Problem 23 - Fundamentals of Mathematics

Question

Find each value. (Section 4.7) \(\frac{2}{9}\) of what number is \(\frac{1}{18} ?\)

Step-by-Step Solution

Answer

The number is \( \frac{1}{4} \).

Step 1: Identify the Unknown

Let the unknown number be represented by the variable \( x \). We need to find what number, when \( \frac{2}{9} \) of it is taken, results in \( \frac{1}{18} \).

Step 2: Set Up the Equation

The given problem states that \( \frac{2}{9} \) of some number \( x \) is \( \frac{1}{18} \). We can express this relationship as an equation: \[ \frac{2}{9} \times x = \frac{1}{18} \]

Step 3: Solve for the Unknown

To solve for \( x \), we need to isolate it. Start by multiplying both sides of the equation by the reciprocal of \( \frac{2}{9} \) which is \( \frac{9}{2} \):\[ x = \frac{1}{18} \times \frac{9}{2} \]

Step 4: Simplify the Expression

First, multiply the numerators and then multiply the denominators:\[ x = \frac{1 \times 9}{18 \times 2} = \frac{9}{36} \]

Step 5: Final Calculation

Now simplify \( \frac{9}{36} \) by finding the greatest common divisor, which is 9:\[ \frac{9}{36} = \frac{1}{4} \]

Step 6: Conclusion

Therefore, the number for which \( \frac{2}{9} \) of it equals \( \frac{1}{18} \) is \( \frac{1}{4} \).

Key Concepts

Solving Equations

To tackle fraction problems like this, it's important to set up an equation that represents the situation. Begin with identifying any known parts and the unknown you need to find. Consider this example:- We know that \( \frac{2}{9} \) of a number equals \( \frac{1}{18} \).- Let \( x \) be the unknown number we are trying to find. This leads us to the equation \( \frac{2}{9} \times x = \frac{1}{18} \). Having equations helps in systematically finding unknown values by applying mathematical operations. The key to solving these equations is maintaining equality while manipulating them. Breaking down the solution into manageable steps can make the problem seem less daunting. Each operation you perform should help you isolate the unknown variable, moving closer to finding its value.

Isolation of Variables

The art of isolating the variable is crucial in solving equations. Isolating the variable means getting it by itself on one side of the equation. In our example, we need to solve for \( x \) in the equation \( \frac{2}{9} \times x = \frac{1}{18} \).To isolate \( x \), multiply both sides by the reciprocal of \( \frac{2}{9} \), which is \( \frac{9}{2} \). This operation will cancel out the \( \frac{2}{9} \) on the left side:- \( x = \frac{1}{18} \times \frac{9}{2} \)Now, perform the multiplication carefully:1. Multiply the numerators: \( 1 \times 9 = 9 \)2. Multiply the denominators: \( 18 \times 2 = 36 \)Thus, \( x = \frac{9}{36} \). The variable \( x \) has been isolated and now represents \( \frac{9}{36} \). The next step is to simplify this fraction to find the exact value of \( x \).

Greatest Common Divisor

Simplifying fractions is a critical skill, and the greatest common divisor (GCD) helps us do that. The GCD of two numbers is the largest number that can divide both numbers evenly.To simplify \( \frac{9}{36} \), determine the GCD of 9 and 36:- The divisors of 9 are 1, 3, and 9.- The divisors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36.- The highest common divisor is 9.Use this GCD to simplify the fraction:- Divide the numerator: \( 9 \div 9 = 1 \)- Divide the denominator: \( 36 \div 9 = 4 \)Thus, \( \frac{9}{36} \) simplifies to \( \frac{1}{4} \). Finding the GCD allows us to reduce fractions to their simplest form, making them easier to work with and interpret. With these skills, you can solve even complex problems involving fractions.