Problem 43 - Fundamentals of Mathematics

Question

For the following problems, determine the missing numerator or denominator. $$\frac{3}{4}=\frac{?}{16}$$

Step-by-Step Solution

Answer

The missing numerator is 12.

Step 1: Understand the Relationship

When two fractions are equal, their cross-products are also equal. This means that if \( \frac{3}{4} = \frac{x}{16} \), then \( 3 \times 16 = 4 \times x \).

Step 2: Set Up the Equation

Write down the equation based on the cross-multiplication principle: \[ 3 \times 16 = 4 \times x \]

Step 3: Calculate the Cross Product

Calculate \(3 \times 16\): \[ 3 \times 16 = 48 \].

Step 4: Solve for the Missing Numerator

Use the equation from Step 2 to solve for \(x\): \[ 4 \times x = 48 \]. Divide both sides by 4 to find \(x\): \[ x = \frac{48}{4} = 12 \].

Step 5: Verify the Solution

Substitute \(x = 12\) back into the original equation to ensure both fractions are equal. Check: \( \frac{3}{4} = \frac{12}{16} \), simplify \( \frac{12}{16} \) which is equal to \( \frac{3}{4} \). Thus, the fractions are indeed equal.

Key Concepts

Cross-Multiplication

Cross-multiplication is a very effective technique used to solve equations involving two fractions. When you have two fractions set equal to each other, like \( \frac{3}{4} = \frac{x}{16} \), you can determine the equality by cross-multiplying. This means you multiply the numerator of each fraction by the denominator of the other fraction.

• In our example, cross-multiplying means calculating \( 3 \times 16 \) and \( 4 \times x \). This gives us the equation \( 3 \times 16 = 4 \times x \).

Cross-multiplication is based on the principle that if two ratios are equal, the product of the means equals the product of the extremes. This method is especially helpful for finding a missing numerator or denominator in fraction equations, as it transforms a fraction comparison into a simple algebraic equation. Once the equation is set, solving for the unknown becomes straightforward.

Equivalent Fractions

Equivalent fractions are different fractions that represent the same value. For instance, \( \frac{3}{4} \) and \( \frac{12}{16} \) are equivalent fractions because both fractions simplify to the same value.

• This can be verified by simplifying \( \frac{12}{16} \) to \( \frac{3}{4} \) by dividing both the numerator and denominator by their greatest common divisor, which is 4.

Understanding equivalent fractions is key to grasping how fractions can look different but have the same value. This concept is crucial when solving problems that involve comparing fractions or finding unknowns in equations like \( \frac{3}{4} = \frac{x}{16} \). By ensuring that both fractions simplify to the same basic form, you can confirm that they are equivalent and that your solution is correct.

Numerator and Denominator

In every fraction, the numerator and denominator play specific roles. The numerator is the top part of the fraction, and it indicates how many parts you have. The denominator is the bottom part and shows how many equal parts make up a whole.

• In \( \frac{3}{4} \), 3 is the numerator which tells us that we have 3 parts out of the 4 that make a whole.
• In the fraction \( \frac{x}{16} \), 16 is the denominator showing the whole is divided into 16 equal parts, and \( x \) is the numerator we need to find out.

Recognizing these roles helps in understanding what is being represented in fraction problems. When determining a missing numerator or denominator, knowing which part of the fraction you are looking for makes applying techniques like cross-multiplication straightforward. This understanding allows you to solve fraction equations and confirm that fractions are equivalent by checking if they reduce to the same value.