Problem 62 - Elementary Algebra

Question

A building lot is \(\frac{1}{6}\) acre. If it is combined with an adjoining lot that is \(\frac{5}{32}\) acre, find the total area of the combined lots.

Step-by-Step Solution

Answer

The total area is \(\frac{31}{96}\) acre.

Step 1: Find a Common Denominator

To add the fractions \(\frac{1}{6}\) and \(\frac{5}{32}\), we first need a common denominator. The least common multiple (LCM) of 6 and 32 is 96.

Step 2: Convert Fractions to Common Denominator

Convert each fraction to an equivalent fraction with a denominator of 96. For \(\frac{1}{6}\), multiply both the numerator and the denominator by 16 to get \(\frac{16}{96}\). For \(\frac{5}{32}\), multiply both the numerator and the denominator by 3 to get \(\frac{15}{96}\).

Step 3: Add the Fractions

Add the two fractions with the common denominator: \(\frac{16}{96} + \frac{15}{96} = \frac{31}{96}\).

Step 4: Simplify if Possible

Check if the fraction \(\frac{31}{96}\) can be simplified. In this case, 31 is a prime number and does not divide 96 evenly, so this is the simplest form.

Key Concepts

least common multiple

When adding fractions, we first need a common denominator. The least common multiple (LCM) of two numbers is the smallest number that both numbers divide into without leaving a remainder. Finding the LCM ensures that both fractions have the same denominator, making them easier to add.

• To find the LCM of two numbers, list the multiples of each number.
• For 6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, 96
• For 32: 32, 64, 96

The smallest common multiple of 6 and 32 is 96.
This is why we use 96 as our common denominator in the original exercise.

equivalent fractions

To add fractions with different denominators, we convert them into equivalent fractions with a common denominator. Equivalent fractions are different fractions that represent the same number. For example, 1/2 is the same as 2/4 or 3/6.
To convert a fraction to an equivalent fraction:

• Multiply both the numerator and the denominator by the same number.
• Example: To convert \(\frac{1}{6}\) to have a denominator of 96, multiply both the numerator and denominator by 16: \(\frac{1 \times 16}{6 \times 16} = \frac{16}{96}\).
• Similarly, convert \(\frac{5}{32}\) to have a denominator of 96 by multiplying both the numerator and denominator by 3: \(\frac{5 \times 3}{32 \times 3} = \frac{15}{96}\).

Now, both fractions have the same denominator, making them easier to add.

simplifying fractions

Simplifying fractions makes them easier to understand and compare. A fraction is simplified when its numerator and denominator have no common factors other than 1. The process involves dividing both the numerator and the denominator by their greatest common divisor (GCD).
However, in the original exercise, the fraction \(\frac{31}{96}\) was already in its simplest form.

• 31 is a prime number, meaning it has no divisors other than 1 and itself.
• Since 31 doesn’t divide evenly into 96, we know \(\frac{31}{96}\) is already simplified.

Recognizing when a fraction is already in simplest form is an important skill in working with fractions.