Problem 47 - Basic College Mathematics

Question

Write the answer using fraction notation. $$ \frac{341}{517} \cdot \frac{209}{349} $$

Step-by-Step Solution

Answer

\( \frac{19}{47} \)

Step 1: Write out the fractions

Given fractions are: \( \frac{341}{517} \boldcdot \frac{209}{349} \)

Step 2: Simplify each fraction if possible

Check if the numerators and denominators of each fraction can be simplified: For \( \frac{341}{517} \): Both 341 and 517 are divisible by 11: \( 341 ÷ 11 = 31 \) and \( 517 ÷ 11 = 47 \) So, \( \frac{341}{517} = \frac{31}{47} \) For \( \frac{209}{349} \): Both 209 and 349 are divisible by 11: \( 209 ÷ 11 = 19 \) and \( 349 ÷ 11 = 31 \) So, \( \frac{209}{349} = \frac{19}{31} \)

Step 3: Multiply the simplified fractions

After simplification, multiply the fractions: \( \frac{31}{47} \boldcdot \frac{19}{31} \)

Step 4: Simplify the resulting fraction

Cancel out common terms in the numerator and denominator: \( \frac{31 \boldcdot 19}{47 \boldcdot 31} = \frac{19}{47} \)

Key Concepts

fraction multiplication

Fraction multiplication is an essential concept in math. It involves multiplying the numerators together and the denominators together. For instance, if you have two fractions \(\frac{a}{b} \times \frac{c}{d}\), the multiplication results in \(\frac{a \cdot c}{b \cdot d}\). Always remember to multiply across the top and across the bottom.
In our example, we are given \(\frac{341}{517} \times \frac{209}{349}\). We first simplify these fractions for ease, but initially, we would still follow the rule of multiplying numerators and denominators.
By multiplying these fractions, we follow: \(\frac{341 \cdot 209}{517 \cdot 349}\). This step ensures all parts of each fraction are appropriately combined according to the multiplication rules.

greatest common divisor

Finding the greatest common divisor (GCD) is crucial in simplifying fractions. The GCD is the highest number that can evenly divide both the numerator and the denominator.
To simplify a fraction, you divide both its numerator and denominator by their GCD. For \(\frac{341}{517}\), both 341 and 517 are divisible by 11, making 11 their GCD. So, \(\frac{341 \div 11}{517 \div 11} = \frac{31}{47}\). Similarly, for \(\frac{209}{349}\), the GCD is again 11, thus simplifying it to \(\frac{19}{31}\).
We can use the Euclidean algorithm to find the GCD efficiently. This involves repeatedly subtracting the smaller number from the larger until a common divisor is found. Regular practice will make finding the GCD feel more natural, leading to easier simplifications.

fraction reduction

Fraction reduction involves simplifying a fraction to its simplest form, where the numerator and denominator share no common factors other than 1. You achieve this by dividing both the numerator and denominator by their greatest common divisor (GCD).
In our step-by-step solution, we first identified the GCD for both fractions. Then we divided both terms of each fraction by the GCD:
For \(\frac{341}{517}\), dividing by 11 gives \(\frac{341 \div 11}{517 \div 11} = \frac{31}{47}\).
For \(\frac{209}{349}\), dividing by 11 gives \(\frac{209 \div 11}{349 \div 11} = \frac{19}{31}\).
Once fractions are simplified, multiplication becomes easier: \(\frac{31}{47} \cdot \frac{19}{31}\). Common factors between the numerators and denominators can then be canceled out, leaving the final simplified fraction: \(\frac{19}{47}\).
Always ensure you simplify intermediate results where possible to make calculations easier.