Problem 12 - Prealgebra

Question

Solve each proportion. $$\frac{7}{45}=\frac{x}{9}$$

Step-by-Step Solution

Answer

x = \( \frac{7}{5} \)

Step 1: Cross-Multiply the Proportions

To solve the proportion \( \frac{7}{45} = \frac{x}{9} \), we first cross-multiply. This means multiplying the numerator of one fraction by the denominator of the other fraction, and doing the same in reverse. Thus, we have: \[ 7 \cdot 9 = 45 \cdot x \] which simplifies to \[ 63 = 45x \].

Step 2: Solve for x

Next, we need to solve for \( x \) by isolating it on one side. To do this, divide both sides of the equation by 45: \[ x = \frac{63}{45} \].

Step 3: Simplify the Fraction

To simplify \( \frac{63}{45} \), we find the greatest common divisor (GCD) of 63 and 45, which is 9. Divide both the numerator and the denominator by 9: \[ x = \frac{63 \div 9}{45 \div 9} = \frac{7}{5} \].

Key Concepts

Cross-Multiplication

Cross-multiplication is a method used to solve proportions, which are equations that state that two ratios are equal. This technique is particularly helpful because it lets you eliminate the fractions in an equation, making it simpler to solve for the unknown variable.
When you cross-multiply, you multiply the numerator (top number) of one ratio by the denominator (bottom number) of the other ratio. You do this for both sides of the equation.
Let's say we have the proportion \( \frac{a}{b} = \frac{c}{d} \). In cross-multiplication:

• Multiply \( a \) by \( d \) (the numerators of opposite fractions).
• Multiply \( b \) by \( c \) (the denominators of opposite fractions).

You end up with the equation \( a \cdot d = b \cdot c \). Now, the problem switches from dealing with fractions to dealing with whole numbers, making it easier to isolate and solve for any variable in the equation.

Simplifying Fractions

Simplifying fractions is all about making a fraction as simple as possible. A fraction is simplified when the numerator and the denominator have no common factors left other than 1.
To simplify a fraction, you need to divide both the numerator and the denominator by their greatest common factor, or GCF. This ensures you're reducing the fraction completely without altering its value.
For example, to simplify \( \frac{6}{8} \):

• Find the GCF of 6 and 8, which is 2.
• Divide both the numerator and the denominator by 2: \( \frac{6 \div 2}{8 \div 2} = \frac{3}{4} \).

You've now simplified \( \frac{6}{8} \) to its simplest form, \( \frac{3}{4} \). Simplifying makes fractions easier to understand and compare, while also tidying up equations when solving problems.

Greatest Common Divisor

The Greatest Common Divisor (GCD), sometimes also called the Greatest Common Factor (GCF), is the largest number that divides two numbers without leaving a remainder.
Finding the GCD is a crucial step in simplifying fractions. The GCD tells us exactly how much we can reduce the numerator and the denominator without changing the fraction's value.
To find the GCD of two numbers, follow these steps:

• List all factors of the first number.
• List all factors of the second number.
• Identify the largest factor that appears in both lists; that's your GCD.

For example, to find the GCD of 45 and 63:

• Factors of 45: 1, 3, 5, 9, 15, 45.
• Factors of 63: 1, 3, 7, 9, 21, 63.
• The largest common factor is 9, so the GCD is 9.

By using the GCD, you can simplify fractions quickly and efficiently, leading to easier problem solving and clearer mathematical expressions.