Solve Each Proportion And Give The Answer In Simplest Form

Solving proportions is a fundamental skill in mathematics with wide applications in everyday life. From scaling recipes to calculating distances on a map, understanding proportions is essential. This article provides a comprehensive guide to solving proportions, focusing on techniques to simplify the process and express answers in their simplest form.

Understanding Proportions

A proportion is an equation stating that two ratios are equal. A ratio compares two quantities, and can be written in several ways: as a fraction (a/b), using a colon (a:b), or using the word "to" (a to b). A proportion, then, looks like this:

a/b = c/d or a:b = c:d

In this proportion, 'a' and 'd' are called the extremes, while 'b' and 'c' are called the means. The fundamental property of proportions is that the product of the means equals the product of the extremes. This is known as the cross-product property.

Why Simplify?

Simplifying proportions and expressing answers in their simplest form makes calculations easier and allows for clearer comparisons. A simplified fraction is one where the numerator and denominator have no common factors other than 1.

Methods for Solving Proportions

Several methods can be used to solve proportions, but the most common and versatile is the cross-product property.

1. Cross-Product Property

As mentioned earlier, the cross-product property states that if a/b = c/d, then ad = bc. This property allows us to transform a proportion into a linear equation, which can then be solved using basic algebraic techniques.

Example 1:

Solve for x in the proportion: 3/4 = x/12

• Apply the cross-product property: 3 * 12 = 4 * x
• Simplify: 36 = 4x
• Divide both sides by 4: x = 9

Example 2:

Solve for y in the proportion: y/5 = 7/10

• Apply the cross-product property: y * 10 = 5 * 7
• Simplify: 10y = 35
• Divide both sides by 10: y = 3.5 or 7/2

2. Scaling (Finding a Common Factor)

Sometimes, you can solve a proportion by recognizing a direct scaling relationship between the numerators or denominators.

Example 1:

Solve for x in the proportion: 2/3 = x/9

• Notice that the denominator 3 is multiplied by 3 to get 9.
• To maintain the proportion, multiply the numerator 2 by 3 as well: x = 2 * 3
• Therefore, x = 6

Example 2:

Solve for y in the proportion: 15/25 = 3/y

• Notice that the numerator 15 is divided by 5 to get 3.
• To maintain the proportion, divide the denominator 25 by 5 as well: y = 25 / 5
• Therefore, y = 5

3. Unit Rate Method

The unit rate method involves finding the value of one unit of a quantity and then using that value to find the unknown quantity. This method is particularly useful in real-world problems.

Example:

If 5 apples cost $2.50, how much do 8 apples cost?

• Find the cost of one apple (unit rate): $2.50 / 5 = $0.50 per apple
• Multiply the cost per apple by the desired number of apples: $0.50 * 8 = $4.00
• Therefore, 8 apples cost $4.00

Simplifying the Solution

After solving for the unknown variable, it is crucial to simplify the answer, especially if the answer is a fraction. Here's how to simplify:

1. Find the Greatest Common Factor (GCF): The GCF is the largest number that divides evenly into both the numerator and the denominator.
2. Divide: Divide both the numerator and the denominator by the GCF.
3. Check: Ensure that the simplified fraction cannot be further reduced.

Example 1:

Solve for x in the proportion: 6/8 = 15/x

• Cross-multiply: 6x = 8 * 15
• Simplify: 6x = 120
• Divide both sides by 6: x = 20. Since 20 is an integer, no further simplification is needed.

Example 2:

Solve for y in the proportion: 4/6 = y/9

• Cross-multiply: 4 * 9 = 6y
• Simplify: 36 = 6y
• Divide both sides by 6: y = 6. Since 6 is an integer, no further simplification is needed.

Example 3:

Solve for z in the proportion: z/12 = 10/16

• Cross-multiply: 16z = 12 * 10
• Simplify: 16z = 120
• Divide both sides by 16: z = 120/16
• Simplify the fraction 120/16. The GCF of 120 and 16 is 8.
• Divide both the numerator and the denominator by 8: z = (120/8) / (16/8) = 15/2
• Therefore, z = 15/2 or 7.5

Complex Proportions

Sometimes, proportions involve more complex expressions, such as those containing variables in both the numerator and denominator or multiple terms. These proportions require careful application of algebraic principles.

