Problem 12 - 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. $$\frac{2}{y}=\frac{6}{10}$$

Step-by-Step Solution

Answer

The missing term is \( \frac{10}{3} \).

Step 1: Cross-multiply the proportions

In a proportion \( \frac{a}{b} = \frac{c}{d} \), the cross-multiplication method involves multiplying \( a \) by \( d \) and \( b \) by \( c \). This results in the equation \( a \times d = b \times c \). For the given proportion \( \frac{2}{y} = \frac{6}{10} \), cross-multiply to get \( 2 \times 10 = y \times 6 \).

Step 2: Solve for the missing term

From the cross-multiplied equation \( 20 = 6y \), solve for \( y \) by dividing both sides by 6. This gives \( y = \frac{20}{6} \).

Step 3: Simplify the fraction

Simplify \( \frac{20}{6} \) by finding the greatest common divisor of 20 and 6, which is 2. Divide both the numerator and the denominator by 2. Therefore, \( \frac{20}{6} \) simplifies to \( \frac{10}{3} \).

Key Concepts

Cross-Multiplication

Cross-multiplication is a powerful tool when it comes to solving proportions. If you're dealing with a proportion, which is an equation that states two ratios are equal, cross-multiplication helps you find an unknown value. In the example given, \( \frac{2}{y} = \frac{6}{10} \), it shows us that two ratios are equal: one involving 2 and \( y \), and the other involving 6 and 10. To use cross-multiplication, multiply across the equals sign diagonally: multiply 2 by 10, and \( y \) by 6. This gives us the equation: - \( 2 \times 10 = 6 \times y \) - Which simplifies to: \( 20 = 6y \)Now, you have a simple equation to solve for \( y \). Cross-multiplication turns complex ratios into manageable equations, helping you solve problems efficiently.

Simplifying Fractions

When you find a fraction, such as
\( \frac{20}{6} \), it's often not in its simplest form. Simplifying a fraction means making it as simple as possible by dividing both the numerator and denominator by their greatest common divisor (GCD). To simplify \( \frac{20}{6} \), identify the common factors of 20 and 6, then find the largest one they share. After calculating, you discover that their GCD is 2. Dividing both the numerator and the denominator of \( \frac{20}{6} \) by 2, you'll get:

• 20 divided by 2 = 10
• 6 divided by 2 = 3

Thus, \( \frac{20}{6} \) simplifies to \( \frac{10}{3} \). Simplifying fractions helps to make math easier and tidier, providing clearer understanding of the value of a fraction.

Greatest Common Divisor

Finding the greatest common divisor (GCD) of two numbers is crucial when simplifying fractions. GCD is the largest number that evenly divides two or more numbers without leaving a remainder.
Consider the fraction \( \frac{20}{6} \). To simplify this fraction, you first find the GCD of 20 and 6. To find the GCD, list the divisors:

• Divisors of 20: 1, 2, 4, 5, 10, 20
• Divisors of 6: 1, 2, 3, 6

The largest number common to both lists of divisors is 2. Thus, 2 is the GCD of 20 and 6. Once discovered, use the GCD to divide both the numerator and the denominator of your fraction. This process makes your fraction simpler and more efficient for further calculations within math problems.