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Chapter 4: Problem 56

Determine whether each pair of fractions is equivalent. \(\frac{4}{10}\) and \(\frac{6}{15}\)

Short Answer

Expert verified
Yes, the fractions are equivalent.

Step by step solution

01

Simplify the First Fraction

To determine if the fractions are equivalent, simplify each fraction. Start by simplifying \( \frac{4}{10} \). Find the greatest common divisor (GCD) for 4 and 10, which is 2. Divide both the numerator and the denominator by 2: \( \frac{4}{10} = \frac{4 \div 2}{10 \div 2} = \frac{2}{5} \). So, \( \frac{4}{10} \) simplifies to \( \frac{2}{5} \).
02

Simplify the Second Fraction

Now simplify \( \frac{6}{15} \). The GCD for 6 and 15 is 3. Divide both the numerator and the denominator by 3: \( \frac{6}{15} = \frac{6 \div 3}{15 \div 3} = \frac{2}{5} \). So, \( \frac{6}{15} \) simplifies to \( \frac{2}{5} \).
03

Compare the Simplified Fractions

Both fractions simplify to the same value, \( \frac{2}{5} \). This indicates that the original fractions \( \frac{4}{10} \) and \( \frac{6}{15} \) are equivalent.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Greatest Common Divisor
The greatest common divisor (GCD) is a useful mathematical concept, especially when dealing with fractions. It refers to the largest positive integer that divides both numbers without any remainder.
To find the GCD of two numbers:
  • List the factors of each number.
  • Identify the largest factor that both numbers share.
For example, to find the GCD for 4 and 10, list their factors:
  • Factors of 4: 1, 2, 4
  • Factors of 10: 1, 2, 5, 10
The common factors are 1 and 2. Therefore, the GCD is 2.
Knowing the GCD is essential in simplifying fractions as it helps in reducing them to their simplest form, making calculations and comparisons easier.
Simplifying Fractions
Simplifying fractions means reducing them to their simplest form. A fraction is in its simplest form when the numerator and the denominator have no common factors other than 1.
To simplify a fraction:
  • Find the greatest common divisor (GCD) of the numerator and the denominator.
  • Divide both the numerator and the denominator by this GCD.
Take the fraction \(\frac{6}{15}\) for instance:
  • The GCD of 6 and 15 is 3.
  • Divide both by 3: \(\frac{6 \div 3}{15 \div 3} = \frac{2}{5}\)
By doing this, \(\frac{6}{15}\) simplifies to \(\frac{2}{5}\). A simpler fraction helps make further operations or comparisons clearer and more manageable.
Numerator and Denominator Comparison
Once fractions are simplified, comparing their values becomes straightforward. The equivalence of fractions depends on them reducing to the same simplest form.
When comparing fractions like \(\frac{4}{10}\) and \(\frac{6}{15}\):
  • Both simplify to \(\frac{2}{5}\).
  • Since they both reduce to the same value, these fractions are equivalent.
This comparison confirms that initially different looking fractions can indeed represent the same portion or value. Understanding this concept is crucial for solving problems involving fraction operations and ensures accurate results during mathematical computations.

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Most popular questions from this chapter

$$\text { Evaluate each expression if } x=\frac{3}{4} \text { and } y=-\frac{4}{7}$$ $$\frac{2+x}{y}$$

Write each improper fraction as a mixed number or a whole number. See Example 21. $$\frac{37}{8}$$

Each expression contains one addition, one subtraction, one multiplication, and one division. Write the operations in the order that they should be performed. Do not actually simplify. $$\frac{1}{3} \div\left(\frac{2}{3}\right)\left(\frac{4}{5}\right)-\frac{1}{4}+\frac{1}{2}$$

Solve. A student asked you to check his work below. Is it correct? If not, where is the error? \(3 \frac{2}{3} \cdot 1 \frac{1}{7} \stackrel{?}{=} 3 \frac{2}{21}\)

Solve. A heart attack patient in rehabilitation walked on a treadmill \(12 \frac{3}{4}\) miles over 4 days. How many miles is this per day on average? 1\. UNDERSTAND the problem. Reread it as many times as needed. 2\. TRANSLATE into an equation. (Fill in the blanks.) Finish with: 3\. SOLVE and 4\. INTERPRET
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