Isabel ordered topsoil and crushed stone, which cost a total of $641, for her garden. The given equation 24.5x +...

1. TRANSLATE the equation components

• Given equation: \(24.5\mathrm{x} + 24.75\mathrm{y} = 641\)
• Where \(\mathrm{x}\) = cubic yards of topsoil, \(\mathrm{y}\) = tons of crushed stone
• This tells us: (cost per cubic yard of topsoil)(cubic yards) + (cost per ton of crushed stone)(tons) = \(\$641\)

2. INFER what the coefficients represent

• In cost equations, coefficients are unit costs
• 24.5 = cost per cubic yard of topsoil = \(\$24.50\)
• 24.75 = cost per ton of crushed stone = \(\$24.75\)

3. SIMPLIFY to find the difference

• Difference = \(\$24.75 - \$24.50 = \$0.25\)
• A ton of crushed stone costs \(\$0.25\) more than a cubic yard of topsoil

Answer: $0.25 (or 0.25 or 1/4)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret what the coefficients 24.5 and 24.75 represent in the equation.

They might think these numbers represent the total amounts Isabel spent on each material, rather than the cost per unit. This leads them to try solving the equation for x and y first, or to perform unnecessary calculations with the total cost of $641. This confusion causes them to get stuck and abandon systematic solution, leading to guessing.

The Bottom Line:

This problem tests whether students understand the structure of linear cost equations. The key insight is recognizing that when an equation is in the form \(\mathrm{(coefficient}_1\mathrm{)(variable}_1\mathrm{) + (coefficient}_2\mathrm{)(variable}_2\mathrm{) = total}\), the coefficients represent unit costs, not total costs. Once you see this, the solution becomes a simple subtraction.