Fraction Arithmetic: Step-by-Step Solutions

by TheNnagam 44 views

Hey guys! Let's break down some fraction problems together. Fractions can seem tricky, but with a few simple steps, they become super manageable. We'll cover addition and division, making sure you understand each step along the way. So, grab your pencils, and let's dive in!

a. 1/2 + 3/4

When adding fractions, the first thing we need to do is find a common denominator. This is a number that both denominators (the bottom numbers) can divide into evenly. In this case, we have denominators of 2 and 4. The smallest number that both 2 and 4 can divide into is 4. So, we'll use 4 as our common denominator.

Now, we need to convert each fraction to have this new denominator. For the first fraction, 1/2, we need to multiply both the numerator (the top number) and the denominator by 2 to get an equivalent fraction with a denominator of 4:

1/2 * (2/2) = 2/4

The second fraction, 3/4, already has a denominator of 4, so we don't need to change it.

Now we can add the fractions:

2/4 + 3/4

When adding fractions with a common denominator, we simply add the numerators and keep the denominator the same:

(2 + 3) / 4 = 5/4

So, 1/2 + 3/4 = 5/4. We can also express this as a mixed number. Since 4 goes into 5 once with a remainder of 1, we can write 5/4 as 1 1/4. Understanding how to manipulate fractions is essential for various mathematical concepts, from basic arithmetic to more complex algebra and calculus.

b. 2/5 + 3/7

Alright, let's tackle another addition problem! This time, we have 2/5 + 3/7. Again, our first step is to find a common denominator. We need a number that both 5 and 7 can divide into. Since 5 and 7 are both prime numbers (they are only divisible by 1 and themselves), the easiest way to find a common denominator is to multiply them together:

5 * 7 = 35

So, our common denominator is 35. Now, we need to convert each fraction to have this denominator. For the first fraction, 2/5, we multiply both the numerator and denominator by 7:

2/5 * (7/7) = 14/35

For the second fraction, 3/7, we multiply both the numerator and denominator by 5:

3/7 * (5/5) = 15/35

Now we can add the fractions:

14/35 + 15/35

Add the numerators and keep the denominator the same:

(14 + 15) / 35 = 29/35

So, 2/5 + 3/7 = 29/35. In this case, 29/35 is already in its simplest form because 29 is a prime number and doesn't share any common factors with 35. Mastering fraction addition is crucial as it lays the groundwork for understanding more advanced topics such as rational expressions and equation solving.

c. Solve fraction problem c

Okay, let's pretend we have a fraction problem 'c' here. Since the specific problem isn't given, let's create a general example to illustrate the process. Suppose problem 'c' is: 1/3 + 2/5. Just like before, we need to find the least common denominator (LCD). In this case, the LCD of 3 and 5 is 15 (since 3 * 5 = 15).

Now, we convert each fraction to an equivalent fraction with a denominator of 15:

1/3 * (5/5) = 5/15

2/5 * (3/3) = 6/15

Next, we add the fractions:

5/15 + 6/15 = (5+6)/15 = 11/15

Therefore, if problem 'c' was 1/3 + 2/5, the answer would be 11/15. The key takeaway here is to always find the common denominator, convert the fractions, and then perform the addition. Practice with different fractions will help you become more comfortable with this process. You can apply the same method to solve any fraction addition problem, regardless of the numbers involved.

d. Solve fraction problem d

Let's imagine fraction problem 'd' is something like: 3/8 + 1/4. Again, we start by identifying the least common denominator (LCD). In this case, the LCD of 8 and 4 is 8, because 8 is divisible by both 8 and 4. Now, we convert each fraction to have a denominator of 8.

The first fraction, 3/8, already has the correct denominator, so we don't need to change it.

For the second fraction, 1/4, we multiply both the numerator and the denominator by 2 to get an equivalent fraction with a denominator of 8:

1/4 * (2/2) = 2/8

Now we can add the fractions:

3/8 + 2/8 = (3+2)/8 = 5/8

So, if problem 'd' was 3/8 + 1/4, the answer would be 5/8. Remember, identifying the LCD correctly is crucial for simplifying the addition process. Once you find the LCD, converting the fractions and adding them becomes straightforward. Regular practice helps you to recognize common denominators quickly and accurately.

e. Solve fraction problem e

Let’s say fraction problem 'e' is: 7/10 + 1/2. Our first step, as always, is to find the least common denominator (LCD). Between 10 and 2, the LCD is 10 since 10 is divisible by both 10 and 2. Now, we need to convert each fraction to have a denominator of 10.

The first fraction, 7/10, already has the correct denominator, so no change is needed.

