To find the equation for the proportionate cost of the roast beef, we start by establishing the relationship between the weight of the roast beef (in pounds) and its cost (in dollars).
Since Nigel paid $10 for 2 1/2 pounds, we first convert 2 1/2 pounds to an improper fraction:
\[ 2 \frac{1}{2} = \frac{5}{2} \text{ pounds} \]
Given that the cost \( C \) is proportional to the weight \( W \), we can express this relationship with the formula:
\[ C = kW \]
where \( k \) is the constant of proportionality (cost per pound).
To find \( k \), we can use the information that Nigel paid $10 for \( \frac{5}{2} \) pounds:
Substituting the values into the equation:
\[ 10 = k \cdot \frac{5}{2} \]
To solve for \( k \), we can multiply both sides by the reciprocal of \( \frac{5}{2} \), which is \( \frac{2}{5} \):
\[ k = 10 \cdot \frac{2}{5} = 4 \]
Thus, the constant of proportionality \( k \) is $4 per pound.
We can now write the equation that represents the relationship between cost and weight:
\[ C = 4W \]
This equation indicates that for every pound of roast beef, the cost is $4.
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