Fill In The Blank To Make The Two Fractions Equivalent.${ \frac{\square}{24} = \frac{5}{8} }$

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Introduction

Fractions are a fundamental concept in mathematics, and understanding how to make them equivalent is crucial for solving various mathematical problems. In this article, we will explore the concept of equivalent fractions and provide a step-by-step guide on how to fill in the blank to make two fractions equivalent.

What are Equivalent Fractions?

Equivalent fractions are fractions that have the same value, but with different numerators and denominators. For example, the fractions 1/2 and 2/4 are equivalent because they both represent the same value. In this article, we will focus on making two fractions equivalent by filling in the blank.

The Concept of Equivalent Fractions

To make two fractions equivalent, we need to find a common denominator. The common denominator is the smallest multiple of both denominators. Once we have the common denominator, we can multiply the numerator and denominator of each fraction by the same number to make them equivalent.

Filling in the Blank to Make Two Fractions Equivalent

Let's consider the following problem:

{ \frac{\square}{24} = \frac{5}{8} \}

To fill in the blank, we need to find a number that, when multiplied by 24, will give us a product that is equivalent to 5 times 8. In other words, we need to find a number that will make the two fractions equivalent.

Step 1: Find the Least Common Multiple (LCM)

The first step is to find the least common multiple (LCM) of 24 and 8. The LCM is the smallest multiple of both numbers. To find the LCM, we can list the multiples of each number and find the smallest number that appears in both lists.

Multiples of 24:

  • 24
  • 48
  • 72
  • 96
  • 120

Multiples of 8:

  • 8
  • 16
  • 24
  • 32
  • 40

The smallest number that appears in both lists is 24, which is the LCM of 24 and 8.

Step 2: Multiply the Numerator and Denominator

Now that we have the LCM, we can multiply the numerator and denominator of each fraction by the same number to make them equivalent. In this case, we need to multiply the numerator and denominator of the first fraction by 3 to make them equivalent to the second fraction.

{ \frac{3 \times \square}{3 \times 24} = \frac{3 \times 5}{3 \times 8} \}

{ \frac{\square}{72} = \frac{15}{24} \}

Step 3: Simplify the Fraction

Now that we have the equivalent fractions, we can simplify the fraction by dividing both the numerator and denominator by their greatest common divisor (GCD). In this case, the GCD of 15 and 24 is 3.

{ \frac{15 \div 3}{24 \div 3} = \frac{5}{8} \}

Conclusion

In this article, we have explored the concept of equivalent fractions and provided a step-by-step guide on how to fill in the blank to make two fractions equivalent. We have seen that to make two fractions equivalent, we need to find a common denominator and multiply the numerator and denominator of each fraction by the same number. We have also seen that we can simplify the fraction by dividing both the numerator and denominator by their greatest common divisor. By following these steps, we can make two fractions equivalent and solve various mathematical problems.

Common Mistakes to Avoid

When making fractions equivalent, there are several common mistakes to avoid. These include:

  • Not finding the least common multiple (LCM) of the denominators
  • Not multiplying the numerator and denominator of each fraction by the same number
  • Not simplifying the fraction by dividing both the numerator and denominator by their greatest common divisor

Real-World Applications

Making fractions equivalent has several real-world applications. For example, in cooking, we often need to convert between different units of measurement, such as cups and ounces. By making fractions equivalent, we can convert between these units and ensure that our recipes are accurate.

Conclusion

Q: What is the purpose of making fractions equivalent?

A: The purpose of making fractions equivalent is to simplify complex fractions and make them easier to work with. Equivalent fractions have the same value, but with different numerators and denominators.

Q: How do I find the least common multiple (LCM) of two numbers?

A: To find the LCM of two numbers, you can list the multiples of each number and find the smallest number that appears in both lists. Alternatively, you can use the following formula:

LCM(a, b) = (a × b) / GCD(a, b)

where GCD(a, b) is the greatest common divisor of a and b.

Q: What is the greatest common divisor (GCD) of two numbers?

A: The GCD of two numbers is the largest number that divides both numbers without leaving a remainder. For example, the GCD of 12 and 18 is 6, because 6 is the largest number that divides both 12 and 18 without leaving a remainder.

Q: How do I multiply the numerator and denominator of a fraction by the same number?

A: To multiply the numerator and denominator of a fraction by the same number, you can simply multiply both numbers by that number. For example, if you want to multiply the numerator and denominator of the fraction 1/2 by 3, you would get:

(1 × 3) / (2 × 3) = 3/6

Q: Can I simplify a fraction by dividing both the numerator and denominator by their GCD?

A: Yes, you can simplify a fraction by dividing both the numerator and denominator by their GCD. This is called reducing the fraction to its simplest form. For example, if you have the fraction 6/8, you can simplify it by dividing both the numerator and denominator by their GCD, which is 2:

(6 ÷ 2) / (8 ÷ 2) = 3/4

Q: What is the difference between equivalent fractions and similar fractions?

A: Equivalent fractions have the same value, but with different numerators and denominators. Similar fractions, on the other hand, have the same numerator and denominator, but with different values. For example, the fractions 1/2 and 2/4 are equivalent, while the fractions 1/2 and 1/3 are similar.

Q: Can I make fractions equivalent by adding or subtracting the numerator and denominator?

A: No, you cannot make fractions equivalent by adding or subtracting the numerator and denominator. To make fractions equivalent, you need to find a common denominator and multiply the numerator and denominator of each fraction by the same number.

Q: What are some real-world applications of making fractions equivalent?

A: Making fractions equivalent has several real-world applications, including:

  • Converting between different units of measurement, such as cups and ounces
  • Simplifying complex fractions in cooking and recipe development
  • Making calculations easier in finance and accounting
  • Solving problems in science and engineering

Q: Can I use a calculator to make fractions equivalent?

A: Yes, you can use a calculator to make fractions equivalent. Many calculators have a fraction mode that allows you to enter fractions and perform calculations with them. Alternatively, you can use a computer program or online tool to make fractions equivalent.

Q: What are some common mistakes to avoid when making fractions equivalent?

A: Some common mistakes to avoid when making fractions equivalent include:

  • Not finding the least common multiple (LCM) of the denominators
  • Not multiplying the numerator and denominator of each fraction by the same number
  • Not simplifying the fraction by dividing both the numerator and denominator by their greatest common divisor
  • Not using a common denominator when adding or subtracting fractions

Conclusion

Making fractions equivalent is a crucial concept in mathematics that has several real-world applications. By following the steps outlined in this article, you can make two fractions equivalent and solve various mathematical problems. Remember to avoid common mistakes and use a calculator or computer program if needed.