For the second fraction, 1/2, we multiply both the numerator and the denominator by 5 to get an equivalent fraction with a denominator of 10:

1/2 * (5/5) = 5/10

Now, we add the two fractions:

7/10 + 5/10 = (7+5)/10 = 12/10

We can simplify 12/10 by dividing both the numerator and the denominator by their greatest common divisor, which is 2:

12/10 = (12 ÷ 2) / (10 ÷ 2) = 6/5

Therefore, if fraction problem 'e' was 7/10 + 1/2, the answer is 6/5, or 1 1/5 as a mixed number. Understanding and simplifying fractions is fundamental in many areas of mathematics, and consistent practice will boost your confidence and accuracy.

f. Solve fraction problem f

Alright, let's assume fraction problem 'f' is: 4/9 + 1/3. Our mission is to find the least common denominator (LCD). In this case, the LCD of 9 and 3 is 9, because 9 is divisible by both 9 and 3. Now, we convert each fraction to have a denominator of 9.

The first fraction, 4/9, already has the correct denominator.

For the second fraction, 1/3, we multiply both the numerator and the denominator by 3 to get an equivalent fraction with a denominator of 9:

1/3 * (3/3) = 3/9

Now we can add the fractions:

4/9 + 3/9 = (4+3)/9 = 7/9

So, if problem 'f' was 4/9 + 1/3, the answer would be 7/9. It's always a good practice to check if the final fraction can be simplified further, but in this case, 7/9 is already in its simplest form since 7 and 9 have no common factors other than 1.

g. Solve fraction problem g

Let’s consider fraction problem 'g' to be: 5/12 + 1/4. To start, we need to find the least common denominator (LCD) of 12 and 4. The LCD is 12, as 12 is divisible by both 12 and 4. Now, let's convert each fraction to have a denominator of 12.

The first fraction, 5/12, already has the correct denominator.

For the second fraction, 1/4, we multiply both the numerator and the denominator by 3 to get an equivalent fraction with a denominator of 12:

1/4 * (3/3) = 3/12

Now, we add the two fractions:

5/12 + 3/12 = (5+3)/12 = 8/12

We can simplify 8/12 by dividing both the numerator and the denominator by their greatest common divisor, which is 4:

8/12 = (8 ÷ 4) / (12 ÷ 4) = 2/3

So, if fraction problem 'g' was 5/12 + 1/4, the answer is 2/3. Consistently applying these steps not only improves your accuracy but also deepens your understanding of fraction arithmetic. Keep practicing to strengthen your skills!

h. Solve fraction problem h

Suppose fraction problem 'h' is: 2/3 + 1/6. The first thing we need to do is find the least common denominator (LCD) for 3 and 6. The LCD is 6, since 6 is divisible by both 3 and 6. Now, we convert each fraction to have a denominator of 6.

The second fraction, 1/6, already has the correct denominator.

For the first fraction, 2/3, we multiply both the numerator and the denominator by 2 to get an equivalent fraction with a denominator of 6:

2/3 * (2/2) = 4/6

Now, we add the fractions:

4/6 + 1/6 = (4+1)/6 = 5/6

Thus, if problem 'h' was 2/3 + 1/6, the answer is 5/6. With practice, you'll find that identifying common denominators and simplifying fractions becomes second nature. Always remember that consistent effort leads to mastery in mathematics.

i. 1/2 : 3/8

Now, let's switch gears to division! Dividing fractions is a little different than adding them. The trick is to remember to "keep, change, flip." This means:

  • Keep the first fraction as it is.
  • Change the division sign to a multiplication sign.
  • Flip the second fraction (reciprocal).

So, for 1/2 : 3/8, we keep 1/2, change the division to multiplication, and flip 3/8 to 8/3:

1/2 * 8/3

Now, we simply multiply the numerators and the denominators:

(1 * 8) / (2 * 3) = 8/6

Finally, we simplify the fraction. Both 8 and 6 are divisible by 2:

8/6 = (8 ÷ 2) / (6 ÷ 2) = 4/3

So, 1/2 : 3/8 = 4/3. We can also express this as a mixed number. Since 3 goes into 4 once with a remainder of 1, we can write 4/3 as 1 1/3. Dividing fractions might seem complex, but with the keep, change, flip method, it becomes straightforward. Keep practicing!

j. 5/9 : 2/7

Let's do one more division problem! We have 5/9 : 2/7. Remember our "keep, change, flip" rule:

  • Keep the first fraction (5/9).
  • Change the division to multiplication.
  • Flip the second fraction (2/7 becomes 7/2).

So, we have:

5/9 * 7/2

Now, multiply the numerators and the denominators:

(5 * 7) / (9 * 2) = 35/18

In this case, 35 and 18 don't share any common factors other than 1, so the fraction is already in its simplest form. However, we can express it as a mixed number. 18 goes into 35 once with a remainder of 17, so 35/18 = 1 17/18.

So, 5/9 : 2/7 = 35/18 or 1 17/18. Consistent application of the keep, change, flip method makes dividing fractions much easier. Remember to always check if your final answer can be simplified!

Hope this helps you understand fraction arithmetic better! Keep practicing, and you'll become a fraction master in no time!