Question

Evaluate each expression. $$2 \frac{1}{3} \cdot 1 \frac{5}{6}$$

Short answer

\(\frac{77}{18}\)

Step-by-step solution

Convert Mixed Numbers to Improper Fractions

First, convert each mixed number to an improper fraction. For the mixed number \(2 \frac{1}{3}\), multiply the whole number (2) by the denominator (3) and add the numerator (1) to get \(\frac{7}{3}\). Similarly, for \(1 \frac{5}{6}\), multiply the whole number (1) by the denominator (6) and add the numerator (5) to get \(\frac{11}{6}\).

Multiply the Improper Fractions

Now, multiply the fractions \(\frac{7}{3}\) and \(\frac{11}{6}\). Multiply the numerators together: \(7 \times 11 = 77\). Multiply the denominators together: \(3 \times 6 = 18\). So, the result of multiplying the improper fractions is \(\frac{77}{18}\).

Simplify the Fraction (if necessary)

Check if the fraction \(\frac{77}{18}\) can be simplified. Since 77 and 18 have no common factors other than 1, the fraction is already in its simplest form.

Convert Back to a Mixed Number (optional)

Convert \(\frac{77}{18}\) back to a mixed number if desired. Divide 77 by 18 to get 4 with a remainder of 5. Therefore, \(\frac{77}{18}\) can be written as the mixed number \(4 \frac{5}{18}\).

Key concepts

Improper Fractions Conversion

Mixed numbers make multiplication tricky, so we first convert them to improper fractions.
An improper fraction has a numerator larger than its denominator. For example, in the case of \(2\frac{1}{3}\), we multiply the whole number (2) by the denominator (3), then add the numerator (1).
This gives \(\frac{7}{3}\). Similarly, \(1\frac{5}{6}\) converts to \(\frac{11}{6}\).

• Step one: Multiply the whole number by the denominator.


• Step two: Add the numerator to this product.


• Step three: The result is the numerator of the improper fraction.

If the steps are followed correctly, you will have successfully converted your mixed numbers to improper fractions, simplifying the multiplication process.

Multiplication of Fractions

Now, let's multiply our improper fractions \(\frac{7}{3}\) and \(\frac{11}{6}\). The multiplication of fractions is straightforward.

• Multiply the numerators:\(7 \times 11 = 77\).


• Multiply the denominators: \(3 \times 6 = 18\).

The result is \(\frac{77}{18}\). Notice, we did not need common denominators for this operation.
Just a simple multiplication of both numerators and denominators will do.

Simplifying Fractions

Once you've got the fraction, you need to see if it can be simplified. Simplifying means reducing the fraction to its simplest form.
In this case, \(\frac{77}{18}\) doesn't have any common factors other than 1.
So, it is already in the simplest form.

• Step one: Determine if the numerator and denominator share any common factors.


• Step two: If there is a common factor, divide both by that number.

Since there are no common factors for 77 and 18, the fraction remains as \(\frac{77}{18}\). Easy!

Mixed Numbers

Sometimes, you might be asked to convert an improper fraction back into a mixed number.
Let's take our result \(\frac{77}{18}\) and convert it back.
Divide the numerator by the denominator: 77 ÷ 18 = 4 remainder 5.
So, \(\frac{77}{18}\) becomes 4 with a remainder of 5. You write the remainder over the original denominator.
The result is the mixed number \(4\frac{5}{18}\).

• Step one: Divide the numerator by the denominator to get the whole number part.


• Step two: The remainder becomes the new numerator.


• Step three: Keep the original denominator.

Converting back to a mixed number can make your final answer more understandable and intuitive.