Calculate Without A Calculator Step-by-Step Solution

Hey guys! Today, we're going to tackle a math problem that might look a little intimidating at first glance, but trust me, we'll break it down step-by-step so it's super easy to understand. We're going to solve the expression 4 1/2 - 1/3 of 4 + 1/6 without reaching for a calculator. That's right, we're going old school and doing it all by hand! This is a fantastic way to sharpen your mental math skills and really understand the order of operations. So, let's dive in and get started!

Understanding the Order of Operations (PEMDAS/BODMAS)

Before we jump into the problem, it's crucial to remember the order of operations. You might have heard of it as PEMDAS or BODMAS, but it's the same thing. This handy acronym tells us the sequence in which we need to perform mathematical operations:

• Parentheses (or Brackets)
• Exponents (or Orders)
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

This order is like the golden rule of math. If you follow it, you'll always arrive at the correct answer. If you ignore it, well, things can get messy pretty quickly!

In our problem, we have subtraction, a fraction "of" (which means multiplication), addition, and fractions. So, according to PEMDAS/BODMAS, we need to handle the "of" (multiplication) part first. This means we'll calculate 1/3 of 4 before we do anything else. Once we've tackled that, we'll move on to the rest of the problem, working from left to right for addition and subtraction.

Remember, guys, understanding the order of operations is half the battle. It's like having a roadmap for solving the problem. Without it, you might end up taking a wrong turn and getting lost. So, keep PEMDAS/BODMAS in mind as we work through the steps!

Step 1: Tackling the "Of" (Multiplication)

Okay, let's zoom in on the heart of the problem: 1/3 of 4. As we discussed, "of" in math terms means multiplication. So, we're really trying to figure out what 1/3 multiplied by 4 is. Now, how do we multiply a fraction by a whole number? It's easier than you think!

To multiply a fraction by a whole number, we can simply rewrite the whole number as a fraction with a denominator of 1. So, 4 becomes 4/1. Now we have a straightforward multiplication problem: (1/3) * (4/1). To multiply fractions, we multiply the numerators (the top numbers) together and the denominators (the bottom numbers) together.

So, 1 multiplied by 4 is 4, and 3 multiplied by 1 is 3. This gives us the fraction 4/3. Ta-da! We've successfully calculated 1/3 of 4. This is a crucial step, guys, because it simplifies our original expression and gets us closer to the final answer.

Now, let's pause for a moment and think about what we've done. We've taken a potentially confusing part of the problem – the "1/3 of 4" – and transformed it into a simple improper fraction, 4/3. This is the power of breaking down a problem into smaller, more manageable steps. It makes the whole process less daunting and more, dare I say, enjoyable! So, let's carry this momentum forward as we move on to the next step.

Step 2: Converting the Mixed Number to an Improper Fraction

Now that we've handled the multiplication, let's turn our attention to the mixed number in the original expression: 4 1/2. Mixed numbers can be a little tricky to work with directly, so the best approach is to convert them into improper fractions. An improper fraction is simply a fraction where the numerator (the top number) is greater than or equal to the denominator (the bottom number).

So, how do we convert 4 1/2 into an improper fraction? Here's the magic formula: we multiply the whole number (4) by the denominator (2) and then add the numerator (1). This gives us the new numerator. The denominator stays the same. Let's do it step-by-step:

1. Multiply the whole number (4) by the denominator (2): 4 * 2 = 8
2. Add the numerator (1) to the result: 8 + 1 = 9
3. Keep the original denominator (2).

So, 4 1/2 is equivalent to 9/2 as an improper fraction. See? Not too scary, right? This conversion is essential because it allows us to perform addition and subtraction with fractions much more easily. We can't really add or subtract mixed numbers directly without going through this process.

Think of it like this: converting to an improper fraction puts all our numbers on the same playing field. It allows us to work with them consistently and avoid any potential errors. Now that we've transformed 4 1/2 into 9/2, we're one step closer to solving the entire problem. Let's keep going, guys!

Step 3: Rewriting the Expression and Finding a Common Denominator

Okay, we've done some crucial groundwork. We've calculated 1/3 of 4, which gave us 4/3, and we've converted the mixed number 4 1/2 into the improper fraction 9/2. Now, let's rewrite the entire expression with these new values:

9/2 - 4/3 + 1/6

Doesn't that look a little more manageable? We've gotten rid of the "of" and the mixed number, leaving us with just fractions to add and subtract. But here's the catch: we can't add or subtract fractions unless they have the same denominator. This is like trying to add apples and oranges – they're different things, and we need a common unit to work with.

