Solve Each Proportion And Give The Answer In Simplest Form

Solve Each Proportion and Give the Answer in Simplest Form: A Complete Guide

Understanding how to solve each proportion and express the answer in simplest form is a foundational mathematical skill with practical applications in everything from cooking and construction to finance and science. A proportion states that two ratios are equal, and mastering its solution involves a straightforward yet powerful algebraic technique. This guide will walk you through the concept, methods, and essential practice to confidently tackle any proportion problem, ensuring your final answers are always reduced to their most basic terms.

What Exactly Is a Proportion?

A proportion is an equation that declares two ratios or fractions are equivalent. It typically appears in the format a/b = c/d, where a, b, c, and d are numbers, and b and d are not zero. The core idea is that the relationship between a and b is identical to the relationship between c and d. For example, 1/2 = 2/4 is a true proportion because both fractions represent the same value, 0.5. When solving a proportion, you are usually given three of the four values and must find the missing fourth, often represented by a variable like x.

The instruction to provide the answer in simplest form means your final fraction must have a numerator and denominator that share no common factors other than 1. For instance, 4/8 simplifies to 1/2. This step is not optional; it is a mandatory part of presenting a complete and correct solution, ensuring clarity and precision in your answer.

The Golden Rule: Cross-Multiplication

The primary tool for solving each proportion is cross-multiplication. This method leverages the fundamental property of proportions: if a/b = c/d, then a * d = b * c. The process involves multiplying the numerator of one fraction by the denominator of the other and setting the two products equal to each other.

Step-by-Step Process:

1. Set up the equation: Write the proportion clearly.
2. Cross-multiply: Multiply the outer terms (a and d) and the inner terms (b and c).
3. Form the new equation: This gives you a*d = b*c.
4. Solve for the variable: Use basic algebra (addition, subtraction, multiplication, division) to isolate the unknown variable.
5. Simplify the result: Reduce the resulting fraction to its simplest form by dividing both the numerator and denominator by their greatest common divisor (GCD).

Solving Basic Proportions with Numbers

Let's start with a straightforward example where all values are known except one. Problem: Solve the proportion 3/5 = x/15.

• Step 1: The proportion is 3/5 = x/15.
• Step 2: Cross-multiply: 3 * 15 = 5 * x.
• Step 3: This simplifies to 45 = 5x.
• Step 4: Solve for x by dividing both sides by 5: x = 45 / 5, so x = 9.
• Step 5: The answer is the whole number 9. In simplest form, this is just 9 or 9/1.

Now, consider a problem where the solution is a fraction. Problem: Solve 7/4 = 14/x.

• Cross-multiply: 7 * x = 4 * 14 → 7x = 56.
• Solve: x = 56 / 7 → x = 8.
• Result: x = 8.

Problem: Solve 2/9 = 6/x.

• Cross-multiply: 2 * x = 9 * 6 → 2x = 54.
• Solve: x = 54 / 2 → x = 27.
• Result: x = 27.

Solving Proportions with Variables in Multiple Places

Sometimes, the variable appears in more than one position. Problem: Solve (x+1)/4 = 3/2.

• Cross-multiply: 2 * (x + 1) = 4 * 3 → 2(x + 1) = 12.
• Distribute and Solve: 2x + 2 = 12 → 2x = 10 → x = 5.
• Check: Substitute x=5: (5+1)/4 = 6/4 = 3/2. Correct.

Problem: Solve 5/(x-2) = 10/6.

• Cross-multiply: 5 * 6 = 10 * (x - 2) → 30 = 10(x - 2).
• Solve: 30 = 10x - 20 → 50 = 10x → x = 5.
• Check: 5/(5-2) = 5/3. 10/6 simplifies to 5/3. Correct.

Applying Proportions to Word Problems

The real power of proportions lies in translating real-world scenarios into mathematical equations.

Example 1: Scaling a Recipe A recipe for 4 people requires 3 cups of flour. How many cups are needed for 10 people?

• **Set up the