Question

explain how to use the distributive property to find the product (3)\left(4\frac{1}{5}\
ight)

Explanation:

Step1: Convert mixed number to improper fraction

First, convert the mixed number \(4\frac{1}{5}\) to an improper fraction. The formula for converting a mixed number \(a\frac{b}{c}\) to an improper fraction is \(\frac{a\times c + b}{c}\). So for \(4\frac{1}{5}\), we have \(a = 4\), \(b = 1\), \(c = 5\). Then \(\frac{4\times5 + 1}{5}=\frac{20 + 1}{5}=\frac{21}{5}\). Now the expression becomes \(3\times\frac{21}{5}\). But wait, actually, using the distributive property, we can think of \(4\frac{1}{5}\) as \(4+\frac{1}{5}\). So the original expression \(3\times(4\frac{1}{5})\) can be rewritten as \(3\times(4+\frac{1}{5})\) according to the distributive property \(a\times(b + c)=a\times b+a\times c\), where \(a = 3\), \(b = 4\), \(c=\frac{1}{5}\).

Step2: Apply distributive property

Now apply the distributive property: \(3\times(4+\frac{1}{5})=3\times4+3\times\frac{1}{5}\). Calculate each term: \(3\times4 = 12\) and \(3\times\frac{1}{5}=\frac{3}{5}\).

Step3: Add the results

Then add the two results together: \(12+\frac{3}{5}\). Convert \(12\) to a fraction with denominator \(5\): \(12=\frac{60}{5}\). Then \(\frac{60}{5}+\frac{3}{5}=\frac{60 + 3}{5}=\frac{63}{5}\). If we want to convert it back to a mixed number, \(\frac{63}{5}=12\frac{3}{5}\).

Answer:

Using the distributive property, we rewrite \(3\times(4\frac{1}{5})\) as \(3\times(4+\frac{1}{5})\), then distribute the 3: \(3\times4 + 3\times\frac{1}{5}=12+\frac{3}{5}=12\frac{3}{5}\) (or \(\frac{63}{5}\)). The product is \(12\frac{3}{5}\) (or \(\frac{63}{5}\)).