Example 1:

Solve for x in the proportion: (x + 1)/4 = 9/12

• Cross-multiply: 12(x + 1) = 4 * 9
• Distribute: 12x + 12 = 36
• Subtract 12 from both sides: 12x = 24
• Divide both sides by 12: x = 2

Example 2:

Solve for y in the proportion: 3/(y - 2) = 5/8

• Cross-multiply: 3 * 8 = 5(y - 2)
• Simplify: 24 = 5y - 10
• Add 10 to both sides: 34 = 5y
• Divide both sides by 5: y = 34/5

Example 3:

Solve for a in the proportion: (a + 2)/(a - 1) = 4/3

• Cross-multiply: 3(a + 2) = 4(a - 1)
• Distribute: 3a + 6 = 4a - 4
• Subtract 3a from both sides: 6 = a - 4
• Add 4 to both sides: 10 = a
• Therefore, a = 10

Real-World Applications

Proportions are incredibly useful in various real-world scenarios. Here are a few examples:

1. Cooking: Scaling recipes up or down requires understanding proportions. If a recipe for 4 people requires 2 cups of flour, how much flour is needed for 10 people?

   • Proportion: 2 cups / 4 people = x cups / 10 people
   • Cross-multiply: 2 * 10 = 4x
   • Simplify: 20 = 4x
   • Divide both sides by 4: x = 5
   • Therefore, 5 cups of flour are needed for 10 people.

2. Maps and Scale Drawings: Maps use scales to represent actual distances. If a map has a scale of 1 inch = 50 miles, and two cities are 3.5 inches apart on the map, what is the actual distance between them?

   • Proportion: 1 inch / 50 miles = 3.5 inches / x miles
   • Cross-multiply: 1 * x = 50 * 3.5
   • Simplify: x = 175
   • Therefore, the actual distance between the cities is 175 miles.

3. Unit Conversion: Converting between units (e.g., inches to centimeters, pounds to kilograms) often involves proportions. Given that 1 inch is equal to 2.54 centimeters, how many centimeters are in 12 inches?

   • Proportion: 1 inch / 2.54 cm = 12 inches / x cm
   • Cross-multiply: 1 * x = 2.54 * 12
   • Simplify: x = 30.48
   • Therefore, 12 inches is equal to 30.48 centimeters.

4. Business and Finance: Proportions are used to calculate percentages, discounts, and interest rates. For example, if an item is on sale for 20% off and the original price is $50, what is the discount amount?

   • Proportion: 20 / 100 = x / 50
   • Cross-multiply: 20 * 50 = 100x
   • Simplify: 1000 = 100x
   • Divide both sides by 100: x = 10
   • Therefore, the discount amount is $10.

Common Mistakes to Avoid

When solving proportions, it's important to avoid these common mistakes:

• Incorrect Cross-Multiplication: Ensure that you correctly multiply the extremes and the means. Double-check your work.
• Forgetting to Distribute: When dealing with complex proportions involving expressions in parentheses, remember to distribute the multiplication across all terms inside the parentheses.
• Not Simplifying: Always simplify the resulting fraction to its simplest form.
• Mixing Units: When applying proportions to real-world problems, ensure that the units are consistent. For example, don't mix inches and feet in the same proportion without converting them first.
• Misinterpreting the Problem: Read the problem carefully to understand what is being asked and set up the proportion correctly.

Practice Problems

To solidify your understanding of solving proportions, try these practice problems:

1. Solve for x: 5/8 = x/24
2. Solve for y: y/7 = 12/21
3. Solve for z: 9/15 = 6/z
4. Solve for a: (a + 3)/5 = 8/10
5. Solve for b: 4/(b - 1) = 6/9
6. If 3 notebooks cost $4.50, how much do 7 notebooks cost?
7. On a map with a scale of 1 cm = 25 km, two cities are 4.2 cm apart. What is the actual distance between them?
8. Convert 36 inches to centimeters, given that 1 inch = 2.54 cm.
9. An item is on sale for 30% off. If the original price is $80, what is the sale price?
10. Solve for x: (2x + 1)/(x - 2) = 5/2

Answer Key:

1. x = 15
2. y = 4
3. z = 10
4. a = 1
5. b = 7
6. $10.50
7. 105 km
8. 91.44 cm
9. $56
10. x = 4

Conclusion

Mastering the skill of solving proportions is essential for success in mathematics and for tackling real-world problems. By understanding the cross-product property, recognizing scaling relationships, and practicing simplification techniques, you can confidently solve proportions and express answers in their simplest form. Remember to avoid common mistakes and always double-check your work. With practice, solving proportions will become a straightforward and valuable skill.