Subtract These Fractions And Simplify Your Answer As Much As Possible:1. \[$\frac{20}{4} - \frac{3}{7}\$\]2. \[$2 \frac{7}{9} - 1 \frac{1}{5}\$\]3. \[$7 - 4 \frac{1}{6}\$\]

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Introduction

Fractions are a fundamental concept in mathematics, and simplifying them is an essential skill to master. In this article, we will explore three different scenarios where we need to subtract fractions and simplify our answer as much as possible. We will break down each problem step by step, using a combination of mathematical techniques and logical reasoning.

Scenario 1: Subtracting Fractions with Different Denominators

Problem

Subtract the following fractions: 204−37\frac{20}{4} - \frac{3}{7}

Solution

To subtract these fractions, we need to find a common denominator. The least common multiple (LCM) of 4 and 7 is 28. We can rewrite each fraction with a denominator of 28:

204=20×74×7=14028\frac{20}{4} = \frac{20 \times 7}{4 \times 7} = \frac{140}{28}

37=3×47×4=1228\frac{3}{7} = \frac{3 \times 4}{7 \times 4} = \frac{12}{28}

Now that we have a common denominator, we can subtract the fractions:

14028−1228=12828\frac{140}{28} - \frac{12}{28} = \frac{128}{28}

We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is 4:

12828=128÷428÷4=327\frac{128}{28} = \frac{128 ÷ 4}{28 ÷ 4} = \frac{32}{7}

Therefore, the simplified answer is 327\frac{32}{7}.

Scenario 2: Subtracting Mixed Numbers

Problem

Subtract the following mixed numbers: 279−1152 \frac{7}{9} - 1 \frac{1}{5}

Solution

To subtract mixed numbers, we need to convert them to improper fractions first. We can do this by multiplying the whole number part by the denominator and then adding the numerator:

279=(2×9)+79=18+79=2592 \frac{7}{9} = \frac{(2 \times 9) + 7}{9} = \frac{18 + 7}{9} = \frac{25}{9}

115=(1×5)+15=5+15=651 \frac{1}{5} = \frac{(1 \times 5) + 1}{5} = \frac{5 + 1}{5} = \frac{6}{5}

Now that we have improper fractions, we can find a common denominator. The LCM of 9 and 5 is 45. We can rewrite each fraction with a denominator of 45:

259=25×59×5=12545\frac{25}{9} = \frac{25 \times 5}{9 \times 5} = \frac{125}{45}

65=6×95×9=5445\frac{6}{5} = \frac{6 \times 9}{5 \times 9} = \frac{54}{45}

Now that we have a common denominator, we can subtract the fractions:

12545−5445=7145\frac{125}{45} - \frac{54}{45} = \frac{71}{45}

We can simplify this fraction by dividing both the numerator and the denominator by their GCD, which is 1:

7145\frac{71}{45}

Therefore, the simplified answer is 7145\frac{71}{45}.

Scenario 3: Subtracting a Whole Number and a Mixed Number

Problem

Subtract the following expression: 7−4167 - 4 \frac{1}{6}

Solution

To subtract a whole number and a mixed number, we need to convert the mixed number to an improper fraction first. We can do this by multiplying the whole number part by the denominator and then adding the numerator:

416=(4×6)+16=24+16=2564 \frac{1}{6} = \frac{(4 \times 6) + 1}{6} = \frac{24 + 1}{6} = \frac{25}{6}

Now that we have an improper fraction, we can find a common denominator. The LCM of 1 and 6 is 6. We can rewrite the whole number 7 as a fraction with a denominator of 6:

7=7×66=4267 = \frac{7 \times 6}{6} = \frac{42}{6}

Now that we have a common denominator, we can subtract the fractions:

426−256=176\frac{42}{6} - \frac{25}{6} = \frac{17}{6}

We can simplify this fraction by dividing both the numerator and the denominator by their GCD, which is 1:

176\frac{17}{6}

Therefore, the simplified answer is 176\frac{17}{6}.

Conclusion

Q: What is the difference between a fraction and a mixed number?

A: A fraction is a way of expressing a part of a whole as a ratio of two numbers. For example, 12\frac{1}{2} is a fraction. A mixed number, on the other hand, is a combination of a whole number and a fraction. For example, 2122 \frac{1}{2} is a mixed number.

Q: How do I simplify a fraction?

A: To simplify a fraction, you need to find the greatest common divisor (GCD) of the numerator and the denominator. You can then divide both the numerator and the denominator by the GCD to simplify the fraction.

Q: What is the least common multiple (LCM)?

A: The least common multiple (LCM) is the smallest multiple that two or more numbers have in common. For example, the LCM of 4 and 6 is 12.

Q: How do I find the LCM of two numbers?

A: To find the LCM of two numbers, you can list the multiples of each number and find the smallest multiple that they have in common. Alternatively, you can use the following formula:

LCM(a, b) = (a × b) / GCD(a, b)

Q: What is the greatest common divisor (GCD)?

A: The greatest common divisor (GCD) is the largest number that divides two or more numbers without leaving a remainder. For example, the GCD of 12 and 18 is 6.

Q: How do I find the GCD of two numbers?

A: To find the GCD of two numbers, you can list the factors of each number and find the largest factor that they have in common. Alternatively, you can use the following formula:

GCD(a, b) = (a × b) / LCM(a, b)

Q: Can I simplify a fraction with a negative sign?

A: Yes, you can simplify a fraction with a negative sign. To do this, you need to follow the same steps as simplifying a fraction with a positive sign, but you need to take into account the negative sign.

Q: How do I subtract fractions with different denominators?

A: To subtract fractions with different denominators, you need to find a common denominator. You can do this by listing the multiples of each denominator and finding the smallest multiple that they have in common. Alternatively, you can use the following formula:

Common denominator = LCM(denominator 1, denominator 2)

Once you have a common denominator, you can subtract the fractions.

Q: Can I subtract a whole number and a fraction?

A: Yes, you can subtract a whole number and a fraction. To do this, you need to convert the whole number to a fraction with the same denominator as the fraction. You can then subtract the fractions.

Q: How do I add fractions with different denominators?

A: To add fractions with different denominators, you need to find a common denominator. You can do this by listing the multiples of each denominator and finding the smallest multiple that they have in common. Alternatively, you can use the following formula:

Common denominator = LCM(denominator 1, denominator 2)

Once you have a common denominator, you can add the fractions.

Q: Can I add a whole number and a fraction?

A: Yes, you can add a whole number and a fraction. To do this, you need to convert the whole number to a fraction with the same denominator as the fraction. You can then add the fractions.

Conclusion

In this article, we have answered some of the most frequently asked questions about simplifying fractions. We have covered topics such as the difference between a fraction and a mixed number, how to simplify a fraction, and how to subtract and add fractions with different denominators. By following these steps, you can simplify fractions and arrive at the correct answer. Whether you are a student or a teacher, mastering the art of simplifying fractions is an essential skill that will serve you well in mathematics and beyond.