Problem 44 - Fundamentals of Mathematics

Question

Find the value of \(\frac{\frac{3}{10}+\frac{4}{12}}{\frac{19}{20}}\).

Step-by-Step Solution

Answer

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

Step 1: Simplify the Fractions in the Numerator

First, simplify each fraction in the numerator. For \(\frac{4}{12}\), divide both numerator and denominator by 4 to get \(\frac{1}{3}\). Thus, the numerator becomes \(\frac{3}{10} + \frac{1}{3}\).

Step 2: Find a Common Denominator for the Numerator

The common denominator for \(\frac{3}{10}\) and \(\frac{1}{3}\) is 30.Convert each fraction:- \(\frac{3}{10} = \frac{3 \times 3}{10 \times 3} = \frac{9}{30}\)- \(\frac{1}{3} = \frac{1 \times 10}{3 \times 10} = \frac{10}{30}\)

Step 3: Add the Fractions in the Numerator

Add \(\frac{9}{30} + \frac{10}{30}\) to get \(\frac{19}{30}\).

Step 4: Write the Entire Expression with Fractions

Now the expression becomes \(\frac{\frac{19}{30}}{\frac{19}{20}}\).

Step 5: Divide the Fractions

To divide by a fraction, multiply by its reciprocal. So, \(\frac{19}{30} \div \frac{19}{20}\) becomes \(\frac{19}{30} \times \frac{20}{19}\).

Step 6: Simplify the Multiplication

Multiply the numerators and the denominators:\(\frac{19 \times 20}{30 \times 19} = \frac{380}{570}\).The 19s cancel each other out, simplifying to \(\frac{20}{30}\).

Step 7: Simplify the Resulting Fraction

Simplify \(\frac{20}{30}\) by dividing the numerator and the denominator by their greatest common divisor, which is 10.This gives \(\frac{2}{3}\).

Key Concepts

Common Denominator

When working with fractions, especially when adding or subtracting them, finding a common denominator is essential. This step ensures the fractions are expressed with the same denominator, making it possible to perform arithmetic operations directly on the numerators. In our exercise, we have the fractions \(\frac{3}{10}\) and \(\frac{1}{3}\). To add these, we need to find a common denominator.

• Identify the least common multiple (LCM) of the denominators 10 and 3. The LCM here is 30.
• Convert each fraction to have this common denominator.

For \(\frac{3}{10}\), multiply both the numerator and the denominator by 3 to get \(\frac{9}{30}\). For \(\frac{1}{3}\), multiply by 10 to obtain \(\frac{10}{30}\). Now, both fractions have a denominator of 30, allowing you to add the numerators to get \(\frac{19}{30}\). This process is vital for simplifying and solving fraction problems smoothly.

Reciprocal Multiplication

Reciprocal multiplication is used when dividing fractions. Instead of performing division directly, you multiply by the reciprocal of the divisor. This simplifies calculations and ensures accurate results. In the exercise, the expression changes from \(\frac{\frac{19}{30}}{\frac{19}{20}}\) to multiplication by the reciprocal.Here’s how it's done:

• The reciprocal of \(\frac{19}{20}\) is \(\frac{20}{19}\).
• Replace the division with multiplication: \(\frac{19}{30} \times \frac{20}{19}\).

Notice how the 19s in the numerator and denominator cancel each other out. You’re left with \(\frac{20}{30}\). This simplification step is key to solving the problem effectively, leading us to a simplified result.

Greatest Common Divisor

The greatest common divisor (GCD) is a helpful tool for simplifying fractions after operations like addition or multiplication. It's the largest number that can divide both the numerator and the denominator without leaving a remainder, reducing the fraction to its simplest form.In our problem, we ended with the fraction \(\frac{20}{30}\). To simplify it:

• Calculate the GCD of 20 and 30, which is 10.
• Divide both the numerator (20) and the denominator (30) by their GCD.

So, \(\frac{20}{10}\) over \(\frac{30}{10}\) simplifies to \(\frac{2}{3}\). Employing the GCD to simplify final results is crucial. It transforms complex fractions into something much more understandable, ensuring a tidy and comprehensible final answer.