Subtract: $\[ 4 \frac{9}{10} - 4 \frac{1}{2} = \\]

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Introduction to Subtracting Mixed Numbers

When dealing with mixed numbers, it's essential to understand the concept of subtracting them. Mixed numbers are a combination of a whole number and a fraction. In this case, we have two mixed numbers: 49104 \frac{9}{10} and 4124 \frac{1}{2}. To subtract these numbers, we need to follow a specific procedure.

Understanding the Concept of Subtracting Mixed Numbers

To subtract mixed numbers, we need to first convert them into improper fractions. An improper fraction is a fraction where the numerator is greater than the denominator. To convert a mixed number into an improper fraction, we multiply the whole number by the denominator and then add the numerator. This gives us a new numerator, which we then write over the denominator.

For example, let's convert 49104 \frac{9}{10} into an improper fraction:

4910=(4×10)+910=40+910=49104 \frac{9}{10} = \frac{(4 \times 10) + 9}{10} = \frac{40 + 9}{10} = \frac{49}{10}

Similarly, let's convert 4124 \frac{1}{2} into an improper fraction:

412=(4×2)+12=8+12=924 \frac{1}{2} = \frac{(4 \times 2) + 1}{2} = \frac{8 + 1}{2} = \frac{9}{2}

Subtracting Improper Fractions

Now that we have converted both mixed numbers into improper fractions, we can subtract them. To subtract improper fractions, we need to find a common denominator. The common denominator is the least common multiple (LCM) of the two denominators.

In this case, the denominators are 10 and 2. The LCM of 10 and 2 is 10. So, we can rewrite the second fraction with a denominator of 10:

92=9×52×5=4510\frac{9}{2} = \frac{9 \times 5}{2 \times 5} = \frac{45}{10}

Now that both fractions have the same denominator, we can subtract them:

4910−4510=49−4510=410\frac{49}{10} - \frac{45}{10} = \frac{49 - 45}{10} = \frac{4}{10}

Simplifying the Result

The result of the subtraction is 410\frac{4}{10}. However, we can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor (GCD). The GCD of 4 and 10 is 2.

410=4÷210÷2=25\frac{4}{10} = \frac{4 \div 2}{10 \div 2} = \frac{2}{5}

Conclusion

In conclusion, subtracting mixed numbers involves converting them into improper fractions, finding a common denominator, and then subtracting the fractions. The result of the subtraction can be simplified by dividing both the numerator and the denominator by their GCD.

Example Problems

Here are a few example problems to help you practice subtracting mixed numbers:

  • 378−238=?3 \frac{7}{8} - 2 \frac{3}{8} = ?
  • 523−413=?5 \frac{2}{3} - 4 \frac{1}{3} = ?
  • 2910−1710=?2 \frac{9}{10} - 1 \frac{7}{10} = ?

Step-by-Step Solutions

Here are the step-by-step solutions to the example problems:

Example Problem 1

378−238=?3 \frac{7}{8} - 2 \frac{3}{8} = ?

  1. Convert the mixed numbers into improper fractions: 378=(3×8)+78=24+78=3183 \frac{7}{8} = \frac{(3 \times 8) + 7}{8} = \frac{24 + 7}{8} = \frac{31}{8} 238=(2×8)+38=16+38=1982 \frac{3}{8} = \frac{(2 \times 8) + 3}{8} = \frac{16 + 3}{8} = \frac{19}{8}
  2. Subtract the fractions: 318−198=31−198=128\frac{31}{8} - \frac{19}{8} = \frac{31 - 19}{8} = \frac{12}{8}
  3. Simplify the result: 128=12÷48÷4=32\frac{12}{8} = \frac{12 \div 4}{8 \div 4} = \frac{3}{2}

Example Problem 2

523−413=?5 \frac{2}{3} - 4 \frac{1}{3} = ?

  1. Convert the mixed numbers into improper fractions: 523=(5×3)+23=15+23=1735 \frac{2}{3} = \frac{(5 \times 3) + 2}{3} = \frac{15 + 2}{3} = \frac{17}{3} 413=(4×3)+13=12+13=1334 \frac{1}{3} = \frac{(4 \times 3) + 1}{3} = \frac{12 + 1}{3} = \frac{13}{3}
  2. Subtract the fractions: 173−133=17−133=43\frac{17}{3} - \frac{13}{3} = \frac{17 - 13}{3} = \frac{4}{3}
  3. Simplify the result: 43\frac{4}{3} cannot be simplified further.

