Real-world problems involving solving equations with fractions Grade 8.

📚 Understanding Equations with Fractions

An equation with fractions is simply an equation where one or more terms are fractions. Solving these equations involves finding the value of the unknown variable that makes the equation true. This often requires performing operations like adding, subtracting, multiplying, or dividing fractions.

📜 A Brief History

The concept of fractions dates back to ancient civilizations, with Egyptians and Mesopotamians using them for dividing land and resources. Equations involving fractions evolved alongside the development of algebra, becoming essential tools in various fields like engineering, physics, and economics. Early mathematicians recognized the need for standardized methods to solve these equations, leading to the algebraic techniques we use today.

key Principles of Solving Equations with Fractions

• ⚖️ Maintaining Balance: The most important rule is to keep the equation balanced. Whatever operation you perform on one side, you must perform on the other.
• ➕ Clearing Fractions: Multiply both sides of the equation by the least common multiple (LCM) of the denominators to eliminate the fractions.
• ➗ Isolating the Variable: Use inverse operations (addition/subtraction, multiplication/division) to isolate the variable on one side of the equation.
• ✅ Checking Your Solution: Always substitute your solution back into the original equation to verify that it is correct.

🍕 Real-World Example 1: Sharing Pizza

Problem: Three friends want to share a pizza. Sarah eats $\frac{1}{3}$ of the pizza, and John eats $\frac{1}{4}$ of the pizza. How much of the pizza is left for Emily?

Solution:

1. Let $x$ be the fraction of pizza left for Emily.
2. The equation is: $\frac{1}{3} + \frac{1}{4} + x = 1$
3. Find a common denominator (12): $\frac{4}{12} + \frac{3}{12} + x = 1$
4. Combine fractions: $\frac{7}{12} + x = 1$
5. Subtract $\frac{7}{12}$ from both sides: $x = 1 - \frac{7}{12}$
6. $x = \frac{12}{12} - \frac{7}{12} = \frac{5}{12}$
7. Emily gets $\frac{5}{12}$ of the pizza.

🧮 Real-World Example 2: Mixing Solutions

Problem: A chemist needs to prepare 500 ml of a 30% acid solution. They have a 20% acid solution and a 50% acid solution. How much of each solution should they mix?

Solution:

1. Let $x$ be the amount (in ml) of the 20% solution.
2. Then $500 - x$ is the amount of the 50% solution.
3. The equation is: $0.20x + 0.50(500 - x) = 0.30(500)$
4. $0.20x + 250 - 0.50x = 150$
5. $-0.30x = -100$
6. $x = \frac{-100}{-0.30} = \frac{1000}{3} \approx 333.33$ ml
7. Therefore, approximately 333.33 ml of the 20% solution and 500 - 333.33 = 166.67 ml of the 50% solution are needed.

📐 Real-World Example 3: Construction Project

Problem: A construction team needs to build a fence. One worker can build $\frac{1}{5}$ of the fence per day, and another worker can build $\frac{1}{8}$ of the fence per day. How long will it take them to build the entire fence working together?

Solution:

1. Let $t$ be the number of days it takes them to build the fence together.
2. The equation is: $\frac{1}{5}t + \frac{1}{8}t = 1$
3. Find a common denominator (40): $\frac{8}{40}t + \frac{5}{40}t = 1$
4. Combine fractions: $\frac{13}{40}t = 1$
5. Multiply both sides by $\frac{40}{13}$: $t = \frac{40}{13} \approx 3.08$ days
6. It will take them approximately 3.08 days to build the fence together.

🏃 Real-World Example 4: Distance, Rate, and Time

Problem: John travels $\frac{2}{5}$ of his journey by train and $\frac{1}{3}$ by bus. If he still has 140 km left to cover, what is the total distance of his journey?

Solution:

1. Let $d$ be the total distance of the journey.
2. John traveled $\frac{2}{5}d$ by train and $\frac{1}{3}d$ by bus.
3. The remaining distance is 140 km.
4. The equation is: $\frac{2}{5}d + \frac{1}{3}d + 140 = d$
5. Find a common denominator (15): $\frac{6}{15}d + \frac{5}{15}d + 140 = d$
6. Combine fractions: $\frac{11}{15}d + 140 = d$
7. Subtract $\frac{11}{15}d$ from both sides: $140 = d - \frac{11}{15}d$
8. $140 = \frac{15}{15}d - \frac{11}{15}d$
9. $140 = \frac{4}{15}d$
10. Multiply both sides by $\frac{15}{4}$: $d = 140 \times \frac{15}{4} = 35 \times 15 = 525$ km
11. The total distance of his journey is 525 km.

💰 Real-World Example 5: Budgeting

Problem: Lisa spends $\frac{1}{4}$ of her monthly salary on rent and $\frac{1}{6}$ on groceries. If she has $700 left, what is her monthly salary?

Solution:

1. Let $s$ be Lisa's monthly salary.
2. Lisa spends $\frac{1}{4}s$ on rent and $\frac{1}{6}s$ on groceries.
3. The remaining amount is $700.
4. The equation is: $\frac{1}{4}s + \frac{1}{6}s + 700 = s$
5. Find a common denominator (12): $\frac{3}{12}s + \frac{2}{12}s + 700 = s$
6. Combine fractions: $\frac{5}{12}s + 700 = s$
7. Subtract $\frac{5}{12}s$ from both sides: $700 = s - \frac{5}{12}s$
8. $700 = \frac{12}{12}s - \frac{5}{12}s$
9. $700 = \frac{7}{12}s$
10. Multiply both sides by $\frac{12}{7}$: $s = 700 \times \frac{12}{7} = 100 \times 12 = 1200$
11. Lisa's monthly salary is $1200.

🧑‍🍳 Real-World Example 6: Baking a Cake

Problem: A recipe calls for $\frac{2}{3}$ cup of sugar, but you only want to make half the recipe. How much sugar do you need?

Solution:

1. Let $x$ be the amount of sugar needed.
2. The equation is: $x = \frac{1}{2} \times \frac{2}{3}$
3. Multiply the fractions: $x = \frac{1 \times 2}{2 \times 3} = \frac{2}{6}$
4. Simplify the fraction: $x = \frac{1}{3}$ cup
5. You need $\frac{1}{3}$ cup of sugar.

🧵 Real-World Example 7: Sewing Project

Problem: You have a piece of fabric that is $3\frac{1}{2}$ feet long. You need to cut it into pieces that are $\frac{3}{4}$ foot long. How many pieces can you cut?

Solution:

1. Convert the mixed number to an improper fraction: $3\frac{1}{2} = \frac{7}{2}$
2. Let $n$ be the number of pieces you can cut.
3. The equation is: $n = \frac{7}{2} \div \frac{3}{4}$
4. To divide fractions, multiply by the reciprocal: $n = \frac{7}{2} \times \frac{4}{3}$
5. Multiply the fractions: $n = \frac{7 \times 4}{2 \times 3} = \frac{28}{6}$
6. Simplify the fraction: $n = \frac{14}{3}$
7. Convert the improper fraction to a mixed number: $n = 4\frac{2}{3}$
8. You can cut 4 whole pieces of fabric.

🎯 Conclusion

Solving equations with fractions is a fundamental skill with numerous practical applications. By understanding the underlying principles and practicing with real-world examples, you can confidently tackle problems involving fractions in various contexts.