Math Problems: Fill In The Blanks To Make Equations True!

Hey guys! Let's dive into some cool math problems where we need to fill in the blanks to make the equations true. It's like solving a puzzle, but with numbers! We'll break down each problem step-by-step, so it's super easy to follow. Get ready to sharpen your math skills!

Understanding the Distributive Property

Before we jump into solving these problems, it's really important to understand the distributive property. What exactly is the distributive property? It's a fundamental concept in mathematics that allows us to simplify expressions involving multiplication and addition (or subtraction) inside parentheses. In simpler terms, the distributive property states that multiplying a number by a sum (or difference) is the same as multiplying the number by each term in the sum (or difference) individually and then adding (or subtracting) the products.

The general form of the distributive property is:

• a × (b + c) = a × b + a × c
• a × (b - c) = a × b - a × c

Where 'a', 'b', and 'c' can be any numbers. This property is incredibly useful for simplifying expressions and solving equations. For example, consider the expression 3 × (4 + 5). Using the distributive property, we can rewrite this as (3 × 4) + (3 × 5), which simplifies to 12 + 15 = 27. You'll find that 3 × (4 + 5) = 3 × 9 = 27 as well, so both methods give you the same result. This property works because multiplication distributes over addition and subtraction.

Why is the distributive property so important? Well, it shows up everywhere in algebra and beyond. Being able to use it fluently will make your math life a whole lot easier. Whether you're simplifying complex expressions or solving equations, the distributive property is your friend. By mastering this concept, you will find solving the problems below easier. Let's move on to those math problems now, applying the distributive property to find the missing numbers and balance those equations!

Problem 1: 4 × (3 + 25) = 4 × 3 + 4 × ?

In this first problem, we're using the distributive property in a straightforward way. We need to figure out what number goes in the blank to make the equation true. Let's break it down step-by-step.

The left side of the equation is 4 × (3 + 25). According to the distributive property, this should be equal to 4 × 3 + 4 × something. So, what's that "something"? Well, it's the other number inside the parentheses, which is 25. Therefore, the equation becomes:

4 × (3 + 25) = 4 × 3 + 4 × 25

Let's verify this. First, calculate the left side:

4 × (3 + 25) = 4 × 28 = 112

Now, calculate the right side:

4 × 3 + 4 × 25 = 12 + 100 = 112

Both sides are equal, so our answer is correct!

So, the missing number is 25. This problem illustrates how the distributive property works by breaking down a multiplication problem into smaller, more manageable parts. When you see an equation in this format, just remember to distribute the number outside the parentheses to each number inside.

Answer: 25

Problem 2: (124 + 32) × 3 = 124 × ? + ? × 3

Okay, let's tackle the next problem. This one might look a bit tricky at first, but don't worry, we'll get through it together! The equation is (124 + 32) × 3 = 124 × ? + ? × 3. We need to find the two missing numbers that make the equation true. Again, this is an application of the distributive property, just like the first problem.

The left side of the equation is (124 + 32) × 3. According to the distributive property, this should be equal to 124 multiplied by something plus something else multiplied by 3. Looking at the left side, we can see that 3 is being multiplied by both 124 and 32. So, we can rewrite the equation as:

(124 + 32) × 3 = 124 × 3 + 32 × 3

Now we can see that the missing numbers are 3 and 32. Let's plug them into the equation:

(124 + 32) × 3 = 124 × 3 + 32 × 3

Let's verify this. Calculate the left side:

(124 + 32) × 3 = (156) × 3 = 468

Now, calculate the right side:

124 × 3 + 32 × 3 = 372 + 96 = 468

Both sides are equal, so our answer is correct!

Answers: 3 and 32

Problem 3: (42 - 20) × 5 = ? × 5 - ? × 5

Alright, time for the next problem! This one involves subtraction, but don't sweat it, the same distributive property rules apply. The equation is (42 - 20) × 5 = ? × 5 - ? × 5. We need to find the two missing numbers that make this equation true. Let's dive in!

The left side of the equation is (42 - 20) × 5. Applying the distributive property, we need to distribute the 5 to both the 42 and the 20. So, the equation becomes:

(42 - 20) × 5 = 42 × 5 - 20 × 5

Now we can easily see what the missing numbers are: 42 and 20. Let's plug them in:

(42 - 20) × 5 = 42 × 5 - 20 × 5

Time to verify. Calculate the left side:

(42 - 20) × 5 = (22) × 5 = 110

Now, calculate the right side:

42 × 5 - 20 × 5 = 210 - 100 = 110

Both sides are equal, so we got it right!

Answers: 42 and 20

Problem 4: (250 - 120) × 12 = 250 × ? - ? × 12

Last but not least, let's solve the final problem! The equation is (250 - 120) × 12 = 250 × ? - ? × 12. We need to find the missing numbers that make this equation true. By now, you're probably getting the hang of this, so let's jump right in.

The left side of the equation is (250 - 120) × 12. Using the distributive property, we need to distribute the 12 to both the 250 and the 120. This gives us:

(250 - 120) × 12 = 250 × 12 - 120 × 12

So, the missing numbers are 12 and 120. Let's plug them in:

(250 - 120) × 12 = 250 × 12 - 120 × 12

Now, let's verify. Calculate the left side:

(250 - 120) × 12 = (130) × 12 = 1560

And the right side:

250 × 12 - 120 × 12 = 3000 - 1440 = 1560

Both sides are equal, so we nailed it!

Answers: 12 and 120

Conclusion

Great job, guys! You've successfully solved all the math problems by filling in the blanks using the distributive property. Remember, the distributive property is a powerful tool that can help you simplify complex equations and make math a whole lot easier. Keep practicing, and you'll become a math whiz in no time!