Problem 32 - Algebra A Combined Function

Question

Find each reciprocal. 1.5

Step-by-Step Solution

Answer

The reciprocal of 1.5 is \( \frac{2}{3} \).

Step 1: Understand the Meaning of Reciprocal

The reciprocal of a number is 1 divided by that number. For any non-zero number a, its reciprocal is \( \frac{1}{a} \).

Step 2: Convert Decimal to Fraction

Convert 1.5 into a fraction. Since 1.5 is equivalent to \( \frac{15}{10} \), simplify this fraction by dividing both the numerator and the denominator by 5, resulting in \( \frac{3}{2} \).

Step 3: Find the Reciprocal

Find the reciprocal of the fraction \( \frac{3}{2} \). The reciprocal is obtained by flipping the numerator and denominator, giving \( \frac{2}{3} \).

Step 4: Verify Your Solution

Multiply the original number (1.5) by the reciprocal (\( \frac{2}{3} \)) to check correctness: \( 1.5 \times \frac{2}{3} = \frac{3}{2} \times \frac{2}{3} = 1 \). Since the product is 1, the solution is verified.

Key Concepts

Decimal to Fraction

Converting a decimal to a fraction might seem tricky at first, but with a clear method, it's very simple. Take the decimal 1.5, which falls between whole numbers 1 and 2. To convert 1.5 to a fraction, recognize that the decimal point shifts the place value of each number. 1.5 can be read as "one and five-tenths." This means the fraction form of 1.5 is \( \frac{15}{10} \).
However, fractions can often be simplified by halving both the top number (numerator) and the bottom number (denominator). You simplify \( \frac{15}{10} \) by dividing both by their greatest common divisor, which is 5.

• Divide 15 by 5 to get 3.
• Divide 10 by 5 to get 2.

This gives you the simplified fraction \( \frac{3}{2} \). Now, 1.5 is fully converted and simplified to \( \frac{3}{2} \).

Simplifying Fractions

Simplifying fractions involves reducing a fraction to its simplest form. This means making both the numerator and denominator as small as possible while still having the same value. For example, with the fraction \( \frac{15}{10} \), you want to determine the greatest common divisor (GCD) of both numbers.

• First, identify common factors of 15 and 10. These are 1, 5, and themselves (15 and 10).
• The greatest common factor is 5.

Divide 15 and 10 by their GCD, which results in 3 and 2. So, \( \frac{15}{10} \) simplifies to \( \frac{3}{2} \). Always double-check your work to ensure you cannot simplify further. In this case, 3 and 2 have no common factors other than 1. So, \( \frac{3}{2} \) is the simplest form.

Multiplication Verification

Verification by multiplication is a method to ensure that the reciprocal was calculated correctly. Reciprocal means that when the number and its reciprocal are multiplied, the result should be 1. Let's apply this with our example of 1.5 and its reciprocal, which is \( \frac{2}{3} \).
To verify, convert 1.5 to its fraction form, \( \frac{3}{2} \), and then multiply by its reciprocal:

• Start with: \( \frac{3}{2} \times \frac{2}{3} \).
• Multiply the numerators: 3 x 2 = 6.
• Multiply the denominators: 2 x 3 = 6.

This results in \( \frac{6}{6} \), which simplifies to 1. This confirms that \( \frac{2}{3} \) is indeed the reciprocal of 1.5. If your verification results in anything other than 1, recheck your calculations for errors.