True Equations: Test Your Math Skills!

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Hey guys! Let's dive into some math and figure out which equations are actually true. This might seem tricky, but we'll break it down step by step. We've got a bunch of division problems involving mixed numbers, and our mission is to identify the ones that are correct. So, grab your calculators (or your pencils and paper, if you're feeling old-school!), and let's get started!

Decoding the Equations: A Step-by-Step Guide

Before we jump into solving the equations, let's talk strategy. When you see mixed numbers in division problems, the easiest way to tackle them is to convert them into improper fractions first. This makes the calculations much smoother. Remember, an improper fraction is when the numerator (the top number) is bigger than the denominator (the bottom number).

Why do we do this? Because dividing fractions is a breeze when they're in improper form! You simply flip the second fraction (the one you're dividing by) and multiply. Easy peasy!

Now, let’s look at each equation one by one. We'll convert the mixed numbers to improper fractions, perform the division, and see if the result matches the answer given in the equation. This is where the real fun begins, guys! So buckle up and let’s get cracking!

Equation 1: 8 2/3 ÷ 4 1/8 = 2 1/10

Okay, let's start with our first equation: 8 2/3 ÷ 4 1/8 = 2 1/10. This is where we put our strategy into action. First, we'll convert those mixed numbers into improper fractions.

  • 8 2/3: To convert this, we multiply the whole number (8) by the denominator (3), which gives us 24. Then, we add the numerator (2), resulting in 26. We keep the same denominator, so 8 2/3 becomes 26/3.
  • 4 1/8: Similarly, we multiply 4 by 8, which is 32, and add 1, giving us 33. Keep the denominator 8, so 4 1/8 becomes 33/8.

Now our equation looks like this: 26/3 ÷ 33/8. Remember, to divide fractions, we flip the second fraction and multiply. So, we change the division to multiplication and flip 33/8 to 8/33. Our new equation is 26/3 * 8/33.

Let’s multiply the numerators: 26 * 8 = 208. Then, multiply the denominators: 3 * 33 = 99. So, we have 208/99. Now, we need to simplify this improper fraction and convert it back to a mixed number to see if it matches the answer given in the original equation.

To convert 208/99 to a mixed number, we divide 208 by 99. It goes in 2 times (2 * 99 = 198), with a remainder of 10. So, 208/99 is equal to 2 10/99.

Now, let's compare this to the original answer given in the equation, which was 2 1/10. 2 10/99 is NOT equal to 2 1/10. Therefore, this equation is FALSE. We’ve tackled the first one, guys! Let's move on to the next!

Equation 2: 11 1/2 ÷ 8 3/4 = 1 11/35

Alright, let's jump into the second equation: 11 1/2 ÷ 8 3/4 = 1 11/35. Same strategy here, guys! We're going to convert those mixed numbers to improper fractions, do the division, and then see if we get the same result.

  • 11 1/2: Multiply 11 by 2, which is 22, and add 1, giving us 23. Keep the denominator 2, so 11 1/2 becomes 23/2.
  • 8 3/4: Multiply 8 by 4, which is 32, and add 3, giving us 35. Keep the denominator 4, so 8 3/4 becomes 35/4.

Now, our equation looks like this: 23/2 ÷ 35/4. Remember the division rule? Flip the second fraction and multiply! So, we flip 35/4 to 4/35, and the equation becomes 23/2 * 4/35.

Multiply the numerators: 23 * 4 = 92. Multiply the denominators: 2 * 35 = 70. So, we have 92/70. We can simplify this fraction before converting it to a mixed number. Both 92 and 70 are divisible by 2, so let’s divide both by 2. This gives us 46/35.

Now, let's convert 46/35 to a mixed number. Divide 46 by 35. It goes in 1 time (1 * 35 = 35), with a remainder of 11. So, 46/35 is equal to 1 11/35.

Let's check if this matches the answer given in the original equation: 1 11/35. Bingo! They are the same! This equation is TRUE! We’ve got one true equation down, guys. Let’s keep going!

Equation 3: 2 5/6 ÷ 3 1/4 = 3 1/9

Time for equation number three: 2 5/6 ÷ 3 1/4 = 3 1/9. We know the drill by now, right? Let’s convert those mixed numbers into improper fractions and get this show on the road!

  • 2 5/6: Multiply 2 by 6, which is 12, and add 5, giving us 17. Keep the denominator 6, so 2 5/6 becomes 17/6.
  • 3 1/4: Multiply 3 by 4, which is 12, and add 1, giving us 13. Keep the denominator 4, so 3 1/4 becomes 13/4.

So, our equation is now: 17/6 ÷ 13/4. Let's flip the second fraction and multiply. We flip 13/4 to 4/13, making the equation 17/6 * 4/13.

Multiply the numerators: 17 * 4 = 68. Multiply the denominators: 6 * 13 = 78. So, we have 68/78. We can simplify this fraction! Both 68 and 78 are divisible by 2, so let’s divide both by 2. This simplifies to 34/39.

Now, let's check if 34/39 is equal to 3 1/9. Hmmm… 34/39 is less than 1 (since the numerator is smaller than the denominator), while 3 1/9 is greater than 3. These are definitely not the same! So, this equation is FALSE. We’re on a roll, guys!

Equation 4: 12 2/3 ÷ ? (Incomplete Equation)

Okay, so we've hit a snag. It looks like the fourth equation, 12 2/3 ÷ ?, is incomplete. We can't determine if it's true or false because we don't know what 12 2/3 is being divided by! It's like trying to bake a cake without all the ingredients – it’s just not gonna work!

Since we can't solve it, we can't include it in our list of true equations. Sometimes in math, you just don’t have enough information to find an answer, and that’s perfectly okay! It’s all part of the learning process.

Conclusion: Finding the Truth in Equations

So, after carefully working through each equation, we've discovered that only one of them is true. Give yourself a pat on the back if you followed along and got the same answer! Math can be challenging, but by breaking down problems step by step and using the right strategies, we can conquer even the trickiest equations. Remember, converting mixed numbers to improper fractions is a game-changer when it comes to division!

The only true equation from the list is:

  • 11 1/2 ÷ 8 3/4 = 1 11/35

Great job, guys! Keep practicing, and you'll become math whizzes in no time. And remember, it's okay to make mistakes – that's how we learn! Now, go forth and conquer more math problems!