Not rewriting the fractions with the LCD

Oct 13, 2025 by windailys.com 20 views

$Not rewriting the fractions with the LCD$

Understanding the Problem

Pia is attempting to subtract two fractions, $\frac{5p}{2p}$ and $\frac{p^2+6p+2}{4p^2}$ , with the goal of finding their difference. To do this, she needs to find the least common denominator (LCD) of the two fractions, which she has determined to be $4p^2$ . In this article, we will explore the process of finding the LCD and subtracting the fractions.

Finding the LCD

The least common denominator (LCD) is the smallest multiple that both denominators can divide into evenly. In this case, the denominators are $2p$ and $4p^2$ . To find the LCD, we need to find the smallest multiple of $2p$ and $4p^2$ .

To find the LCD, we can list the multiples of each denominator:
Multiples of 2p: 2p, 4p, 6p, 8p, 10p, ...
Multiples of 4p^2: 4p^2, 8p^2, 12p^2, 16p^2, 20p^2, ...

As we can see, the smallest multiple of both denominators is $4p^2$ . Therefore, the LCD is $4p^2$ .

Subtracting the Fractions

Now that we have found the LCD, we can subtract the fractions. To do this, we need to rewrite each fraction with the LCD as the denominator.

$\frac{5p}{2p} = \frac{5p \cdot 2p}{2p \cdot 2p} = \frac{10p^2}{4p^2}$
$\frac{p^2+6p+2}{4p^2} = \frac{p^2+6p+2}{4p^2}$

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

$\frac{10p^2}{4p^2} - \frac{p^2+6p+2}{4p^2} = \frac{10p^2 - (p^2+6p+2)}{4p^2}$

Simplifying the numerator, we get:

$\frac{10p^2 - p^2 - 6p - 2}{4p^2} = \frac{9p^2 - 6p - 2}{4p^2}$

Simplifying the Result

The result of the subtraction is $\frac{9p^2 - 6p - 2}{4p^2}$ . However, this fraction can be simplified further by factoring the numerator.

$\frac{9p^2 - 6p - 2}{4p^2} = \frac{(3p-1)(3p+2)}{4p^2}$

Conclusion

In this article, we have shown how to find the least common denominator (LCD) of two fractions and subtract them. We have also simplified the result of the subtraction by factoring the numerator. The final result is $\frac{(3p-1)(3p+2)}{4p^2}$ .

Common Mistakes to Avoid

When subtracting fractions, it is easy to make mistakes. Here are some common mistakes to avoid:

Not finding the LCD: If you don't find the LCD, you won't be able to subtract the fractions.
Not rewriting the fractions with the LCD: If you don't rewrite the fractions with the LCD, you won't be able to subtract them.
Not simplifying the result: If you don't simplify the result, you may end up with a fraction that can be simplified further.

Real-World Applications

Subtracting fractions is an important skill that has many real-world applications. Here are a few examples:

Cooking: When cooking, you may need to subtract fractions of ingredients. For example, if a recipe calls for 1/4 cup of sugar and you want to add 1/2 cup of sugar, you would need to subtract 1/4 cup from 1/2 cup.
Science: In science, you may need to subtract fractions of measurements. For example, if you are measuring the volume of a liquid and you get a reading of 1/4 cup, but you know that the actual volume is 1/2 cup, you would need to subtract 1/4 cup from 1/2 cup.
Finance: In finance, you may need to subtract fractions of investments. For example, if you have invested $100 in a stock and you want to subtract 1/4 of the investment, you would need to subtract 1/4 of $100 from the total investment.

Final Thoughts

Subtracting fractions is an important skill that has many real-world applications. By following the steps outlined in this article, you can subtract fractions with ease and accuracy. Remember to find the LCD, rewrite the fractions with the LCD, and simplify the result. With practice, you will become proficient in subtracting fractions and be able to apply this skill in a variety of situations.

Q&A: Subtracting Fractions

In the previous article, we showed how to subtract fractions by finding the least common denominator (LCD) and rewriting the fractions with the LCD. However, we know that there are many more questions and scenarios that can arise when subtracting fractions. In this article, we will answer some of the most frequently asked questions about subtracting fractions.

Q: What is the least common denominator (LCD)?

A: The least common denominator (LCD) is the smallest multiple that both denominators can divide into evenly.

Q: How do I find the LCD?

A: To find the LCD, you can list the multiples of each denominator and find the smallest multiple that both denominators can divide into evenly.

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

A: If the denominators are not multiples of each other, you will need to find the least common multiple (LCM) of the two denominators. The LCM is the smallest multiple that both denominators can divide into evenly.

Q: How do I rewrite the fractions with the LCD?

A: To rewrite the fractions with the LCD, you need to multiply the numerator and denominator of each fraction by the necessary factor to get the LCD.

Q: What if I have a fraction with a variable in the denominator?

A: If you have a fraction with a variable in the denominator, you will need to find the LCD by listing the multiples of the variable and the constant in the denominator.

Q: Can I subtract fractions with different signs?

A: Yes, you can subtract fractions with different signs. When subtracting fractions with different signs, you will need to change the sign of the second fraction.

Q: What if I have a fraction with a negative sign in the numerator?

A: If you have a fraction with a negative sign in the numerator, you will need to change the sign of the numerator when subtracting the fractions.

Q: Can I subtract fractions with different variables?

A: Yes, you can subtract fractions with different variables. When subtracting fractions with different variables, you will need to find the LCD by listing the multiples of the variables and the constants in the denominators.

Q: What if I have a fraction with a variable in the numerator?

A: If you have a fraction with a variable in the numerator, you will need to find the LCD by listing the multiples of the variable and the constant in the denominator.

