Question

operations with mixed numbers
divide: $2\frac{3}{4} \div 1\frac{1}{2}$
question 1 of 8
options: $1\frac{2}{3}$, $2\frac{5}{6}$, $1\frac{1}{4}$, $1\frac{5}{6}$

Explanation:

Step1: Convert mixed numbers to improper fractions

To convert \(2\frac{3}{4}\) to an improper fraction, we use the formula \(a\frac{b}{c}=\frac{a\times c + b}{c}\). So, \(2\frac{3}{4}=\frac{2\times4 + 3}{4}=\frac{8 + 3}{4}=\frac{11}{4}\).

To convert \(1\frac{1}{2}\) to an improper fraction, we have \(1\frac{1}{2}=\frac{1\times2+1}{2}=\frac{2 + 1}{2}=\frac{3}{2}\).

Step2: Divide the improper fractions

Dividing fractions is the same as multiplying by the reciprocal. So, \(\frac{11}{4}\div\frac{3}{2}=\frac{11}{4}\times\frac{2}{3}\).

Step3: Multiply the fractions

Multiply the numerators and denominators: \(\frac{11\times2}{4\times3}=\frac{22}{12}\).

Step4: Simplify the fraction

Simplify \(\frac{22}{12}\) by dividing numerator and denominator by their greatest common divisor, which is 2. So, \(\frac{22\div2}{12\div2}=\frac{11}{6}\).

Step5: Convert back to a mixed number

Convert \(\frac{11}{6}\) to a mixed number. \(11\div6 = 1\) with a remainder of \(5\), so \(\frac{11}{6}=1\frac{5}{6}\)? Wait, no, wait. Wait, let's check again. Wait, \(\frac{11}{4}\times\frac{2}{3}=\frac{11\times2}{4\times3}=\frac{22}{12}=\frac{11}{6}\). Wait, \(11\div6 = 1\) with remainder \(5\), so \(1\frac{5}{6}\)? But wait, let's check the calculation again. Wait, \(2\frac{3}{4}\) is \(\frac{11}{4}\), \(1\frac{1}{2}\) is \(\frac{3}{2}\). Dividing \(\frac{11}{4}\) by \(\frac{3}{2}\) is \(\frac{11}{4}\times\frac{2}{3}=\frac{11\times2}{4\times3}=\frac{22}{12}=\frac{11}{6}\approx1.833\). Wait, but let's check the options. Wait, maybe I made a mistake. Wait, \(2\frac{3}{4}\div1\frac{1}{2}\). Let's do it again. \(2\frac{3}{4}=\frac{11}{4}\), \(1\frac{1}{2}=\frac{3}{2}\). So \(\frac{11}{4}\div\frac{3}{2}=\frac{11}{4}\times\frac{2}{3}=\frac{11\times2}{4\times3}=\frac{22}{12}=\frac{11}{6}=1\frac{5}{6}\)? But the options have \(1\frac{2}{3}\), \(2\frac{5}{6}\), \(1\frac{1}{4}\), \(1\frac{5}{6}\). Wait, but let's check the calculation again. Wait, maybe I messed up the conversion. Wait, \(2\frac{3}{4}\) is \(\frac{11}{4}\), \(1\frac{1}{2}\) is \(\frac{3}{2}\). So \(\frac{11}{4}\div\frac{3}{2}=\frac{11}{4}\times\frac{2}{3}=\frac{11\times2}{4\times3}=\frac{22}{12}=\frac{11}{6}=1\frac{5}{6}\). Wait, but let's check the options. The options are \(1\frac{2}{3}\), \(2\frac{5}{6}\), \(1\frac{1}{4}\), \(1\frac{5}{6}\). So the correct answer is \(1\frac{5}{6}\)? Wait, but let's check with another method. Let's convert to decimals. \(2\frac{3}{4}=2.75\), \(1\frac{1}{2}=1.5\). Then \(2.75\div1.5 = 1.833\ldots\), and \(1\frac{5}{6}\approx1.833\), \(1\frac{2}{3}\approx1.666\), \(2\frac{5}{6}\approx2.833\), \(1\frac{1}{4}=1.25\). So the correct answer is \(1\frac{5}{6}\). Wait, but let's check the calculation again. Wait, \(\frac{11}{4}\div\frac{3}{2}=\frac{11}{4}\times\frac{2}{3}=\frac{11\times2}{4\times3}=\frac{22}{12}=\frac{11}{6}=1\frac{5}{6}\). So the correct option is the one with \(1\frac{5}{6}\).

Answer:

\(1\frac{5}{6}\) (the option with \(1\frac{5}{6}\))