So, our next task is to find a common denominator for 2, 3, and 6. The easiest way to do this is to find the least common multiple (LCM) of these numbers. The LCM is the smallest number that all three denominators divide into evenly. Let's think about the multiples of each number:

• Multiples of 2: 2, 4, 6, 8, 10...
• Multiples of 3: 3, 6, 9, 12...
• Multiples of 6: 6, 12, 18...

Aha! The least common multiple of 2, 3, and 6 is 6. This means we need to convert all our fractions so they have a denominator of 6. This is like finding a common language for our fractions so they can all communicate and play nicely together.

Now, let's move on to the next step and actually convert those fractions!

Step 4: Converting Fractions to the Common Denominator

We've identified that 6 is our magic number – the common denominator that will allow us to add and subtract our fractions. Now, we need to convert each fraction in our expression (9/2, 4/3, and 1/6) so that it has a denominator of 6. Let's take them one at a time:

• 9/2: To get a denominator of 6, we need to multiply the current denominator (2) by 3. But remember, whatever we do to the denominator, we must also do to the numerator to keep the fraction equivalent. So, we multiply both the numerator and denominator by 3: (9 * 3) / (2 * 3) = 27/6
• 4/3: To get a denominator of 6, we need to multiply the current denominator (3) by 2. So, we multiply both the numerator and denominator by 2: (4 * 2) / (3 * 2) = 8/6
• 1/6: This fraction already has a denominator of 6, so we don't need to do anything to it! It remains 1/6.

Now, look at what we've accomplished! We've transformed our fractions into equivalent fractions that all share the same denominator. This is a huge step forward because it means we can finally perform the addition and subtraction operations. Our expression now looks like this:

27/6 - 8/6 + 1/6

See how much simpler things become when we have a common denominator? It's like building a bridge that connects all the fractions, allowing us to easily travel between them. Now, let's cross that bridge and get to the final calculation!

Step 5: Performing the Addition and Subtraction

We've reached the final stretch, guys! We have our expression with a common denominator: 27/6 - 8/6 + 1/6. Now, we can simply perform the addition and subtraction operations from left to right. This is the beauty of having a common denominator – we can just focus on the numerators.

First, let's subtract 8/6 from 27/6: 27/6 - 8/6 = (27 - 8) / 6 = 19/6

Now, we have 19/6 + 1/6. Let's add these together: 19/6 + 1/6 = (19 + 1) / 6 = 20/6

So, the result of our calculation is 20/6. We've done it! We've successfully solved the problem without using a calculator. But, before we celebrate too much, let's take one more step to simplify our answer.

Step 6: Simplifying the Improper Fraction (If Possible)

We've arrived at the answer 20/6, which is an improper fraction. While it's a perfectly valid answer, it's often good practice to simplify fractions to their lowest terms. This means finding the greatest common factor (GCF) of the numerator and denominator and dividing both by it.

In this case, the greatest common factor of 20 and 6 is 2. So, let's divide both the numerator and denominator by 2:

• 20 / 2 = 10
• 6 / 2 = 3

This gives us the simplified improper fraction 10/3. This fraction is in its simplest form because 10 and 3 have no common factors other than 1.

And there you have it, guys! The final answer to our problem, 4 1/2 - 1/3 of 4 + 1/6, is 10/3. We did it! We tackled a multi-step math problem without a calculator, and we learned a lot along the way. We reviewed the order of operations, converted mixed numbers to improper fractions, found common denominators, and simplified fractions. These are all essential skills in mathematics, and you should be proud of yourselves for mastering them!

Conclusion: You're a Math Rockstar!

Congratulations, you've successfully navigated this mathematical journey! By breaking down the problem into manageable steps and applying the principles of PEMDAS/BODMAS, fraction manipulation, and simplification, you've proven that you can tackle complex calculations without relying on a calculator. Remember, the key to success in math (and in many other areas of life) is to approach challenges with a systematic and methodical mindset. So, the next time you encounter a tricky problem, take a deep breath, break it down, and remember the skills you've learned today. You've got this!