Problem 58 - Prealgebra

Question

Write as equivalent fractions with denominator 24. a. \(\frac{3}{4}\) b. \(\frac{7}{8}\) c. \(\frac{5}{8}\) d. \(\frac{3}{8}\)

Step-by-Step Solution

Answer

a. \( \frac{18}{24} \), b. \( \frac{21}{24} \), c. \( \frac{15}{24} \), d. \( \frac{9}{24} \).

Step 1: Convert \( \frac{3}{4} \) to denominator 24

Find a number that you can multiply the denominator 4 by to get 24. Since 4 times 6 equals 24, multiply both the numerator and the denominator of \( \frac{3}{4} \) by 6: \[ \frac{3 \times 6}{4 \times 6} = \frac{18}{24} \].

Step 2: Convert \( \frac{7}{8} \) to denominator 24

Determine the number needed to multiply the denominator 8 by to obtain 24. Since 8 times 3 equals 24, multiply both the numerator and the denominator of \( \frac{7}{8} \) by 3: \[ \frac{7 \times 3}{8 \times 3} = \frac{21}{24} \].

Step 3: Convert \( \frac{5}{8} \) to denominator 24

Multiply the denominator 8 by 3 to get 24. This means multiplying both the numerator and the denominator of \( \frac{5}{8} \) by 3: \[ \frac{5 \times 3}{8 \times 3} = \frac{15}{24} \].

Step 4: Convert \( \frac{3}{8} \) to denominator 24

Again, since 8 times 3 equals 24, multiply both the numerator and the denominator of \( \frac{3}{8} \) by 3: \[ \frac{3 \times 3}{8 \times 3} = \frac{9}{24} \].

Key Concepts

Denominator Conversion

Changing the denominator of a fraction is a fundamental skill in fraction manipulation. The idea is to rewrite the fraction so that it has a new denominator, while still representing the same value. For example, converting fractions to have a common denominator of 24 helps in comparing or adding them.

The key step is finding what number you need to multiply the original denominator by to achieve the new denominator. This process is often called the least common denominator technique.

Here's a step-by-step process:

• Identify the current denominator of the fraction, e.g., in the fraction \(\frac{3}{4}\), the denominator is 4.
• Find the number that when multiplied to this denominator gives the desired denominator. For \(\frac{3}{4}\), we need to multiply 4 by 6 to get 24.
• Multiply both the numerator and the denominator by this number. If the original fraction is \(\frac{3}{4}\), you multiply both 3 and 4 by 6, resulting in \(\frac{18}{24}\).

The conversion keeps the fraction's value unchanged because you're simply multiplying by 1 (in the form of \(\frac{6}{6}\)). This ensures the fractions remain equivalent.

Multiplying Fractions

Multiplication of fractions is a straightforward operation. It involves multiplying the numerators together and denominators together directly. When converting fractions to equivalent forms, this operation is used to adjust the terms to meet desired needs like a common denominator.

For example, if you need to convert \(\frac{7}{8}\) into a fraction with a denominator of 24, you need to multiply the fraction’s current numerator and denominator by the same number.

Here’s a quick guide:

• Determine what you need the new denominator to be. If it’s 24 and the original is 8, calculate that 8 must be multiplied by 3 to reach 24.
• Correspondingly, multiply the numerator by 3. For \(\frac{7}{8}\), that’s 7 times 3, resulting in \(\frac{21}{24}\).
• This approach ensures the fraction’s value stays the same by effectively scaling it.

This manipulation does not alter the value of the fraction because multiplying the numerator and the denominator by the same number keeps the fraction identical in value, maintaining the equivalence.

Prealgebra

Prealgebra forms the foundational skills necessary for high school algebra and beyond. Understanding equivalent fractions, such as those explored above, is a part of this crucial groundwork.

In prealgebra, converting fractions by adjusting denominators or finding equivalencies is a common activity. These activities help students:

• Understand proportional reasoning by seeing how different expressions can represent the same value.
• Prepare for more complex problem-solving tasks that involve fractions.
• Strengthen number sense by manipulating and understanding numbers flexibly.

The comfort with multiplying fractions and converting denominators cleanly works to set a strong numerical foundation.

Comprehending these processes builds more than just arithmetic dexterity; it contributes to mathematical fluency and problem-solving abilities fundamental for more advanced math topics.