Example Problem 3

2910−1710=?2 \frac{9}{10} - 1 \frac{7}{10} = ?

  1. Convert the mixed numbers into improper fractions: 2910=(2×10)+910=20+910=29102 \frac{9}{10} = \frac{(2 \times 10) + 9}{10} = \frac{20 + 9}{10} = \frac{29}{10} 1710=(1×10)+710=10+710=17101 \frac{7}{10} = \frac{(1 \times 10) + 7}{10} = \frac{10 + 7}{10} = \frac{17}{10}
  2. Subtract the fractions: 2910−1710=29−1710=1210\frac{29}{10} - \frac{17}{10} = \frac{29 - 17}{10} = \frac{12}{10}
  3. Simplify the result: 1210=12÷210÷2=65\frac{12}{10} = \frac{12 \div 2}{10 \div 2} = \frac{6}{5}

Frequently Asked Questions

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

A: A mixed number is a combination of a whole number and a fraction, while an improper fraction is a fraction where the numerator is greater than the denominator.

Q: How do I convert a mixed number into an improper fraction?

A: To convert a mixed number into an improper fraction, you multiply the whole number by the denominator and then add the numerator. This gives you a new numerator, which you then write over the denominator.

Q: What is the common denominator?

A: The common denominator is the least common multiple (LCM) of the two denominators. It is the smallest number that both denominators can divide into evenly.

Q: How do I find the common denominator?

A: To find the common denominator, you need to list the multiples of each denominator and find the smallest number that appears in both lists.

Q: Can I simplify the result of the subtraction?

A: Yes, you can simplify the result of the subtraction by dividing both the numerator and the denominator by their greatest common divisor (GCD).

Q: What is the GCD?

A: The GCD is the largest number that divides both the numerator and the denominator evenly.

Q: How do I find the GCD?

A: To find the GCD, you can list the factors of each number and find the largest number that appears in both lists.

Q: Can I subtract mixed numbers with different denominators?

A: Yes, you can subtract mixed numbers with different denominators by finding a common denominator and then subtracting the fractions.

Q: What if the denominators are not multiples of each other?

A: If the denominators are not multiples of each other, you can find the least common multiple (LCM) of the two denominators and then rewrite the fractions with the LCM as the denominator.

Q: Can I subtract mixed numbers with negative numbers?

A: Yes, you can subtract mixed numbers with negative numbers by following the same steps as subtracting mixed numbers with positive numbers.

Q: How do I handle fractions with zero as the numerator?

A: If the numerator is zero, the fraction is equal to zero, and you can simplify the result of the subtraction accordingly.

Q: Can I subtract mixed numbers with decimals?

A: Yes, you can subtract mixed numbers with decimals by converting the decimals to fractions and then following the same steps as subtracting mixed numbers.

Q: What if I get a negative result?

A: If you get a negative result, it means that the second mixed number is greater than the first mixed number.

Q: Can I add and subtract mixed numbers with different signs?

A: Yes, you can add and subtract mixed numbers with different signs by following the same rules as adding and subtracting integers.

Q: How do I handle fractions with negative denominators?

A: If the denominator is negative, you can rewrite the fraction with a positive denominator by multiplying the numerator and the denominator by -1.

Q: Can I subtract mixed numbers with fractions that have different signs?

A: Yes, you can subtract mixed numbers with fractions that have different signs by following the same rules as subtracting integers.

Q: What if I get a fraction with a negative numerator and a positive denominator?

A: If you get a fraction with a negative numerator and a positive denominator, you can rewrite the fraction with a positive numerator by multiplying the numerator and the denominator by -1.

Q: Can I subtract mixed numbers with fractions that have different signs and denominators?

A: Yes, you can subtract mixed numbers with fractions that have different signs and denominators by following the same rules as subtracting integers.

Q: How do I handle fractions with zero as the denominator?

A: If the denominator is zero, the fraction is undefined, and you cannot subtract mixed numbers with this fraction.

Q: Can I subtract mixed numbers with fractions that have different signs and denominators and numerators?

A: Yes, you can subtract mixed numbers with fractions that have different signs and denominators and numerators by following the same rules as subtracting integers.

Conclusion

Subtracting mixed numbers involves converting them into improper fractions, finding a common denominator, and then subtracting the fractions. The result of the subtraction can be simplified by dividing both the numerator and the denominator by their GCD. By following these steps and understanding the concepts of mixed numbers, improper fractions, and common denominators, you can confidently subtract mixed numbers and solve problems involving fractions.