- Identify the known values: 18 pennies and a 6:1 ratio of pennies to nickels.
- Set up the proportion \(\frac{P}{N} = \frac{6}{1}\), where \(P=18\).
- Solve the equation \(\frac{18}{N} = 6\) for \(N\).
- The number of nickels is \(\boxed{3}\).
Explanation
- Analysis of the Problem
We are given that a bag contains pennies and nickels. The number of pennies is \(18\), and the ratio of pennies to nickels is \(6:1\). We need to determine the number of nickels in the bag. Let \(P\) be the number of pennies and \(N\) be the number of nickels. We are given \(P = 18\) and the ratio \(\frac{P}{N} = \frac{6}{1}\).
- Calculating the Number of Nickels
The ratio \(\frac{P}{N} = \frac{6}{1}\) tells us that for every 6 pennies, there is 1 nickel. We can set up the equation:
\[\frac{18}{N} = \frac{6}{1}\]
To solve for \(N\), we can cross-multiply:
\[6 \times N = 18 \times 1\]
\[6N = 18\]
Dividing both sides by 6, we get:
\[N = \frac{18}{6} = 3\]
So, there are 3 nickels in the bag.
- Conclusion
We have determined that with 18 pennies and a ratio of 6:1 (pennies to nickels), the number of nickels is 3. We can verify this: 18 divided by 3 equals 6, which confirms the 6:1 ratio. Thus, there are 3 nickels in the bag. The final answer is \(\boxed{3}\).
Examples
Understanding ratios is essential for many real-life applications, such as mixing ingredients for a recipe or preparing a chemical solution in a lab. For instance, if a cookie recipe calls for 6 cups of flour for every 1 cup of sugar, and you have 18 cups of flour, you would need to calculate how much sugar to use. Using the ratio method, \(\frac{18 \text{ cups of flour}}{N \text{ cups of sugar}} = \frac{6}{1}\), you find you need 3 cups of sugar. This helps ensure your recipe turns out perfectly!