Find Equivalent Ratios: Converting 3 Bags to 21 Popcorn Packages

To solve this problem, we'll proceed with the following steps:

• Step 1: Simplify the initial ratio given in the problem.
• Step 2: Simplify the ratios of each choice and compare them with the simplified initial ratio.

Let's go through these steps in detail:

Step 1: Simplifying the ratio given in the problem:

The initial ratio is 3 bags of corn to 21 packages of popcorn. We can express this as the ratio $\frac{3}{21}$.

To simplify, divide both the numerator and the denominator by their greatest common divisor, which is 3:

$\frac{3 \div 3}{21 \div 3} = \frac{1}{7}$.

So, the simplified ratio is 1 bag of corn to 7 packages of popcorn.

Step 2: Now, let's simplify each of the given choices to see which one matches the simplified ratio of $\frac{1}{7}$.

• Choice 1: 6 bags of corn to 3 packages of popcorn
  Expressed as a ratio: $\frac{6}{3} = 2$, which simplifies to 2, not $\frac{1}{7}$.
• Choice 2: 1 bag of corn to 4 packages of popcorn
  Expressed as a ratio: $\frac{1}{4} = 0.25$, not $\frac{1}{7}$.
• Choice 3: 1 bag of corn to 7 packages of popcorn
  This ratio is clearly already $\frac{1}{7}$, the same as our simplified initial ratio.
• Choice 4: 2 bags of corn to 10 packages of popcorn
  Expressed as a ratio: $\frac{2}{10} = \frac{1}{5}$, not $\frac{1}{7}$.

Comparing each of the choices to the simplified ratio $\frac{1}{7}$, we find that Choice 3 is the correct match.

Therefore, the solution to the problem is 1 bag of corn, 7 packages of popcorn, which corresponds to Choice 3.