Problem 94 - Algebra 1: Concepts and Skills

Question

Evaluate the expression. Write the answer as a fraction or mixed number in simplest form. \(\frac{7}{3}+\frac{1}{5}-\frac{2}{15}\)

Step-by-Step Solution

Answer

The simplified form of the expression \(\frac{7}{3}+\frac{1}{5}-\frac{2}{15}\) is \(2\frac{2}{5}\).

Step 1: Find the Common Denominator

The denominators are 3, 5, and 15. Their least common multiple is 15. So, the common denominator we shall use here is 15.

Step 2: Convert Fractions to Equivalent Fractions with Common Denominator

\(\frac{7}{3}\) can be converted to \(\frac{35}{15}\) by multiplying both the numerator and the denominator by 5. \(\frac{1}{5}\) can be converted to \(\frac{3}{15}\) by multiplying both the numerator and the denominator by 3. \(\frac{2}{15}\) already has 15 as the denominator so it doesn't need to be changed.

Step 3: Add and Subtract the Converted Fractions

Now add and subtract the fractions: \(\frac{35}{15} + \frac{3}{15} - \frac{2}{15} = \frac{36}{15}\)

Step 4: Simplify the Result to its Lowest Terms

Simplify \(\frac{36}{15}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 3. This gives us \(\frac{12}{5}\). Furthermore, \(\frac{12}{5}\) can be written as a mixed number \(2\frac{2}{5}\).

Key Concepts

Least Common Denominator

When we encounter fraction addition or subtraction, having the same denominator is crucial. That common number is known as the "least common denominator" (LCD). It is the smallest number that can be divided evenly by each of the denominators involved in the fractions.
The LCD helps standardize the fractions so they can be easily added or subtracted without altering their respective values. To find the LCD, look for the least common multiple (LCM) of the denominators.

For example, in

• \(rac{7}{3}\), \(rac{1}{5}\), and \(rac{2}{15}\),

The denominators are 3, 5, and 15. Their LCM is 15 because:

• 15 is a multiple of 3 (since 3 x 5 = 15)
• 15 is a multiple of 5 (since 5 x 3 = 15)
• And obviously, 15 is already a multiple of 15

Once we find the least common denominator, we use it to adjust each fraction, making it possible to perform the addition or subtraction seamlessly.

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest form. This means both the numerator and denominator are divided by their greatest common factor (GCF) until they cannot be reduced any further.
When a fraction is in its simplest form, it is easier to understand and often more useful in subsequent calculations. It represents the same value with the least complexity.

For instance, take

• \(rac{36}{15}\).

The GCF of 36 and 15 is 3. By dividing both the numerator and denominator by 3, we get:

• \(rac{36}{15} \div \frac{3}{3} = \frac{12}{5}\)

Now, \(rac{12}{5}\) is in its simplest form. Simplified fractions are more straightforward to interpret, making them essential in mathematics.

Mixed Numbers

A mixed number is a great way to express improper fractions, which have numerators larger than their denominators, in a more understandable form. It combines a whole number with a proper fraction, which helps in visualizing the actual value.
For instance,

• In our example, after simplifying to \(rac{12}{5}\), we can convert this improper fraction to a mixed number.

Here's how:

• Divide the numerator (12) by the denominator (5). The quotient is 2, which is the whole number part.
• The remainder is 2, which becomes the numerator of the fractional part.
• Place the remainder over the original denominator: \(2 \frac{2}{5}\).

So \(\frac{12}{5}\) can be represented as a mixed number: \(2\frac{2}{5}\).
This form is more intuitive, as it clearly identifies how many whole units are present along with the fraction that remains.