Solve: If A/4 = 5/b, Find A * B

by ADMIN 32 views

Let's dive into solving this math problem together, guys! We're given the equation a/4 = 5/b, where b is not equal to zero, and our mission is to find the value of the product a * b. Don't worry, it's simpler than it looks! This is a classic problem that involves cross-multiplication and a bit of algebraic thinking. So, grab your thinking caps, and let's get started!

Understanding the Problem

Before we jump into solving, let's make sure we fully grasp what the problem is asking. We have a proportion, which is essentially two fractions that are equal to each other. The proportion is a/4 = 5/b. Our goal is to find the value of a * b, which means we need to figure out what happens when we multiply 'a' and 'b' together. The condition b ≠ 0 is important because division by zero is undefined in mathematics, so we need to make sure our solution respects this rule.

Key Points to Remember:

  • We are dealing with a proportion: a/4 = 5/b.
  • We need to find the value of the product a * b.
  • The condition b ≠ 0 is given to ensure the equation is valid.

Solving the Proportion

The most straightforward way to solve this type of problem is by using cross-multiplication. Cross-multiplication involves multiplying the numerator of the first fraction by the denominator of the second fraction, and vice versa. In our case, this means multiplying 'a' by 'b' and '4' by '5'. Let's do it step by step:

  1. Start with the given proportion: a/4 = 5/b.
  2. Cross-multiply: a * b = 4 * 5.
  3. Simplify: a * b = 20.

And there you have it! The value of the product a * b is 20. This method works because, in essence, we are multiplying both sides of the equation by 4b to eliminate the denominators. As long as b is not zero, this operation is perfectly valid.

Why Cross-Multiplication Works

For those who are curious, let's briefly explain why cross-multiplication works. When we have a proportion a/4 = 5/b, we can think of it as an equation. To solve for a * b, we want to eliminate the fractions. We can do this by multiplying both sides of the equation by the least common multiple (LCM) of the denominators, which in this case is 4b.

So, we have:

(a/4) * (4b) = (5/b) * (4b)

On the left side, the '4's cancel out, leaving us with a * b. On the right side, the 'b's cancel out, leaving us with 5 * 4. Therefore, we get:

a * b = 20

This is exactly what we found using cross-multiplication! Cross-multiplication is just a shortcut that combines these steps into one easy operation.

Checking Our Answer

It's always a good idea to check our answer to make sure it makes sense. We found that a * b = 20. Let's see if we can find values for 'a' and 'b' that satisfy both the original equation and this condition. For example, if a = 5, then b = 4. Plugging these values into the original equation:

5/4 = 5/4

This is true! So, our answer is consistent with the given information. Another example: if a = 10, then b = 2. Plugging these values into the original equation:

10/4 = 5/2

Simplifying 10/4 gives us 5/2, so this also works. No matter what values we choose for 'a' and 'b' as long as their product is 20, the original equation will hold true. This confirms that our answer is correct.

Common Mistakes to Avoid

When solving problems like this, there are a few common mistakes that students often make. Here are some pitfalls to watch out for:

  1. Forgetting the Condition b ≠ 0: Always remember that b cannot be zero, as division by zero is undefined. This condition is crucial for the validity of the equation.
  2. Incorrect Cross-Multiplication: Make sure you multiply the numerator of each fraction by the denominator of the other fraction. It's easy to mix this up if you're not careful.
  3. Algebraic Errors: Double-check your algebra to avoid simple mistakes like incorrect simplification or sign errors. These can lead to the wrong answer.
  4. Not Checking Your Answer: Always take a moment to check your answer to make sure it makes sense in the context of the problem. This can help you catch any errors you might have made.

Real-World Applications

While this problem might seem purely theoretical, proportions and ratios have countless real-world applications. Here are a few examples:

  • Cooking: When scaling recipes up or down, you need to maintain the correct ratios of ingredients. For example, if a recipe calls for 1 cup of flour and 2 eggs, and you want to double the recipe, you need to double both the flour and the eggs to maintain the correct proportion.
  • Map Reading: Maps use scales to represent distances on the ground. A scale of 1:10,000 means that 1 unit on the map represents 10,000 units in the real world. Proportions are essential for calculating actual distances from map measurements.
  • Finance: Financial ratios, such as the debt-to-equity ratio, are used to assess the financial health of a company. These ratios are based on proportions and help investors make informed decisions.
  • Engineering: Engineers use proportions extensively in designing structures and machines. For example, when designing a bridge, engineers need to ensure that the proportions of different components are correct to ensure the bridge can support the intended load.

Practice Problems

To solidify your understanding of proportions and ratios, here are a few practice problems for you to try:

  1. If x/3 = 7/y, find the value of x * y.
  2. If 2/p = q/5, find the value of p * q.
  3. If a/6 = 8/b, find the value of a * b.

Try solving these problems on your own, and feel free to share your answers in the comments below! Practicing is the best way to improve your skills and build confidence.

Conclusion

So, to wrap it up, when we're given a problem like a/4 = 5/b and asked to find the value of a * b, the key is to use cross-multiplication. This simple technique allows us to quickly and easily solve for the desired product. Remember to always double-check your work and be mindful of any given conditions, such as b ≠ 0. With a little practice, you'll become a pro at solving proportions and ratios! Keep up the great work, guys, and happy problem-solving!