Question

If 8 braccia of cloth are worth 11 florins, what are 97 braccia worth?

Short answer

Answer: 133.375 florins.

Step-by-step solution

Identify the known values

We are given that 8 braccia of cloth is worth 11 florins. So, we know the value of cloth per braccia. We need to find the value of 97 braccia of cloth.

Set up a proportion

We can set up a proportion using the known values and the unknown value we want to find. Let x be the value of 97 braccia of cloth. The proportion can be written as: \[\frac{8 \text{ braccia}}{11 \text{ florins}} = \frac{97 \text{ braccia}}{x \text{ florins}}\]

Solve the proportion

To solve the proportion for x, we can cross-multiply: \[8 \text{ braccia} \cdot x \text{ florins} = 97 \text{ braccia} \cdot 11 \text{ florins}\] Now, divide both sides by 8 braccia to isolate x: \[x \text{ florins} = \frac{97 \text{ braccia} \cdot 11 \text{ florins}}{8 \text{ braccia}}\]

Calculate the value of x

Next, calculate the value of x: \[x \text{ florins} = \frac{1067 \text{ florins}}{8}\]

Simplify the answer

Now, we can simplify by calculating the value of the fraction: \[x \text{ florins} = 133.375 \text{ florins}\] So, 97 braccia of cloth are worth 133.375 florins.

Key concepts

Proportion Problem Solving

Proportion problem solving is a fundamental aspect of algebra and can be applied to various real-world situations. When faced with a question such as determining the value of one quantity based on the proportionality of another, the key is to set up a ratio that compares the two. In our example, we have 8 braccia of cloth worth 11 florins. To find out what 97 braccia are worth, we set up a proportion where the ratios of braccia to florins from the known condition must equal the ratio of braccia to florins in the unknown condition. We express this relationship using the formula:

\[\frac{8 \text{ braccia}}{11 \text{ florins}} = \frac{97 \text{ braccia}}{x \text{ florins}}\]

The principle of proportion assumes that the value per braccia remains constant. This constant relationship helps us find the unknown value by scaling up or down the known value proportionally. To improve understanding, it is important to remember:

• Proportions are equations that assert two ratios are equivalent.
• To solve a proportion, you can often find a unit rate (value per single unit) and then multiply it by the quantity of interest.
• Ensuring the units are consistent across the proportion is crucial for accuracy.

Working through proportions in this structured way helps develop an efficient and reliable technique for problem solving that can be applied to a variety of contexts.

Cross-Multiplication Method

The cross-multiplication method is an effective technique used to solve proportions. Once we've established our proportion, as shown in the exercise, we can apply this method to find the unknown variable. To cross-multiply, we take the numerator of one fraction and multiply it by the denominator of the other, and do the same with the remaining numerator and denominator. The two products are then set equal to each other, which forms an equation that we can solve for the unknown. The cross-multiplying step would look as follows:

\[8 \text{ braccia} \times x \text{ florins} = 97 \text{ braccia} \times 11 \text{ florins}\]

This process effectively 'crosses out' the units of braccia, simplifying the equation to one where we only need to isolate the variable that represents the unknown value in florins. Students should visualize this step to understand the 'crossing' aspect:

• The numerators and denominators diagonally across from each other multiply together.
• The resulting products are two numbers without fractions, making it easier to solve for the variable.
• Reach the simplest form of the equation before attempting to solve for the variable, to minimize the chance of error.

Cross-multiplication is a foundational tool in algebra that simplifies the process of solving for unknowns within proportional relationships.

Mathematical Reasoning

Mathematical reasoning involves the ability to think logically about the relationships between concepts and to apply mathematical principles to derive conclusions. It's the thread that ties together the understanding of how and why particular math strategies work. In the context of our cloth and florins problem, we use mathematical reasoning to understand why proportionality allows us to extrapolate the unknown value of 97 braccia from the known value of 8 braccia. A strong grasp on mathematical reasoning is crucial as it enables students to:

• Identify relevant information and disregard extraneous details.
• Understand the underlying principles, like the constancy of the value per braccia in this price context.
• Recognize patterns, such as the linear relationship between the amount of cloth and its cost.
• Apply learned techniques, like cross-multiplication, effectively and understand when and why they work.

Ultimately, mathematical reasoning is about making sense of quantities and their interrelationships. It empowers students to approach problems systematically, leading them to conclusions that are logically sound. This skill is invaluable not only in mathematics but in everyday problem solving and decision making.