Salami Cost: Calculate Price Per Kilogram

Hey there! Ever wondered how to figure out the price of something per unit when you're given a total cost and a fractional amount? Let's break down a real-world math problem involving calculating the cost per kilogram of delicious sliced salami. This is a practical skill that can help you in everyday situations, from grocery shopping to comparing prices. So, grab your mental calculator, and let's dive in!

Understanding the Problem

The key question we're tackling today is: If Hailey paid $13 for $1 \frac{3}{7}$ kg of sliced salami, what was the cost per kilogram? This means we need to find out how much just one kilogram of salami costs. To solve this, we'll use our knowledge of fractions and division.

Before we jump into the solution, let’s make sure we understand what the problem is asking. We know the total cost ($13) and the total weight ($1 \frac{3}{7}$ kg). The phrase "cost per kilogram" tells us we need to find the cost for one unit of weight, which in this case is one kilogram. This is a classic unit rate problem, where we divide the total cost by the total weight to find the cost per unit weight. Keep this concept in mind as we proceed.

It's also important to note the instruction: "Enter an exact answer, without rounding." This means we need to be precise in our calculations and avoid any approximations. We'll need to work with fractions carefully to get the exact cost per kilogram. This attention to detail is crucial in mathematics and in real-life scenarios where accuracy matters.

Converting Mixed Numbers to Improper Fractions

The first step to solving this problem is converting the mixed number $1 \frac{3}{7}$ into an improper fraction. This will make it easier to perform the division later on. Remember, a mixed number is a whole number combined with a fraction, while an improper fraction has a numerator that is greater than or equal to the denominator.

To convert a mixed number to an improper fraction, we use the following method: Multiply the whole number by the denominator of the fraction, and then add the numerator. This result becomes the new numerator, and the denominator stays the same. Let’s apply this to our mixed number, $1 \frac{3}{7}$. The whole number is 1, the denominator is 7, and the numerator is 3. So, we calculate: (1 × 7) + 3 = 7 + 3 = 10. Therefore, the improper fraction is $\frac{10}{7}$.

Now that we have converted $1 \frac{3}{7}$ to $\frac{10}{7}$, we can rewrite our problem as: Hailey paid $13 for $\frac{10}{7}$ kg of sliced salami. What was the cost per kilogram? This conversion is a fundamental step in working with fractions, and it sets us up for the next operation: division.

Dividing the Total Cost by the Total Weight

Now that we've expressed the weight as an improper fraction, we can find the cost per kilogram by dividing the total cost by the total weight. This is where our understanding of dividing fractions comes into play. Remember, dividing by a fraction is the same as multiplying by its reciprocal.

In our problem, we need to divide $13 by $\frac{10}{7}$. To do this, we first rewrite $13$ as a fraction, which is $\frac{13}{1}$. Then, we find the reciprocal of $\frac{10}{7}$, which is $\frac{7}{10}$. Now, we can multiply $\frac{13}{1}$ by $\frac{7}{10}$. When multiplying fractions, we multiply the numerators together and the denominators together.

So, we have: $\frac{13}{1} × \frac{7}{10} = \frac{13 × 7}{1 × 10} = \frac{91}{10}$. This fraction represents the cost per kilogram of salami. We've done the hard part of the calculation, but we're not quite finished yet. The fraction $\frac{91}{10}$ is an improper fraction, and while it's a correct answer, it's often more useful to express it as a mixed number or a decimal.

Expressing the Answer as a Mixed Number and Decimal

We've calculated the cost per kilogram as $\frac{91}{10}$, but let's make this answer even more understandable by converting it into a mixed number and a decimal. This will give us a clearer sense of the price.

To convert the improper fraction $\frac{91}{10}$ to a mixed number, we divide the numerator (91) by the denominator (10). The quotient becomes the whole number part of the mixed number, and the remainder becomes the numerator of the fractional part. The denominator stays the same. When we divide 91 by 10, we get a quotient of 9 and a remainder of 1. So, the mixed number is $9 \frac{1}{10}$.

Now, let's convert $\frac{91}{10}$ to a decimal. To do this, we simply divide 91 by 10, which gives us 9.1. Since we're dealing with money, we express this as $9.10.

Therefore, the cost per kilogram of salami is $9 \frac{1}{10}$ or $9.10. This gives us a clear and practical understanding of the price: each kilogram of salami costs $9.10.

The Final Answer

We've journeyed through converting mixed numbers, dividing fractions, and expressing our answer in different forms. Now, let's state our final answer clearly and confidently. The cost per kilogram of sliced salami is $9.10.

To arrive at this answer, we first converted the mixed number $1 \frac{3}{7}$ to the improper fraction $\frac{10}{7}$. Then, we divided the total cost ($13) by the total weight ($\frac{10}{7}$ kg) by multiplying $13$ by the reciprocal of $\frac{10}{7}$, which is $\frac{7}{10}$. This gave us the fraction $\frac{91}{10}$, which we then converted to the decimal $9.10.

This problem demonstrates the practical application of fractions and division in everyday situations. By breaking down the problem into manageable steps, we were able to find the exact cost per kilogram of salami. Remember, understanding these fundamental math concepts can empower you to make informed decisions and solve real-world problems.

Why This Matters

Understanding how to calculate unit costs, like the cost per kilogram of salami, is a valuable skill that extends beyond the classroom. It's a practical tool that can help you make informed decisions in various aspects of your life. Whether you're grocery shopping, comparing prices, or managing a budget, the ability to calculate unit costs can save you money and ensure you're getting the best value for your purchases.

In the context of grocery shopping, knowing how to calculate the cost per kilogram (or per pound, or per ounce) allows you to compare the prices of different products and sizes. For example, a larger package might have a lower price per unit than a smaller package, even if the total cost is higher. By doing the math, you can determine which option is the most cost-effective. This skill is particularly useful when buying items in bulk or when comparing different brands.

Beyond shopping, the concept of unit cost is essential in business and finance. Businesses use unit cost calculations to determine the profitability of their products and services. They also use it to set prices and manage expenses. In personal finance, understanding unit costs can help you budget effectively and track your spending. For instance, you can calculate the cost per use of a product or service to see if it's worth the investment.

Moreover, the ability to work with fractions and decimals is a fundamental skill in many fields, including science, engineering, and technology. These concepts are the building blocks for more advanced mathematical topics, so mastering them early on can set you up for success in future studies and careers.

In conclusion, the problem we solved today isn't just about salami; it's about developing a valuable skill that can benefit you in countless ways. By understanding how to calculate unit costs, you're empowering yourself to make smarter decisions and navigate the world with confidence.

Conclusion

We've successfully navigated the process of calculating the cost per kilogram of sliced salami, and hopefully, you've gained a better understanding of how to approach similar problems. Remember, the key is to break down the problem into smaller steps, convert mixed numbers to improper fractions, and apply the principles of division. Whether you're shopping for groceries or tackling a math problem, these skills will serve you well.

Math isn't just about numbers and formulas; it's about problem-solving and critical thinking. By practicing these skills, you're building a foundation for success in many areas of life. So, keep exploring, keep learning, and keep applying your math knowledge to the world around you.

For further learning on fractions and unit rates, check out resources like Khan Academy's lessons on fractions and ratios. Happy calculating!