Question

27÷(-3)=(-27)÷□

Explanation:

Step1: Recall division property

For non - zero numbers \(a\), \(b\), \(a\div b=\frac{a}{b}\) and \((-a)\div(-b)=\frac{-a}{-b}=\frac{a}{b}\), \(a\div(-b)=\frac{a}{-b}=-\frac{a}{b}\), \((-a)\div b=\frac{-a}{b}=-\frac{a}{b}\). Also, if \(a\div b = c\div d\), then \(a\times d=b\times c\) (cross - multiplication property of equal fractions).

Given \(27\div(-3)=(-27)\div x\) (let the unknown be \(x\)). Using the rule of division: \(27\div(-3)=\frac{27}{-3}=- 9\). And \((-27)\div x=\frac{-27}{x}\).

Step2: Solve for \(x\)

Since \(\frac{27}{-3}=\frac{-27}{x}\), cross - multiply: \(27\times x=(-3)\times(-27)\)

Calculate the right - hand side: \((-3)\times(-27) = 81\)

So, \(27x = 81\)

Divide both sides by 27: \(x=\frac{81}{27}=3\)

We can also think about the property of division: \(a\div b = c\div d\) is equivalent to \(a\times d=b\times c\), or we can use the fact that \(27\div(-3)=-9\), and we want to find \(x\) such that \((-27)\div x=-9\). Then, by the definition of division, \(x = (-27)\div(-9)=3\) (because dividing a negative number by a negative number gives a positive number, and \(27\div9 = 3\)).

Answer:

\(3\)