A rectangle was altered by doubling its length and decreasing the width by \(p\) percent. If these alterations decreased the area of the rectangle by 40 percent, what is the value of \(p\) ?

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The area of the initial rectange is \(A=l\cdot w\). The area of the modified rectange is 60 percent of the original (40% decrease):

$$ A_{mod}=0.60A $$

Modifications were made to the length and width:

$$ l_{new}=2\cdot l $$ $$ w_{new}=w \cdot (p \text{ percent}) $$

We want to multiply the new length and width so that it becomes 60% of the original:

$$ 2\cdot l \cdot w \cdot (\text{p percent}) = 0.60A $$ $$ 2\cdot l \cdot w \cdot (\text{p percent}) = 0.60\cdot l\cdot w $$ $$ 2 \cdot \text{ (p percent)} = 0.60 $$ $$ \text{ (p percent)} = 0.30 $$

We can interpret this as:

$$ w_{new}=w \cdot 0.30 $$

Taking \(30\%\) of \(w\) is equivalent to a \(\boxed{70\%}\) decrease in \(w\).

Let's investigate by using hypothetical numbers. Draw a \(10\) by \(10\) rectangle with area \(100\).

If we double one side and decrease the area by \(40\) percent, this is the result:

The missing side for the modified rectangle must be \(3\). This is a \(\boxed{70\%}\) decrease from the original, \(10\).