Problem 47 - Fundamentals of Mathematics

Question

For the following problems, determine the missing numerator or denominator. $$\frac{9}{25}=\frac{27}{?}$$

Step-by-Step Solution

Answer

The missing numerator is 75.

Step 1: Understand the Problem

We have the proportion \( \frac{9}{25} = \frac{27}{?} \). In this exercise, one fraction equals another, and we need to find the missing part of the second fraction.

Step 2: Use Cross-Multiplication

To solve the proportion, use cross-multiplication: multiply the numerator of the first fraction by the denominator of the second fraction and set it equal to the denominator of the first fraction times the numerator of the second fraction. That gives us: \[ 9 \times ? = 25 \times 27 \]

Step 3: Calculate the Known Product

Calculate \( 25 \times 27 \). Breaking this down, you can multiply step-by-step: \[ 25 \times 27 = 25 \times (20 + 7) = (25 \times 20) + (25 \times 7) \] \[ 25 \times 20 = 500 \] and \[ 25 \times 7 = 175 \] Adding these products gives: \[ 500 + 175 = 675 \]

Step 4: Solve for the Missing Numerator

Using \( 9 \times ? = 675 \), solve for \(?\): Divide both sides by 9 to isolate the question mark: \[ ? = \frac{675}{9} \]

Step 5: Simplify the Division

Perform the division: \[ 675 \div 9 = 75 \] Thus, \(? = 75\).

Step 6: Verify the Solution

Check your work by verifying if \( \frac{9}{25} = \frac{27}{75} \) still holds true by simplifying the second fraction: Divide both the numerator and the denominator by their greatest common divisor (GCD), which is 3. \[ \frac{27 \div 3}{75 \div 3} = \frac{9}{25} \] This confirms our solution is correct.

Key Concepts

Understanding Fractions

Fractions are a way to represent parts of a whole. Think of a fraction as being split into two main parts: the numerator and the denominator. The numerator is the number on the top, and it tells us how many parts we have. The denominator is the number on the bottom, telling us how many equal parts the whole is divided into. Let's look at the fraction \( \frac{9}{25} \). Here:

• 9 is the numerator, showing we have 9 parts.
• 25 is the denominator, showing these parts come from a whole that is divided into 25 parts.

Fractions can be used in equations, like proportions, to solve unknowns. Understanding this is essential as it allows us to tackle many real-world problems and mathematical equations with better ease.
Understanding fractions fully means recognizing their role in expressing division and being comfortable with operations involving them, such as addition, subtraction, multiplication, and division.

Cross-Multiplication Technique

Cross-multiplication is a key technique used to solve equations involving proportions. When two fractions are set equal to each other, they form a proportion, such as \( \frac{9}{25} = \frac{27}{?} \). The secret to solving these proportions quickly is cross-multiplication. Here’s how it works:

• Multiply the numerator of the first fraction by the denominator of the second fraction.
• This result equals the product of the denominator of the first fraction and the numerator of the second fraction.

For example, with \( \frac{9}{25} = \frac{27}{?} \), we cross-multiply: \( 9 \times ? = 25 \times 27 \). This principle helps create a single equation to find the unknown part. It simplifies solving for missing values, especially in comparisons and scaling problems.

Finding the Greatest Common Divisor

The Greatest Common Divisor (GCD) is a useful tool in simplifying fractions. The GCD of two numbers is the largest number that divides both without leaving a remainder. Simplifying fractions by dividing both the numerator and the denominator by their GCD gives a simpler, equivalent fraction. For example, take \( \frac{27}{75} \). To simplify it:

• Determine the GCD of 27 and 75. The number is 3.
• Divide both the numerator and denominator by 3: \( \frac{27 \div 3}{75 \div 3} = \frac{9}{25} \).

Simplifying fractions helps compare them easily or integrate them better into calculations. Knowing how to find the GCD and use it for simplification is important in mathematics for streamlining problem-solving processes.