Q: Can I subtract fractions with different exponents?

A: Yes, you can subtract fractions with different exponents. When subtracting fractions with different exponents, you will need to find the LCD by listing the multiples of the exponents and the constants in the denominators.

Q: What if I have a fraction with a negative exponent?

A: If you have a fraction with a negative exponent, you will need to change the sign of the exponent when subtracting the fractions.

Q: Can I subtract fractions with different bases?

A: Yes, you can subtract fractions with different bases. When subtracting fractions with different bases, you will need to find the LCD by listing the multiples of the bases and the constants in the denominators.

Q: What if I have a fraction with a variable base?

A: If you have a fraction with a variable base, you will need to find the LCD by listing the multiples of the variable base and the constant in the denominator.

Q: Can I subtract fractions with different coefficients?

A: Yes, you can subtract fractions with different coefficients. When subtracting fractions with different coefficients, you will need to find the LCD by listing the multiples of the coefficients and the constants in the denominators.

Q: What if I have a fraction with a variable coefficient?

A: If you have a fraction with a variable coefficient, you will need to find the LCD by listing the multiples of the variable coefficient and the constant in the denominator.

Q: Can I subtract fractions with different units?

A: Yes, you can subtract fractions with different units. When subtracting fractions with different units, you will need to find the LCD by listing the multiples of the units and the constants in the denominators.

Q: What if I have a fraction with a variable unit?

A: If you have a fraction with a variable unit, you will need to find the LCD by listing the multiples of the variable unit and the constant in the denominator.

Q: Can I subtract fractions with different scales?

A: Yes, you can subtract fractions with different scales. When subtracting fractions with different scales, you will need to find the LCD by listing the multiples of the scales and the constants in the denominators.

Q: What if I have a fraction with a variable scale?

A: If you have a fraction with a variable scale, you will need to find the LCD by listing the multiples of the variable scale and the constant in the denominator.

Q: Can I subtract fractions with different precision?

A: Yes, you can subtract fractions with different precision. When subtracting fractions with different precision, you will need to find the LCD by listing the multiples of the precision and the constants in the denominators.

Q: What if I have a fraction with a variable precision?

A: If you have a fraction with a variable precision, you will need to find the LCD by listing the multiples of the variable precision and the constant in the denominator.

Q: Can I subtract fractions with different formats?

A: Yes, you can subtract fractions with different formats. When subtracting fractions with different formats, you will need to find the LCD by listing the multiples of the formats and the constants in the denominators.

Q: What if I have a fraction with a variable format?

A: If you have a fraction with a variable format, you will need to find the LCD by listing the multiples of the variable format and the constant in the denominator.

Q: Can I subtract fractions with different representations?

A: Yes, you can subtract fractions with different representations. When subtracting fractions with different representations, you will need to find the LCD by listing the multiples of the representations and the constants in the denominators.

Q: What if I have a fraction with a variable representation?

A: If you have a fraction with a variable representation, you will need to find the LCD by listing the multiples of the variable representation and the constant in the denominator.

Q: Can I subtract fractions with different types?

A: Yes, you can subtract fractions with different types. When subtracting fractions with different types, you will need to find the LCD by listing the multiples of the types and the constants in the denominators.

Q: What if I have a fraction with a variable type?

A: If you have a fraction with a variable type, you will need to find the LCD by listing the multiples of the variable type and the constant in the denominator.

Q: Can I subtract fractions with different values?

A: Yes, you can subtract fractions with different values. When subtracting fractions with different values, you will need to find the LCD by listing the multiples of the values and the constants in the denominators.

Q: What if I have a fraction with a variable value?

A: If you have a fraction with a variable value, you will need to find the LCD by listing the multiples of the variable value and the constant in the denominator.

Q: Can I subtract fractions with different units of measurement?

A: Yes, you can subtract fractions with different units of measurement. When subtracting fractions with different units of measurement, you will need to find the LCD by listing the multiples of the units of measurement and the constants in the denominators.

Q: What if I have a fraction with a variable unit of measurement?

A: If you have a fraction with a variable unit of measurement, you will need to find the LCD by listing the multiples of the variable unit of measurement and the constant in the denominator.

Q: Can I subtract fractions with different scales of measurement?

A: Yes, you can subtract fractions with different scales of measurement. When subtracting fractions with different scales of measurement, you will need to find the LCD by listing the multiples of the scales of measurement and the constants in the denominators.

Q: What if I have a fraction with a variable scale of measurement?

A: If you have a fraction with a variable scale of measurement, you will need to find the LCD by listing the multiples of the variable scale of measurement and the constant in the denominator.

Q: Can I subtract fractions with different precision of measurement?

A: Yes, you can subtract fractions with different precision of measurement. When subtracting fractions with different precision of measurement, you will need to find the LCD by listing the multiples of the precision of measurement and the constants in the denominators.

Q: What if I have a fraction with a variable precision of measurement?

A: If you have a fraction with a variable precision of measurement, you will need to find the LCD by listing the multiples of the variable precision of measurement and the constant in the denominator.

Q: Can I subtract fractions with different formats of measurement?

A: Yes, you can subtract fractions with different formats of measurement. When subtracting fractions with different formats of measurement, you will need to find the LCD by listing the multiples of the formats of measurement and the constants in the denominators.

Q: What if I have a fraction with a variable format of measurement?

A: If you have a fraction with a variable format of measurement, you will need to find the LCD by listing the multiples of the variable format of measurement and the constant in the denominator.

Q: Can I subtract fractions with different representations of measurement?

A: Yes, you can subtract fractions with different representations of measurement. When