Math Problems: Practice Your Fraction Skills
Hey math whizzes! Today, we're diving deep into the awesome world of fractions and exponents. Get ready to flex those mathematical muscles as we tackle some super engaging problems that will really help you get a grip on these concepts. We've got five juicy problems for you, covering squares of fractions and mixed numbers multiplied by other fractions. It's going to be a fun ride, so grab your calculators (or just your brilliant brains!) and let's get started!
Problem 1: Cracking the Code of (5/7)² × 7/15
Alright guys, let's kick things off with our first challenge: Calculate (5/7)² × 7/15. This problem is a fantastic way to practice squaring a fraction and then multiplying it by another fraction. Remember, when you square a fraction, you square both the numerator and the denominator. So, (5/7)² means (5/7) × (5/7), which equals (5²)/(7²), or 25/49. Now we need to multiply this result by 7/15. So, we have (25/49) × (7/15). When multiplying fractions, you multiply the numerators together and the denominators together: (25 × 7) / (49 × 15). Before we do the multiplication, let's see if we can simplify. We know that 25 and 15 share a common factor of 5 (25 = 5 × 5, and 15 = 3 × 5). Also, 49 and 7 share a common factor of 7 (49 = 7 × 7, and 7 = 1 × 7). So, we can rewrite our multiplication as: ( (5 × 5) × 7 ) / ( (7 × 7) × (3 × 5) ). Now, we can cancel out a 7 from the numerator and the denominator, and a 5 from the numerator and the denominator. This leaves us with (5 × 1) / (7 × 3). Multiplying these out gives us 5/21. Awesome job if you got that! This process of simplifying before multiplying is a super useful trick that makes calculations much easier.
Problem 2: Tackling (3/4)² × 32/45
Next up, we've got Calculate (3/4)² × 32/45. This problem is pretty similar to the first one, just with different numbers. First things first, let's square our fraction (3/4). That means (3/4) × (3/4), which is (3²)/(4²), resulting in 9/16. Now, we need to multiply this by 32/45. So, we have (9/16) × (32/45). Again, we multiply the numerators and denominators: (9 × 32) / (16 × 45). Let's look for simplifications. We can see that 16 is a factor of 32 (32 = 16 × 2). We can also see that 9 is a factor of 45 (45 = 9 × 5). So, we can rewrite our expression as: (9 × (16 × 2)) / (16 × (9 × 5)). Now we can cancel out the 16s and the 9s. This leaves us with (1 × 2) / (1 × 5), which equals 2/5. Way to go, team! This problem reinforces how important it is to spot those common factors.
Problem 3: The Simplicity of (1/2)² × 4/5
Let's move on to an easier one, shall we? Problem 3 is Calculate (1/2)² × 4/5. This is a great problem to build your confidence. First, we square the fraction (1/2). (1/2)² is (1²/2²) which is 1/4. Now, we multiply 1/4 by 4/5. So, (1/4) × (4/5). Following our rule, we multiply the numerators and denominators: (1 × 4) / (4 × 5). Look at that! We have a 4 in the numerator and a 4 in the denominator. We can cancel them both out, leaving us with 1/5. So, the answer is 1/5. See? Sometimes the simplest approach is the best. This problem shows how even a small fraction can get bigger when you square it, but then return to a simpler form when multiplied correctly.
Problem 4: Conquering Mixed Numbers: (2 5/7)² × 14/19
Alright folks, this is where things get a little more interesting! We need to Calculate (2 5/7)² × 14/19. The first step with mixed numbers is always to convert them into improper fractions. To convert 2 5/7, we multiply the whole number (2) by the denominator (7) and add the numerator (5): (2 × 7) + 5 = 14 + 5 = 19. So, 2 5/7 is the same as 19/7. Now, we need to square this improper fraction: (19/7)². This means (19/7) × (19/7), which is (19²)/(7²). 19² is 361, and 7² is 49. So, we have 361/49. Now, we multiply this by 14/19: (361/49) × (14/19). Let's simplify before multiplying. We know 19 is a factor of 361 (361 = 19 × 19). We also know that 7 is a factor of both 49 (49 = 7 × 7) and 14 (14 = 7 × 2). So, we can rewrite our expression: ( (19 × 19) / (7 × 7) ) × ( (7 × 2) / 19 ). Now we can cancel out one of the 19s and one of the 7s. This leaves us with (19 / 7) × (2 / 1). Multiplying across gives us (19 × 2) / (7 × 1), which is 38/7. Alternatively, we could express this as a mixed number: 5 3/7. Great work tackling those mixed numbers!
Problem 5: The Grand Finale: (2 1/3)² × 27/98
We've reached the final boss, guys! It's time to Calculate (2 1/3)² × 27/98. Just like before, the first step is to convert the mixed number 2 1/3 into an improper fraction. Multiply the whole number (2) by the denominator (3) and add the numerator (1): (2 × 3) + 1 = 6 + 1 = 7. So, 2 1/3 is equal to 7/3. Now, we square this fraction: (7/3)² which is (7²/3²) = 49/9. Our final step is to multiply 49/9 by 27/98. So, we have (49/9) × (27/98). Let's simplify! We can see that 9 is a factor of 27 (27 = 9 × 3). We also know that 49 is a factor of 98 (98 = 49 × 2). So, we can rewrite our expression: (49 / 9) × ( (9 × 3) / (49 × 2) ). Now, cancel out the 49s and the 9s. This leaves us with (1 / 1) × (3 / 2). Multiplying these together gives us 3/2. As a mixed number, this is 1 1/2. You absolutely crushed it! Solving these problems takes practice, and you guys are doing fantastic.
Wrapping It Up
So there you have it, math adventurers! We've worked through five problems involving squaring fractions and mixed numbers, and then multiplying them by other fractions. The key takeaways here are: always convert mixed numbers to improper fractions first, remember the rules for squaring fractions (square the numerator and the denominator), and don't forget to simplify before you multiply whenever possible. These techniques will make your math life so much easier. Keep practicing, and you'll become a fraction and exponent master in no time. Until next time, stay curious and keep those mathematical gears turning!