Question

Evaluate each expression. $$2 \frac{3}{4} \div \frac{3}{8}$$

Short answer

The simplified result is \(\frac{22}{3}\).

Step-by-step solution

- Convert the Mixed Number to an Improper Fraction

First, convert the mixed number \(2 \frac{3}{4}\) into an improper fraction. Multiply the whole number part by the denominator of the fraction and add the numerator. So, we have:\[2 \times 4 + 3 = 8 + 3 = 11\]Thus, \(2 \frac{3}{4}\) becomes \(\frac{11}{4}\).

- Write the Problem with Improper Fraction

After converting, our problem now looks like this:\[\frac{11}{4} \div \frac{3}{8}\]

- Apply the Division Rule for Fractions

To divide by a fraction, multiply by its reciprocal. The reciprocal of \(\frac{3}{8}\) is \(\frac{8}{3}\). So our problem becomes:\[\frac{11}{4} \times \frac{8}{3}\]

- Multiply the Fractions

Multiply the numerators together and the denominators together:\[\frac{11 \times 8}{4 \times 3} = \frac{88}{12}\]

- Simplify the Fraction

To simplify \(\frac{88}{12}\), find the greatest common divisor (GCD) of 88 and 12, which is 4. Divide both the numerator and the denominator by 4:\[\frac{88 \div 4}{12 \div 4} = \frac{22}{3}\]

Key concepts

improper fractions

An improper fraction is a fraction where the numerator is greater than or equal to the denominator.
For instance, in the problem we solved, after converting the mixed number 2\(\frac{3}{4}\) to an improper fraction, we obtained \(\frac{11}{4}\).
This is because 11 (the numerator) is greater than 4 (the denominator).
Improper fractions are useful in mathematical operations because they simplify many processes.
To convert a mixed number to an improper fraction, perform these steps:

• Multiply the whole number by the denominator.
• Add the result to the numerator.
• Write the final sum above the original denominator.

Let's see it in action with 2\(\frac{3}{4}\):
First, multiply 2 by 4 to get 8.
Next, add 3 to 8, resulting in 11.
Finally, place 11 over 4 to form \(\frac{11}{4}\).
This new fraction is easier to work with during division.

reciprocal

The reciprocal of a fraction is obtained by swapping its numerator and denominator.
If you have a fraction \( \frac{a}{b} \), its reciprocal is \( \frac{b}{a} \).
This is crucial when dividing fractions.
Instead of dividing by a fraction, multiply by its reciprocal.
In our example, we needed to divide by \(\frac{3}{8}\).
So, we found its reciprocal, \(\frac{8}{3}\), and changed the division to multiplication.
Why is this helpful? Because multiplying fractions is more straightforward than dividing them.
Remember these steps to find the reciprocal:

• Take the denominator of the fraction and make it the numerator.
• Take the numerator of the fraction and make it the denominator.

In our exercise, the reciprocal of \(\frac{3}{8}\) is \(\frac{8}{3}\).
This makes the math simpler.

simplifying fractions

Simplifying fractions makes them easier to understand and work with.
To simplify a fraction, divide the numerator and the denominator by their Greatest Common Divisor (GCD).
In our example, we were left with \(\frac{88}{12}\) after multiplying the numerators and denominators.
We found the GCD of 88 and 12, which is 4.
Then we divided both numerator and denominator by 4, simplifying \(\frac{88}{12}\) to \(\frac{22}{3}\).
Steps to simplify a fraction:

• Find the GCD of the numerator and denominator.
• Divide both by this GCD.

But why simplify? Simplifying provides a cleaner, more easily understandable fraction.
Imagine trying to work with fractions like \(\frac{176}{24}\) instead of a simpler \(\frac{22}{3}\).
The latter is easier to recognize and use in further calculations.

mixed numbers

A mixed number combines a whole number with a fraction.
For example, 2 \(\frac{3}{4}\) means you have 2 whole units and an additional \(\frac{3}{4}\) of another unit.
Mixed numbers are useful for representing quantities that are more than a whole but not quite another whole.
To work with mixed numbers in operations like division, it's often best to convert them to improper fractions first.
Let's see why.
Mixed numbers can complicate multiplication and division.
So, converting 2\(\frac{3}{4}\) to \(\frac{11}{4}\) simplifies these operations.
This way, you can directly apply arithmetic operations to fractions.
To convert back, divide the numerator by the denominator.

• The quotient becomes the whole number.
• The remainder becomes the new numerator.
• The original denominator remains the same.

So for \(\frac{22}{3}\), since 22 divided by 3 is 7 with a remainder of 1, it converts back to 7\(\frac{1}{3}\).
This makes the concept more intuitive.