Problem 41 - Prealgebra

Question

Find the missing term in each of the following proportions. Set up each problem like the examples in this section. Write your answers as fractions in lowest terms. $$\frac{n}{39}=\frac{533}{507}$$

Step-by-Step Solution

Answer

The missing term is \( n = \frac{1603}{39} \).

Step 1: Set up a Cross Multiplication Equation

To find the missing term in the proportion \( \frac{n}{39} = \frac{533}{507} \), we can use cross multiplication. Multiply the numerator of the first fraction by the denominator of the second fraction and set it equal to the product of the denominator of the first fraction and the numerator of the second fraction. This gives: \( n \times 507 = 39 \times 533 \).

Step 2: Calculate the Products

Calculate the products from the cross multiplication: \( 39 \times 533 = 20847 \). So, the equation becomes \( 507n = 20847 \).

Step 3: Solve for n

To find the value of \( n \), divide both sides of the equation by 507: \( n = \frac{20847}{507} \).

Step 4: Simplify the Fraction

Simplify the fraction \( \frac{20847}{507} \) by finding the greatest common divisor (GCD) of 20847 and 507. By using the Euclidean algorithm, we find that the GCD is 13. Divide both numerator and denominator by 13: \( \frac{20847}{507} = \frac{20847 \div 13}{507 \div 13} = \frac{1603}{39} \). Therefore, \( n = \frac{1603}{39} \).

Key Concepts

Understanding Proportions

Proportions are a way to compare two ratios or fractions. When two ratios are set equal, they form a proportion. For example, the fractions \( \frac{a}{b} = \frac{c}{d} \) indicate a proportion between \( a/b \) and \( c/d \). Cross multiplication is a useful technique for solving for a missing term in a proportion.
In our original exercise, we had the proportion \( \frac{n}{39} = \frac{533}{507} \). Cross multiplying helps us find the missing term \( n \). Using cross multiplication, we form the equation \( n \times 507 = 39 \times 533 \), allowing us to solve for \( n \).
It's important to correctly set up proportions and carefully multiply to maintain the equality between the ratios. This method helps not only in finding missing terms but also ensures that the relationship between the ratios remains balanced.

• Set up the equation using cross multiplication.
• Calculate each product.
• Solve for the unknown variable.

Simplifying Fractions

Simplifying fractions is a crucial mathematical skill that makes working with proportions and equations easier. To simplify a fraction, divide its numerator and denominator by their greatest common divisor (GCD).
In our example, after finding \( n = \frac{20847}{507} \), the next step is to simplify this fraction. By locating the GCD of 20847 and 507 (which is 13 using the Euclidean algorithm), we can divide each by 13:
\(\frac{20847}{507} = \frac{20847 \div 13}{507 \div 13} = \frac{1603}{39}.\)
This reduces the fraction to its simplest form, making it easier to work with in further calculations.

• Find the GCD of the numerator and denominator.
• Divide both terms by the GCD.
• Express the fraction in simplest form.

Finding the Greatest Common Divisor (GCD)

The greatest common divisor (GCD) is the largest number that divides both the numerator and the denominator of a fraction without a remainder. Finding the GCD is essential for simplifying fractions.
One effective method to find the GCD is the Euclidean algorithm, which involves a process of repeated division. For our example with 20847 and 507, you perform the following steps:
1. Divide 20847 by 507 and get a remainder.
2. Replace the larger number with the smaller number (507) and the smaller number with the remainder.
3. Repeat the process until the remainder is zero.
The last non-zero remainder is the GCD, which in our case was 13.
Using the GCD allows us to simplify fractions efficiently, which provides a cleaner and more manageable form for solving equations and performing further calculations.

• Use division and remainders to find the GCD.
• Repeat the process with remainders until reaching zero.
• The last non-zero remainder is the GCD.