- Identify the ratio of pennies to nickels: \(\frac{P}{N} = \frac{2}{7}\).
- Substitute the number of pennies (\(P=8\)) into the ratio equation: \(\frac{8}{N} = \frac{2}{7}\).
- Solve the resulting equation for the number of nickels (\(N\)) by cross-multiplying and dividing.
- The number of nickels is \(\boxed{28}\).
Explanation
- Analysis of the Problem
The problem asks us to determine the number of nickels in a bag, given that there are 8 pennies and the ratio of pennies to nickels is 2:7. We denote the number of pennies as \(P\) and the number of nickels as \(N\). The ratio of pennies to nickels is given by the expression \(\frac{P}{N} = \frac{2}{7}\). We know the value of \(P\) is 8, so we can substitute this into our ratio equation to solve for \(N\).
- Solving for the Number of Nickels
We have the equation \(\frac{8}{N} = \frac{2}{7}\). To solve for \(N\), we can use cross-multiplication, which gives us \(2 \times N = 8 \times 7\). Calculating the right side of the equation, we get \(2 \times N = 56\). Finally, dividing both sides of the equation by 2, we find \(N = \frac{56}{2} = 28\). Therefore, there are 28 nickels in the bag.
Examples
Ratios are commonly used in daily life for tasks like mixing recipes. For instance, if a cookie recipe calls for a ratio of 2 cups of flour to 7 tablespoons of sugar, and you only have 8 cups of flour, you can use the same ratio to determine that you need 28 tablespoons of sugar to maintain the correct consistency of your cookie dough. Understanding how to set up and solve these proportions ensures that you can scale recipes or resources accurately for any situation.