Problem 68 - Algebra 2

Question

PREREQUISITE SKILL Solve each proportion. $$ \frac{7}{25}=\frac{a}{5} $$

Step-by-Step Solution

Answer

The solution is \( a = \frac{7}{5} \).

Step 1: Identify the Cross Products

In a proportion \( \frac{a}{b} = \frac{c}{d} \), the cross products are equal, meaning \( a \cdot d = b \cdot c \). Let's express the cross products for our proportion \( \frac{7}{25} = \frac{a}{5} \). This means that \( 7 \times 5 = 25 \times a \).

Step 2: Calculate the Cross Products

Multiply the known values to find one of the cross products: \( 7 \times 5 = 35 \). Now, you have the equation \( 35 = 25 \times a \).

Step 3: Solve for the Unknown Variable

To find \( a \), divide both sides of the equation by 25: \( a = \frac{35}{25} \).

Step 4: Simplify the Result

Simplify \( \frac{35}{25} \) by finding the greatest common divisor of 35 and 25, which is 5. Divide the numerator and denominator by 5: \( a = \frac{35 \div 5}{25 \div 5} = \frac{7}{5} \).

Key Concepts

Cross Multiplication

Cross multiplication is a simple technique used to solve equations involving proportions. When you have a proportion like \( \frac{a}{b} = \frac{c}{d} \), cross multiplication allows you to set these fractions equal by multiplying crosswise. This means you multiply the numerator of one fraction by the denominator of the other.

• For \( \frac{7}{25} = \frac{a}{5} \), cross multiply like this: \( 7 \times 5 \) and \( 25 \times a \).
• The result is the equation \( 35 = 25a \).

By doing this, you're able to isolate the unknown variable \( a \) and solve for it. Cross multiplication transforms the proportion into a simple algebraic equation that is much easier to handle. This powerful method is helpful whenever you deal with any problem that involves proportions.

Simplifying Fractions

Simplifying fractions is an important step in solving proportion problems, as it makes the answer easy to understand. A fraction is in its simplest form when the numerator and the denominator have no common factors other than 1. To simplify a fraction, you divide both the numerator and the denominator by their greatest common divisor (GCD).Take the fraction \( \frac{35}{25} \) for our problem, for instance:

• First, identify the GCD of the numerator and the denominator, which is 5.
• Then, divide both the numerator (35) and the denominator (25) by 5.
• The result is the fraction \( \frac{7}{5} \), which is already simplified.

Simplifying fractions not only makes your answers more concise, but it'll also make calculating further operations involving the fraction clearer and more accurate.

Greatest Common Divisor

The greatest common divisor (GCD) is a crucial concept when simplifying fractions, especially in mathematical problems involving proportions. The GCD is the largest number that divides both the numerator and the denominator without leaving a remainder.Here's how you find the GCD for the fraction \( \frac{35}{25} \):

• List the factors of 35 (1, 5, 7, 35) and the factors of 25 (1, 5, 25).
• Find the largest number that appears in both lists. Here, that number is 5.

Once you identify the GCD, you can use it to simplify the fraction by dividing the numerator and the denominator. This technique is very useful in ensuring fractions are easy to interpret and use in further calculations. The GCD makes working with complex fractions more manageable and efficient.