Solve Multiplications & Real Proof: Step-by-Step!

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Solve Multiplications & Real Proof: Step-by-Step!

Hey guys! Today, we're diving into some multiplication problems and, just to be super sure, we'll do the "prova real" (real proof) to check our answers. Let's get started and make math a little less scary, shall we? We'll break down each problem step by step so it's easy to follow. Remember, math is like building blocks – once you get the basics, you can build anything!

689 * 9

First up, we have 689 multiplied by 9. This is a classic multiplication problem, and we'll go through it nice and slow. Let's set it up:

  689
*   9
-----

Now, let's multiply each digit of 689 by 9:

  • 9 * 9 = 81. Write down 1 and carry over 8.
  • 9 * 8 = 72. Add the carried-over 8, so 72 + 8 = 80. Write down 0 and carry over 8.
  • 9 * 6 = 54. Add the carried-over 8, so 54 + 8 = 62. Write down 62.

So, our multiplication looks like this:

  689
*   9
-----
 6201

Therefore, 689 * 9 = 6201.

Real Proof (Prova Real)

To check if our answer is correct, we'll use the "prova real," which involves division. We'll divide our answer (6201) by 9 and see if we get 689.

6201 ÷ 9 = ?

Let's perform the division:

  • 9 goes into 62 six times (6 * 9 = 54). 62 - 54 = 8. Bring down the 0.
  • 9 goes into 80 eight times (8 * 9 = 72). 80 - 72 = 8. Bring down the 1.
  • 9 goes into 81 nine times (9 * 9 = 81). 81 - 81 = 0.

So, 6201 ÷ 9 = 689. This confirms that our multiplication was correct!

In summary: 689 * 9 = 6201, and the real proof (6201 ÷ 9 = 689) checks out. Great job on this one! You've nailed the basics of multiplication and verification. Keep practicing, and you'll become a math whiz in no time. Remember, every problem is a chance to learn something new and sharpen your skills. The more you practice, the easier it gets!

623 * 17

Next, we're tackling 623 multiplied by 17. This one's a bit bigger, but don't worry, we'll take it step by step. Set it up like this:

  623
*  17
-----

Now, let's multiply 623 by 7 and then by 1:

  • 7 * 3 = 21. Write down 1 and carry over 2.
  • 7 * 2 = 14. Add the carried-over 2, so 14 + 2 = 16. Write down 6 and carry over 1.
  • 7 * 6 = 42. Add the carried-over 1, so 42 + 1 = 43. Write down 43.

So, the first part of our multiplication looks like this:

  623
*  17
-----
 4361

Now, let's multiply 623 by 1. Remember to add a 0 as a placeholder because we're multiplying by 10 (1 is in the tens place):

  • 1 * 3 = 3. Write down 3.
  • 1 * 2 = 2. Write down 2.
  • 1 * 6 = 6. Write down 6.
  623
*  17
-----
 4361
6230

Now, add these two results together:

  4361
+ 6230
-----
10591

Therefore, 623 * 17 = 10591.

Real Proof (Prova Real)

To verify our answer, we'll divide 10591 by 17 and see if we get 623.

10591 ÷ 17 = ?

Let's do the division:

  • 17 goes into 105 six times (6 * 17 = 102). 105 - 102 = 3. Bring down the 9.
  • 17 goes into 39 two times (2 * 17 = 34). 39 - 34 = 5. Bring down the 1.
  • 17 goes into 51 three times (3 * 17 = 51). 51 - 51 = 0.

So, 10591 ÷ 17 = 623. This confirms our multiplication is correct!

In conclusion: 623 * 17 = 10591, and the real proof (10591 ÷ 17 = 623) validates our answer. Fantastic work! You're getting the hang of these more complex multiplications. Keep practicing these steps, and soon you'll be able to do them in your head!

357 * 12

Lastly, let's solve 357 multiplied by 12. We're on a roll now! Set it up like this:

  357
*  12
-----

Multiply 357 by 2 and then by 1:

  • 2 * 7 = 14. Write down 4 and carry over 1.
  • 2 * 5 = 10. Add the carried-over 1, so 10 + 1 = 11. Write down 1 and carry over 1.
  • 2 * 3 = 6. Add the carried-over 1, so 6 + 1 = 7. Write down 7.

So, the first part of our multiplication is:

  357
*  12
-----
  714

Now, multiply 357 by 1. Add a 0 as a placeholder:

  • 1 * 7 = 7. Write down 7.
  • 1 * 5 = 5. Write down 5.
  • 1 * 3 = 3. Write down 3.
  357
*  12
-----
  714
 3570

Add these two results together:

  714
+3570
-----
4284

Therefore, 357 * 12 = 4284.

Real Proof (Prova Real)

To double-check, we'll divide 4284 by 12 and see if we get 357.

4284 ÷ 12 = ?

Let's divide:

  • 12 goes into 42 three times (3 * 12 = 36). 42 - 36 = 6. Bring down the 8.
  • 12 goes into 68 five times (5 * 12 = 60). 68 - 60 = 8. Bring down the 4.
  • 12 goes into 84 seven times (7 * 12 = 84). 84 - 84 = 0.

So, 4284 ÷ 12 = 357. This confirms our answer!

In summary: 357 * 12 = 4284, and the real proof (4284 ÷ 12 = 357) confirms our calculation. You're doing amazing! You've successfully solved all three multiplication problems and verified your answers using the "prova real." Keep up the great work, and you'll become a multiplication master!

Conclusion

So, there you have it! We've solved three multiplication problems and used the "prova real" to make sure our answers were spot on. Remember, the key to mastering math is practice, practice, practice! Don't be afraid to make mistakes – that's how we learn. Keep challenging yourself, and you'll be surprised at how much you can achieve. Until next time, happy calculating! Keep your mind sharp and always double-check your work!