Problem 43 - Prealgebra

Question

Find the LCD for each of the following; then use the methods developed in this section to add or subtract as indicated. $$\frac{25}{84}+\frac{41}{90}$$

Step-by-Step Solution

Answer

The sum is \(\frac{949}{1260}\).

Step 1: Identify Denominators

The denominators are 84 and 90. We need to find the least common denominator (LCD) to add the fractions.

Step 2: Prime Factorize Denominators

Prime factorize each denominator: \(84 = 2^2 \times 3 \times 7\) and \(90 = 2 \times 3^2 \times 5\).

Step 3: Find the LCD

Determine the LCD by finding the highest power of each prime number that appears in any of the factorizations. For \(2\), use \(2^2\); for \(3\), use \(3^2\); for \(5\), use \(5^1\); for \(7\), use \(7^1\). The LCD is \(2^2 \times 3^2 \times 5 \times 7 = 1260\).

Step 4: Convert Fractions to Have the Same Denominator

Convert each fraction to have the denominator 1260. For \(\frac{25}{84}\), multiply the numerator and denominator by \(\frac{15}{15}\): \(\frac{25 \times 15}{84 \times 15} = \frac{375}{1260}\). For \(\frac{41}{90}\), multiply the numerator and denominator by \(\frac{14}{14}\): \(\frac{41 \times 14}{90 \times 14} = \frac{574}{1260}\).

Step 5: Add the Fractions

With the fractions having a common denominator, add them: \(\frac{375}{1260} + \frac{574}{1260} = \frac{949}{1260}\).

Step 6: Simplify the Resulting Fraction

Check if \(\frac{949}{1260}\) can be simplified further. Since 949 is a prime number and doesn't divide 1260, the fraction is already in the simplest form.

Key Concepts

Prime Factorization

Prime factorization is a method used to express a number as a product of its prime numbers. Understanding this process is crucial, especially when finding the least common denominator (LCD) in fractions. You start by breaking down each number into the smallest building blocks—prime numbers. A prime number is any number greater than 1 that only has two divisors: 1 and itself.
To do this, find the smallest prime number that divides into your number and continue this process with the quotient until you can't go any further. For example, when we factorize 84, we start with the smallest prime that divides 84, which is 2, and continue the process:

• Start with 84: divided by 2 gives 42
• 42: divided by 2 gives 21
• 21: divided by 3 gives 7
• 7 is a prime number, so stop

Thus, 84 = 2^2 × 3 × 7. Doing the same for 90 gives us 90 = 2 × 3^2 × 5. Understanding prime factorization helps immensely in determining the greatest and least common multiples, making it easier to work with fractions.

Fractions Addition

Adding fractions can be tricky if they don't share a common denominator. Before adding, both fractions need to be converted to have a shared denominator. Here's how you do it:

• Find the least common denominator (LCD) by using the highest powers of all the primes involved.
• For example, with denominators 84 and 90, the LCD is found by using the prime factors: 2, 3, 5, 7.
• The LCD here would be 2^2 × 3^2 × 5 × 7 = 1260.
• Convert each fraction to have this LCD by multiplying both numerator and denominator with necessary factors.

Once they have the same denominator, their numerators can be added directly. So, \[ \frac{25}{84} + \frac{41}{90} = \frac{375}{1260} + \frac{574}{1260} = \frac{949}{1260} \].
Fractions addition requires patience in setting up for the operation, but the arithmetic becomes straightforward once commonality is achieved.

Simplifying Fractions

After performing operations such as addition or subtraction, it is important to check if the resulting fraction can be simplified. Simplifying a fraction means reducing it to its most basic form, so the numerator and denominator have no common factors other than 1.
For example, consider the fraction \( \frac{949}{1260} \), which results from adding two fractions. Check the greatest common factor (GCF) of the numerator and the denominator. If no other number divides them, the fraction is already in its simplest form. In this case, 949 is a prime number, meaning it cannot be divided further (besides by 1 and 949), and since it doesn't divide 1260, it's already simplified.
Simplifying fractions not only makes them easier to read and understand, it often helps in identifying equivalent values and in future calculations.