Problem 31
Question
Write the quotient in simplest form. $$\frac{25 x^{2}}{10 x} \div \frac{5 x}{10 x}$$
Step-by-Step Solution
Verified Answer
The simplified form of the given expression is \(5x^{2}\)
1Step 1: The Division of Fractions
To divide two fractions, one needs to multiply the first fraction by the reciprocal of the second fraction. The reciprocal of a fraction is obtained by swapping the numerator and the denominator. Therefore, \(\frac{25 x^{2}}{10 x} \div \frac{5 x}{10 x}\) becomes \(\frac{25 x^{2}}{10 x} * \frac{10 x}{5 x}\).
2Step 2: Multiplication of Fractions
When multiplying fractions, just multiply the numerators with each other and the denominators with each other. This leads to \(\frac{25 x^{2} * 10 x}{10 x * 5 x}\).
3Step 3: Simplification of the Result
Next, it's important to simplify expressions, especially when dealing with exponents and common factors in a fraction. This gives \(\frac{250 x^{3}}{50 x^{2}}\). Now, we can further simplify this by cancelling out the common factors. The x in the numerator and one of the x in the denominator can be cancelled out: \(\frac{250 x^{2}}{50}\). We can further simplify this by dividing the \(x^{2}\) term and the numbers with their Greatest Common Divisor (gcd). gcd of 250 and 50 is 50, leading to a simplification to \(\frac{5x^{2}}{1}\).
4Step 4: Final Answer
This division simplifies to 5x^2.
Key Concepts
Simplify ExpressionsReciprocal of a FractionMultiplying FractionsGreatest Common Divisor
Simplify Expressions
Simplifying expressions in algebra often means reducing them to their most compact, easiest-to-understand form. For instance, consider the expression \(\frac{250 x^{3}}{50 x^{2}}\). It looks complex but can be made simpler. To do this, look for common factors in the numerator and the denominator that can be divided out.
In the case of our expression, both 250 and 50 have a common factor, which is 50. When we divide both of these by their greatest common divisor (GCD), we get a simpler fraction. Also, the variable terms \(x^3\) in the numerator and \(x^2\) in the denominator can also be simplified because \(x^3\) divided by \(x^2\) is \(x\). After simplifying both the numerical and variable parts, we are left with \(5x^{2}\), which is much easier to work with than the original expression.
In the case of our expression, both 250 and 50 have a common factor, which is 50. When we divide both of these by their greatest common divisor (GCD), we get a simpler fraction. Also, the variable terms \(x^3\) in the numerator and \(x^2\) in the denominator can also be simplified because \(x^3\) divided by \(x^2\) is \(x\). After simplifying both the numerical and variable parts, we are left with \(5x^{2}\), which is much easier to work with than the original expression.
Reciprocal of a Fraction
The reciprocal of a fraction is simply what you get when you switch its numerator and denominator. For example, the reciprocal of \(\frac{a}{b}\) is \(\frac{b}{a}\).
Understanding reciprocals is essential when dividing fractions. To divide by a fraction, you multiply by its reciprocal. So, in our original exercise, to divide \(\frac{25 x^{2}}{10 x}\) by \(\frac{5 x}{10 x}\), you multiply \(\frac{25 x^{2}}{10 x}\) by the reciprocal of \(\frac{5 x}{10 x}\), which is \(\frac{10 x}{5 x}\). This is crucial because without flipping the second fraction, you cannot proceed to the multiplication process—the main operation in fraction division.
Understanding reciprocals is essential when dividing fractions. To divide by a fraction, you multiply by its reciprocal. So, in our original exercise, to divide \(\frac{25 x^{2}}{10 x}\) by \(\frac{5 x}{10 x}\), you multiply \(\frac{25 x^{2}}{10 x}\) by the reciprocal of \(\frac{5 x}{10 x}\), which is \(\frac{10 x}{5 x}\). This is crucial because without flipping the second fraction, you cannot proceed to the multiplication process—the main operation in fraction division.
Multiplying Fractions
When multiplying fractions, the process is direct and straightforward: multiply the numerators (the top numbers) together and the denominators (the bottom numbers) together.
In the example \(\frac{25 x^{2}}{10 x} * \frac{10 x}{5 x}\), we multiply the numerators, 25 \(x^{2}\) and 10 \(x\), to get 250 \(x^{3}\). The denominators, 10 \(x\) and 5 \(x\), multiply to give us 50 \(x^{2}\). This step is vital in obtaining the simplified version of a fraction multiplied by another fraction. After the multiplication, we can then move on to the simplification process, as we did in the first section.
In the example \(\frac{25 x^{2}}{10 x} * \frac{10 x}{5 x}\), we multiply the numerators, 25 \(x^{2}\) and 10 \(x\), to get 250 \(x^{3}\). The denominators, 10 \(x\) and 5 \(x\), multiply to give us 50 \(x^{2}\). This step is vital in obtaining the simplified version of a fraction multiplied by another fraction. After the multiplication, we can then move on to the simplification process, as we did in the first section.
Greatest Common Divisor
The greatest common divisor (GCD), also known as the greatest common factor, is the largest number that divides two or more numbers without leaving a remainder. It's a powerful tool in simplifying fractions.
For example, the GCD of 250 and 50 is 50. When we divide both the numerator and the denominator of \(\frac{250 x^{3}}{50 x^{2}}\) by 50, we significantly reduce the fraction. The idea is to break down the numbers to their simplest form, which in our case, results in the fraction \(\frac{5x^{2}}{1}\), or simply \(5x^{2}\). Identifying the GCD is an essential skill in algebra, as it can greatly simplify expressions and make them much more manageable.
For example, the GCD of 250 and 50 is 50. When we divide both the numerator and the denominator of \(\frac{250 x^{3}}{50 x^{2}}\) by 50, we significantly reduce the fraction. The idea is to break down the numbers to their simplest form, which in our case, results in the fraction \(\frac{5x^{2}}{1}\), or simply \(5x^{2}\). Identifying the GCD is an essential skill in algebra, as it can greatly simplify expressions and make them much more manageable.
Other exercises in this chapter
Problem 31
Write the difference in simplest form. $$ \frac{x-1}{6 x^{2}}-\frac{2}{3 x} $$
View solution Problem 31
Solve the equation. Check your solutions. $$ \frac{x}{2}=\frac{5}{x+3} $$
View solution Problem 31
Simplify the expression. If not possible, write already in simplest form. $$\frac{x^{2}+x-20}{x^{2}+2 x-15}$$
View solution Problem 32
Find and correct the error. $$ \begin{aligned} &\frac{3 n^{2}}{n^{2}-144}-\frac{36 n}{n^{2}-144}=\frac{3 n^{2}-36 n}{n^{2}-144}\\\ &=\frac{3 n(n-12)}{(n-12)(n-1
View solution