Question

michael took 2\\(\frac{1}{6}\\) hours to clean the den. then he took 3\\(\frac{3}{5}\\) hours to clean the bathroom. how much total time did michael take to clean the two rooms?
write your answer as a mixed number in simplest form.
\\(\square\\) hours

Explanation:

Step1: Convert mixed numbers to improper fractions

First, convert \(2\frac{1}{6}\) 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 \(2\frac{1}{6}\), we have \(a = 2\), \(b = 1\), \(c = 6\). Then \(\frac{2\times6 + 1}{6}=\frac{12 + 1}{6}=\frac{13}{6}\).

Next, convert \(3\frac{2}{5}\) to an improper fraction. Here, \(a = 3\), \(b = 2\), \(c = 5\). So \(\frac{3\times5+2}{5}=\frac{15 + 2}{5}=\frac{17}{5}\).

Step2: Find a common denominator and add the fractions

The denominators are 6 and 5, so the least common denominator (LCD) is \(6\times5 = 30\).

Rewrite \(\frac{13}{6}\) with denominator 30: \(\frac{13\times5}{6\times5}=\frac{65}{30}\).

Rewrite \(\frac{17}{5}\) with denominator 30: \(\frac{17\times6}{5\times6}=\frac{102}{30}\).

Now add the two fractions: \(\frac{65}{30}+\frac{102}{30}=\frac{65 + 102}{30}=\frac{167}{30}\).

Step3: Convert the improper fraction back to a mixed number and simplify

Convert \(\frac{167}{30}\) to a mixed number. Divide 167 by 30: \(30\times5 = 150\), \(167-150 = 17\). So \(\frac{167}{30}=5\frac{17}{30}\). Wait, that seems wrong. Wait, no, wait: \(2\frac{1}{6}+3\frac{2}{5}\). Let's add the whole numbers and the fractions separately. The whole numbers are 2 and 3, so \(2 + 3=5\). The fractions are \(\frac{1}{6}\) and \(\frac{2}{5}\). Find LCD of 6 and 5, which is 30. \(\frac{1}{6}=\frac{5}{30}\), \(\frac{2}{5}=\frac{12}{30}\). Then \(\frac{5}{30}+\frac{12}{30}=\frac{17}{30}\). So total time is \(5+\frac{17}{30}=5\frac{17}{30}\)? Wait, no, wait the original mixed numbers: \(2\frac{1}{6}\) and \(3\frac{2}{5}\). Wait, \(2 + 3 = 5\), and \(\frac{1}{6}+\frac{2}{5}=\frac{5 + 12}{30}=\frac{17}{30}\). So total time is \(5\frac{17}{30}\) hours? Wait, but let's check the calculation again. Wait, \(2\frac{1}{6}\) is \(2 + \frac{1}{6}\), \(3\frac{2}{5}\) is \(3+\frac{2}{5}\). So adding the whole parts: \(2 + 3 = 5\). Adding the fractional parts: \(\frac{1}{6}+\frac{2}{5}\). The common denominator is 30. \(\frac{1\times5}{6\times5}=\frac{5}{30}\), \(\frac{2\times6}{5\times6}=\frac{12}{30}\). Then \(\frac{5 + 12}{30}=\frac{17}{30}\). So total time is \(5\frac{17}{30}\) hours. Wait, but maybe I made a mistake in the first step. Wait, no, the problem says "Write your answer as a mixed number in simplest form." So \(2\frac{1}{6}+3\frac{2}{5}=(2 + 3)+(\frac{1}{6}+\frac{2}{5}) = 5+\frac{5 + 12}{30}=5+\frac{17}{30}=5\frac{17}{30}\).

Wait, but let's check the arithmetic again. \(2\frac{1}{6}\) is \(\frac{13}{6}\), \(3\frac{2}{5}\) is \(\frac{17}{5}\). To add \(\frac{13}{6}+\frac{17}{5}\), find a common denominator of 30. \(\frac{13\times5}{30}+\frac{17\times6}{30}=\frac{65}{30}+\frac{102}{30}=\frac{167}{30}\). Now, \(\frac{167}{30}\) as a mixed number: 30*5=150, 167 - 150=17, so \(\frac{167}{30}=5\frac{17}{30}\). Yes, that's correct.

Answer:

\(5\frac{17}{30}\)