[tex]$\frac{2 \times 3 \times 10}{6 \times 5 \times 4} \quad \frac{8 \times 12 \times 15}{4 \times 24 \times 40} \quad \frac{18 \times 10 \times 14}{35 \times 27 \times 15}$[/tex]
Answer (2)
The simplified forms of the given fractions are 1, 8 3 , and 45 8 . The simplification involved finding and cancelling common factors. Each fraction was carefully reduced to its simplest form step-by-step.
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Simplify the first fraction 6 × 5 × 4 2 × 3 × 10 to 2 1 .
Simplify the second fraction 4 × 24 × 40 8 × 12 × 15 to 8 3 .
Simplify the third fraction 35 × 27 × 15 18 × 10 × 14 to 45 8 .
The simplified fractions are 2 1 , 8 3 , 45 8 .
Explanation
Understanding the Problem We are given three fractions to simplify. Our goal is to reduce each fraction to its simplest form by canceling out common factors in the numerator and the denominator.
Simplifying the First Fraction Let's simplify the first fraction: 6 × 5 × 4 2 × 3 × 10 .
We can rewrite the denominator as 6 × 5 × 4 = ( 2 × 3 ) × 5 × ( 2 × 2 ) .
So the fraction becomes 2 × 3 × 5 × 2 × 2 2 × 3 × 10 .
Now we cancel out the common factors: 2 × 3 × 5 × 2 × 2 2 × 3 × 10 = 5 × 2 × 2 10 = 20 10 = 2 1 .
Simplifying the Second Fraction Next, let's simplify the second fraction: 4 × 24 × 40 8 × 12 × 15 .
We can rewrite the numerator and denominator with their prime factorizations: 2 2 × ( 2 3 × 3 ) × ( 2 3 × 5 ) 2 3 × ( 2 2 × 3 ) × ( 3 × 5 ) = 2 8 × 3 × 5 2 5 × 3 2 × 5 .
Now we cancel out the common factors: 2 8 × 3 × 5 2 5 × 3 2 × 5 = 2 3 3 = 8 3 .
Simplifying the Third Fraction Finally, let's simplify the third fraction: 35 × 27 × 15 18 × 10 × 14 .
We can rewrite the numerator and denominator with their prime factorizations: ( 5 × 7 ) × ( 3 3 ) × ( 3 × 5 ) ( 2 × 3 2 ) × ( 2 × 5 ) × ( 2 × 7 ) = 3 4 × 5 2 × 7 2 3 × 3 2 × 5 × 7 .
Now we cancel out the common factors: 3 4 × 5 2 × 7 2 3 × 3 2 × 5 × 7 = 3 2 × 5 2 3 = 45 8 .
Final Answer Therefore, the simplified fractions are 2 1 , 8 3 , and 45 8 .
Examples
Simplifying fractions is a fundamental skill in mathematics with numerous real-world applications. For instance, when baking, you might need to adjust ingredient quantities based on a recipe that serves more people than you intend to. Simplifying fractions allows you to easily determine the correct proportions of each ingredient. Similarly, in construction or engineering, simplifying ratios and fractions is crucial for scaling blueprints or calculating material requirements accurately. These skills ensure precision and efficiency in various practical tasks.
