, and 45. **Stat.(2) → IS → CE**.
Correct.
[[snippet]]
According to Stat. (1), you can find the ratio of pizzas to hot dogs to brownies, but then
what? "At least 10 pizzas" allows for endless trios of real numbers for
the three entities. **Stat.(1) → IS → BCE**.
According to Stat. (2), from the question you know two ratios—hot dogs to pizzas and brownies to hot dogs:
| | Hot dogs | : | Pizzas | : | Brownies |
|-----------|---|---|---|----|----|
| 1st Ratio | $$\hspace{0.3in} 3$$ | : | $$\hspace{0.1in} 1$$ | | |
| 2nd Ratio | $$\hspace{0.3in} 1$$ | | | : | $$\hspace{0.2in} 5$$ |
Arrange the two ratios so that the common entity (hot dogs) is in the middle:
| | Pizzas | : | Hot dogs | : | Brownies |
|-----------|---|---|----|----|----|
| 1st Ratio | $$\hspace{0.2in} 1$$ | : | $$\hspace{0.3in} 3$$ | | |
| 2nd Ratio | | | $$\hspace{0.3in} 1$$ | : | $$\hspace{0.2in} 5$$ |
The least common multiple of 1 and 3 is their product, 3. Thus, you
have to expand the second ratio by 3:
| | Pizzas | : | Hot dogs | : | Brownies |
|-----------|---|---|----|----|----|
| 1st Ratio | $$\hspace{0.2in} 1$$ | : | $$\hspace{0.3in} 3$$ | | |
| 2nd Ratio | | | $$\hspace{0.3in} 3$$ | : | $$\hspace{0.2in} 15$$ |
Thus, the ratio of Pizzas to Hot dog to Brownies is 1:3:15.
It does not help
to know that there are 32 hot dogs at most, since it is still possible
that the numbers of pizzas, hot dogs, and brownies are 1, 3, and 15, or
maybe 2, 6, 30, or maybe even 3, 9, and 45. **Stat.(2) → IS → CE**.
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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