Question

Find the difference. $$4 \frac{2}{3}-2 \frac{1}{5}$$

Short answer

So, the difference of \(4 \frac{2}{3}\) and \(2 \frac{1}{5}\) is \(2 \frac{7}{15}\).

Step-by-step solution

Convert to Improper Fractions

First, convert both of the mixed numbers into improper fractions to simplify calculation. An improper fraction is a fraction whose numerator is greater than or equal to its denominator. For \(4 \frac{2}{3}\), multiply 4 (the whole number) with 3 (the denominator of fraction) and add 2 (the numerator of fraction). This equals to \(4*3+2=14\). Thus, \(4 \frac{2}{3}\) is equivalent to \(\frac{14}{3}\). Similarly, for \(2 \frac{1}{5}\), multiply 2 (the whole number) with 5 (the denominator of fraction) and add 1 (the numerator of fraction). This equals to \(2*5+1=11\). So, \(2 \frac{1}{5}\) is equivalent to \(\frac{11}{5}\). So, the problem becomes \(\frac{14}{3} - \frac{11}{5}\).

Simplify by finding a common denominator

Since the two denominators, 3 and 5, are not the same, a common denominator needs to be found in order to subtract the two fractions. The least common denominator (LCD) of 3 and 5 is 15. Multiply the first fraction \(\frac{14}{3}\) by \(\frac{5}{5}\) and the second fraction \(\frac{11}{5}\) by \(\frac{3}{3}\). Now the two fractions become \(\frac{70}{15}\) and \(\frac{33}{15}\) respectively.

Subtract the two simplifed fractions

Now that the two fractions have the same denominator (15), they can be easily subtracted: \(\frac{70}{15} - \frac{33}{15} = \frac{37}{15}\).

Convert to Mixed Number

Since the numerator of the resulting fraction (\(\frac{37}{15}\)) is larger than the denominator, it can be converted back to a mixed number. By dividing 37 by 15, the quotient is 2 and the remainder is 7. This makes the mixed number: \(2 \frac{7}{15}\).

Key concepts

Improper Fractions

Understanding improper fractions is essential when working with mixed numbers. An improper fraction is a type of fraction where the numerator (the top number) is equal to or greater than the denominator (the bottom number). This often occurs during calculations when fractions are manipulated, such as when moving from a mixed number to an equivalent fraction form.

To convert a mixed number to an improper fraction, you multiply the whole number by the denominator of the fractional part, then add the numerator of the fractional part to that product. For example, with the mixed number \(4 \frac{2}{3}\), you multiply 4 by 3 and add 2, resulting in the improper fraction \(\frac{14}{3}\).

Students might confuse this step, so remember to emphasize the correct order: multiply first, then add. This process creates a single fraction that represents the mixed number, simplifying further operations like addition, subtraction, or comparison.

• Multiply the whole number by the denominator.
• Add the numerator.
• Write the sum as the new numerator over the original denominator.

Common Denominator

To subtract fractions effectively, they must share the same denominator, referred to as the common denominator. The challenge lies in finding the least common denominator (LCD), which is the smallest number that both denominators can divide into without leaving a remainder.

For instance, with fractions \(\frac{14}{3}\) and \(\frac{11}{5}\), the denominators are 3 and 5. You can find the LCD by looking for the smallest number that both 3 and 5 will divide into evenly. In this case, it is 15. Thus, to subtract the given fractions, we convert them to have the same denominator of 15.

Steps to Find a Common Denominator:

• Identify the denominators of the fractions to be subtracted.
• Find the least common multiple (LCM) of these denominators.
• Adjust each fraction to make the denominator equal to the LCM by multiplying both the numerator and the denominator by the appropriate factors.

Ensuring each fraction has this common denominator enables direct subtraction of the numerators, facilitating the computation process.

Mixed Number Subtraction

Subtracting mixed numbers can be tricky, but with a proper understanding of the underlying concepts, it simplifies the process. The initial approach involves converting any mixed numbers into improper fractions. Once you've accomplished this, the next step is to ensure both improper fractions have a common denominator, as we've discussed in the previous sections.

After aligning to the common denominator, subtract the numerators while keeping the denominator intact. The resulting fraction may often be an improper fraction, which you may need to convert back into a mixed number. This is done by dividing the numerator by the denominator to find the whole number, and then writing the remainder over the original denominator.

Key Steps for Mixed Number Subtraction:

• Convert mixed numbers to improper fractions.
• Find a common denominator and adjust fractions accordingly.
• Subtract the numerators and carry the common denominator.
• Convert the result back to a mixed number if necessary.

Using these clear steps can help alleviate confusion, ensuring that students can manage mixed number subtraction with confidence and accuracy.