Problem 2 - Basic Engineering Mathematics

Question

Find the value of \(3 \frac{2}{3}-2 \frac{1}{6}\)

Step-by-Step Solution

Answer

The value is \(\frac{3}{2}\).

Step 1: Convert Mixed Numbers to Improper Fractions

Start by converting mixed numbers into improper fractions. A mixed number consists of a whole number and a fraction. To convert a mixed number to an improper fraction, multiply the whole number by the denominator and add it to the numerator. For \(3 \frac{2}{3}\), multiply \(3 \times 3 = 9\) and add \(2\) to get \(11\) as the new numerator, making it \(\frac{11}{3}\). For \(2 \frac{1}{6}\), multiply \(2 \times 6 = 12\) and add \(1\) to get \(13\) as the new numerator, resulting in \(\frac{13}{6}\).

Step 2: Find a Common Denominator

To subtract fractions, they must have a common denominator. Here, the denominators are \(3\) and \(6\). The least common multiple of \(3\) and \(6\) is \(6\). Now, convert \(\frac{11}{3}\) to a fraction with a denominator of \(6\). Multiply both the numerator and the denominator by \(2\) to get \(\frac{22}{6}\).

Step 3: Subtract the Fractions

Now that both fractions have a common denominator, you can subtract them directly. Subtract \(\frac{13}{6}\) from \(\frac{22}{6}\). This is done by subtracting the numerators: \(22 - 13 = 9\). The denominator remains \(6\), resulting in \(\frac{9}{6}\).

Step 4: Simplify the Resulting Fraction

Lastly, simplify \(\frac{9}{6}\). Find the greatest common divisor of \(9\) and \(6\), which is \(3\). Divide both the numerator and the denominator by \(3\): \(9 \div 3 = 3\) and \(6 \div 3 = 2\). This gives the simplified fraction \(\frac{3}{2}\).

Key Concepts

Mixed Numbers

A mixed number combines a whole number with a fraction. In this form, it indicates how many whole parts there are in addition to a fractional part.
It's a convenient way to represent numbers when a whole part is supplemented by a leftover fraction.

• For example, in the mixed number \(3 \frac{2}{3}\), "3" is the whole part, and "\(\frac{2}{3}\)" is the fraction.
• This form makes it easy to visualize quantities which are greater than a whole but less than another whole number.

Converting mixed numbers to improper fractions is the first step in many math problems, enabling easier operations such as addition or subtraction.

Improper Fractions

Improper fractions have numerators larger than or equal to their denominators.
Transforming mixed numbers into improper fractions allows for straightforward arithmetic operations.

• For instance, the mixed number \(3 \frac{2}{3}\) becomes \(\frac{11}{3}\) when converted to an improper fraction.
• This conversion helps because it treats the whole and fractional parts as parts of a single fraction, facilitating easy calculations.

The process involves multiplying the whole number by the denominator and then adding the numerator. Convert the entire number into a fractional expression with a single numerator, making further math operations more straightforward.

Common Denominator

When dealing with the addition or subtraction of fractions, finding a common denominator is essential.
This is because fractions must share the same denominator to allow numerators to be directly added or subtracted.

• Determine the lowest common denominator using the least common multiple (LCM) of the original denominators.
• For example, when dealing with \(\frac{11}{3}\) and \(\frac{13}{6}\), the common denominator is "6".
• Adjust fractions to this new denominator by appropriately multiplying both the numerator and denominator to maintain their value.

Using a common denominator aligns the fractions on a shared scale, making them directly comparable and operable in addition or subtraction tasks.

Simplifying Fractions

After arithmetic operations, simplifying fractions provides the answer in its most reduced form.
Simplification involves finding the greatest common divisor (GCD) of the numerator and denominator to make the fraction as simple as possible.

• For instance, with the fraction \(\frac{9}{6}\), the GCD is "3".
• Dividing both the numerator and the denominator by "3" simplifies \(\frac{9}{6}\) to the reduced fraction \(\frac{3}{2}\).

This process not only provides a cleaner answer but also makes further use of the fraction more effortless in future calculations or comparisons. Always aim to simplify at the end of any calculations to ensure results are clear and concise.