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, $10$.