The Unified Field of Rational Numbers: Equivalence and Inverses | Core Mathematics SHS 1 SEM 1 WEEK 5 (WASSCE & NaCCA Aligned)

The Unity of Rational Representation: Fractions, Decimals, and Percentages

In mathematics, clarity often comes from viewing the same quantity through different lenses. Fractions, decimals, and percentages are not distinct entities; they are merely three different languages used to describe a part of a whole. Understanding their fundamental connection is key to mastering proportional reasoning.

Converting Decimals to Fractions: Harnessing Place Value

Decimals are inherently rooted in the power of ten. When you see 0.75, you are reading ‘seventy-five hundredths.’ The number of digits after the decimal point tells you the denominator (10, 100, 1000, etc.). Think of Ghana’s currency: 0.75 GHS is 75 pesewas out of 100 pesewas (1 Cedi). The conversion steps are mechanical but must always conclude with simplification to the lowest terms.

• Step 1: Identify the Place Value. Count the decimal places. If there are two places (e.g., 0.65), the denominator is 100.
• Step 2: Form the Fraction. Write the decimal digits over the appropriate power of ten (e.g., 0.65 becomes 65/100).
• Step 3: Simplify. Divide the numerator and denominator by their Greatest Common Divisor (GCD). For 65/100, the GCD is 5, resulting in 13/20.

Converting Fractions to Decimals: The Power of Division

A fraction bar (/) fundamentally represents division. To convert a fraction like 3/8 to its decimal equivalent, we simply perform the operation: 3 divided by 8. This often requires careful long division, which is a fundamental skill. For certain fractions, we can use the method of equivalent fractions by manipulating the denominator to be a power of ten (e.g., 3/5 is equivalent to 6/10, or 0.6). Mastery involves recognizing when a fraction will terminate (like 1/4) and when it will repeat (like 1/3).

The Percentage Bridge: Relating to 100

The term ‘percent’ comes from the Latin per centum, meaning ‘out of one hundred.’ This establishes 100 as the ultimate reference point. If a trader in Makola offers a discount of 20%, it means for every 100 Cedis spent, 20 Cedis are removed.

• Fraction to Percentage: Multiply the fraction by 100%. (Example: 3/4 × 100% = 75%).
• Percentage to Decimal: Divide the percentage by 100 (move the decimal point two places left). (Example: 75% = 0.75).

Understanding Number Inverses: The Nullifying Forces

The concept of an inverse defines a relationship where two numbers combine to yield a fundamental identity element. We study two critical types of inverses:

1. Additive Inverse (The Opposite)

The additive inverse of a number a is the number we must add to a to result in the additive identity, which is zero (0). Zero represents neutrality—the state of having nothing added or taken away. If you owe a peer 5 Cedis (–5), the additive inverse is receiving 5 Cedis (+5), bringing your debt balance back to zero.

• Rule: The additive inverse of a is –a.
• Fractions: The additive inverse of 2/3 is –2/3.

2. Multiplicative Inverse (The Reciprocal)

The multiplicative inverse, also known as the reciprocal, of a non-zero number a is the number we must multiply a by to result in the multiplicative identity, which is one (1). One represents the whole, or the unit quantity.

This concept is paramount for solving algebraic equations involving fractions, such as isolating a variable. If you have 3/5 of a quantity x, you must multiply by its reciprocal, 5/3, to find the full quantity x. This ‘flipping’ mechanism is what makes the product equal to 1, neutralizing the coefficient.

• Rule: The multiplicative inverse of a/b is b/a.
• Crucial Step for Mixed Numbers: Before finding the reciprocal of a mixed number (e.g., $2 \frac{1}{4}$), you must convert it completely to an improper fraction ($9/4$). The reciprocal is then $4/9$. Failure to convert first leads to mathematical error.

The multiplicative inverse also allows us to define division by fractions. Instead of dividing by $1/2$, we multiply by its reciprocal, $2/1$. This reciprocal relationship forms the bedrock of rational number manipulation, enabling efficient problem-solving in areas ranging from calculating tax percentages on goods to dividing resources among groups in a community farming project. By treating fractions, decimals, and percentages as interchangeable representations and mastering the concept of the inverse, learners establish a robust foundation for all subsequent work in algebra and proportional reasoning. The Multi-Purpose Fractional Chart beautifully visualizes this truth: when the fraction grid (the number) and its reciprocal grid (the inverse) interact, their combined effect defines the fundamental unit (1).

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