Question

Calculate the following, expressing your answer as a mixed fraction: (a) \(3 \frac{2}{3} \div 1 \frac{1}{2}\) (b) \(6 \frac{1}{4} \div 2 \frac{2}{3}\) (c) \(10 \frac{3}{4} \div 2 \frac{1}{5}\) (d) \(-12 \frac{2}{5} \div 3 \frac{1}{4}\) (e) \(10 \frac{5}{6} \div\left(-4 \frac{1}{2}\right)\)

Short answer

Question: Divide the following mixed fractions: (a) \(3 \frac{2}{3} \div 1 \frac{1}{2}\) (b) \(6 \frac{1}{4} \div 2 \frac{2}{3}\) (c) \(10 \frac{3}{4} \div 2 \frac{1}{5}\) (d) \(-12 \frac{2}{5} \div 3 \frac{1}{4}\) (e) \(10 \frac{5}{6} \div\left(-4 \frac{1}{2}\right)\) Answer: (a) \(2\frac{4}{9}\) (b) \(2\frac{11}{32}\) (c) \(4\frac{47}{44}\) (d) \(-3\frac{53}{65}\) (e) \(-2\frac{11}{27}\)

Step-by-step solution

Convert mixed fractions into improper fractions

To convert the mixed fraction to an improper fraction, multiply the whole number by the denominator and add the numerator. In this case, we have: \(3\frac{2}{3} = \frac{3(3) + 2}{3} = \frac{11}{3}\) and \(1\frac{1}{2} = \frac{1(2) + 1}{2} = \frac{3}{2}\).

Divide the improper fractions

Divide the improper fractions by multiplying the first fraction by the reciprocal of the second fraction: \(\frac{11}{3} \div \frac{3}{2} = \frac{11}{3} \cdot \frac{2}{3} = \frac{22}{9}\).

Convert the improper fraction to a mixed fraction

To convert the improper fraction back to a mixed fraction, divide the numerator by the denominator and express the remainder as a fraction: \(\frac{22}{9} = 2\frac{4}{9}.\) #(b) \(6 \frac{1}{4} \div 2 \frac{2}{3}\)

Convert mixed fractions into improper fractions

We have: \(6\frac{1}{4} = \frac{6(4) + 1}{4} = \frac{25}{4}\) and \(2\frac{2}{3} = \frac{2(3) + 2}{3} = \frac{8}{3}\).

Divide the improper fractions

Divide the improper fractions: \(\frac{25}{4} \div \frac{8}{3} = \frac{25}{4} \cdot \frac{3}{8} = \frac{75}{32}\).

Convert the improper fraction to a mixed fraction

We have: \(\frac{75}{32} = 2\frac{11}{32}\). #(c) \(10 \frac{3}{4} \div 2 \frac{1}{5}\)

Convert mixed fractions into improper fractions

We have: \(10\frac{3}{4} = \frac{10(4) + 3}{4} = \frac{43}{4}\) and \(2\frac{1}{5} = \frac{2(5) + 1}{5} = \frac{11}{5}\).

Divide the improper fractions

Divide the improper fractions: \(\frac{43}{4} \div \frac{11}{5} = \frac{43}{4} \cdot \frac{5}{11} = \frac{215}{44}\).

Convert the improper fraction to a mixed fraction

We have: \(\frac{215}{44} = 4\frac{47}{44}\). #(d) \(-12 \frac{2}{5} \div 3 \frac{1}{4}\)

Convert mixed fractions into improper fractions

We have: \(-12\frac{2}{5} = -\frac{12(5) + 2}{5} = -\frac{62}{5}\) and \(3\frac{1}{4} = \frac{3(4) + 1}{4} = \frac{13}{4}\).

Divide the improper fractions

Divide the improper fractions: \(-\frac{62}{5} \div \frac{13}{4} = -\frac{62}{5} \cdot \frac{4}{13} = -\frac{248}{65}\).

Convert the improper fraction to a mixed fraction

We have: \(-\frac{248}{65} = -3\frac{53}{65}\). #(e) \(10 \frac{5}{6} \div\left(-4 \frac{1}{2}\right)\)

Convert mixed fractions into improper fractions

We have: \(10\frac{5}{6} = \frac{10(6) + 5}{6} = \frac{65}{6}\) and \(-4\frac{1}{2} = -\frac{4(2) + 1}{2} = -\frac{9}{2}\).

Divide the improper fractions

Divide the improper fractions: \(\frac{65}{6} \div -\frac{9}{2} = \frac{65}{6} \cdot -\frac{2}{9} = -\frac{130}{54}\).

Simplify the improper fraction

Reduce the improper fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2: \(-\frac{130}{54} = -\frac{65}{27}\).

Convert the improper fraction to a mixed fraction

We have: \(-\frac{65}{27} = -2\frac{11}{27}\).

Key concepts

Improper Fraction Conversion

When encountering mixed fractions in division, the first step is usually converting them to improper fractions. An improper fraction is one where the numerator (the top number) is larger than or equal to the denominator (the bottom number).

For example, let's look at the mixed fraction \(3 \frac{2}{3}\). To convert this to an improper fraction, first multiply the whole number (3) by the denominator (3), which gives us 9. Then add the numerator (2) to this result, arriving at 11, making the improper fraction \(\frac{11}{3}\). This conversion is crucial because it simplifies the operation of division into a more straightforward multiplication by the reciprocal in the following step.

It's important to ensure students understand this conversion process because it forms the foundation of handling divisions involving mixed fractions. Breaking it down into a two-part process—multiplication followed by addition—helps clarify the steps involved.

Reciprocal of a Fraction

After converting mixed fractions to improper fractions, the next step in division is to multiply by the reciprocal of the divisor fraction. The reciprocal of a fraction is simply flipping the numerator and the denominator.

For instance, if we have \(\frac{3}{2}\), its reciprocal is \(\frac{2}{3}\). Multiplying by the reciprocal is the same as dividing by the original fraction. This concept is often one of those 'aha' moments for students, as it turns division into multiplication, which is generally a more familiar operation. It's helpful to remind students that division by a fraction is the same as multiplication by its reciprocal, an insight that simplifies many complex fraction problems.

Simplifying Fractions

The final step is often to simplify the fraction, if possible. Simplifying a fraction means reducing it to its simplest form, with the smallest possible whole numbers in the numerator and denominator. This is done by finding the greatest common divisor (GCD) for the numerator and denominator and dividing both by that number.

For instance, in the example \(\frac{130}{54}\), the GCD of 130 and 54 is 2. When both the numerator and the denominator are divided by 2, the fraction simplifies to \(\frac{65}{27}\). Simplifying is an excellent opportunity to reinforce prime factorization and the search for common factors. Simplifying not only makes the fraction more understandable but is often required in math problems, tests, or real-life applications.