Problem 32
Question
For the following problems, replace \(N\) with the proper quantity. $$ \frac{8 x^{2} y}{5 a^{3}}=\frac{N}{25 a^{3} x^{2}} $$
Step-by-Step Solution
Verified Answer
Answer: N = 8x^2y
1Step 1: Find the common denominator
In this case, both fractions already have the same denominator: \(25a^3x^2\). We can proceed to set the numerators equal to each other.
2Step 2: Set the numerators equal to each other
$$
8x^2y = N
$$ We can see that the equation is already simplified, and N is directly equal to the numerator of the first fraction.
3Step 3: Find the value of N
Since the numerators are equal, we can find the value of N by simply replacing N with the numerator of the first fraction:$$
N = 8x^2y
$$
Key Concepts
Common DenominatorsEquivalent FractionsAlgebraic Expressions
Common Denominators
Understanding common denominators is crucial when working with algebraic fractions. A denominator is the bottom part of a fraction that tells us into how many parts the whole is divided. When we have multiple fractions that we want to combine or compare, they need to have the same denominator, which we refer to as the 'common denominator'.
In our exercise, we have two fractions: \( \frac{8 x^{2} y}{5 a^{3}} \) and a fraction with a missing numerator represented by \( N \) over \(25 a^{3} x^{2} \). These fractions can be easily compared because they share a common denominator of \( 25a^3x^2 \). This allows us to directly compare and manipulate the numerators, which is essential in finding the value of \( N \).
In real-world applications, common denominators are used to add, subtract, or compare fractions. For example, if you're trying to combine ingredients from different recipes, ensuring the measurements are in a common unit, like cups or grams, is similar to finding a common denominator.
In our exercise, we have two fractions: \( \frac{8 x^{2} y}{5 a^{3}} \) and a fraction with a missing numerator represented by \( N \) over \(25 a^{3} x^{2} \). These fractions can be easily compared because they share a common denominator of \( 25a^3x^2 \). This allows us to directly compare and manipulate the numerators, which is essential in finding the value of \( N \).
In real-world applications, common denominators are used to add, subtract, or compare fractions. For example, if you're trying to combine ingredients from different recipes, ensuring the measurements are in a common unit, like cups or grams, is similar to finding a common denominator.
Equivalent Fractions
Equivalent fractions are different fractions that represent the same part of a whole. For example, \( \frac{2}{4} \) and \( \frac{1}{2} \) are equivalent because they both represent half of something, even though their numbers are different. To find equivalent fractions, we can multiply or divide the numerator and the denominator by the same non-zero number.
In algebraic terms, when we encounter an expression like \( \frac{8 x^{2} y}{5 a^{3}} \) being equivalent to \( \frac{N}{25 a^{3} x^{2}} \), we're dealing with algebraic equivalent fractions. The presence of variables makes it a bit more complex than with numerical fractions, but the principle is the same. Both fractions actually represent the same 'amount' when it comes to their simplified form.
Understanding equivalent fractions is key to solving algebraic equations involving fractions. It helps in simplifying expressions, solving for variables, and in this case, determining the missing numerator \( N \).
In algebraic terms, when we encounter an expression like \( \frac{8 x^{2} y}{5 a^{3}} \) being equivalent to \( \frac{N}{25 a^{3} x^{2}} \), we're dealing with algebraic equivalent fractions. The presence of variables makes it a bit more complex than with numerical fractions, but the principle is the same. Both fractions actually represent the same 'amount' when it comes to their simplified form.
Understanding equivalent fractions is key to solving algebraic equations involving fractions. It helps in simplifying expressions, solving for variables, and in this case, determining the missing numerator \( N \).
Algebraic Expressions
Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols. Variables, such as \( x \) and \( y \) in our example, represent unknown quantities and allow us to describe general mathematical relationships. Algebraic expressions are the building blocks of algebra and enable us to model and solve real-life problems.
In this exercise, \( 8x^2y \) is an algebraic expression that forms the numerator of the first fraction. It's made up of variables and coefficients. The coefficient \( 8 \) is a constant that multiplies the variables \( x^2 \) and \( y \), which are raised to powers, indicating repeated multiplication. Algebraic expressions can take various forms, be manipulated using algebraic rules, and are used to find the values of variables that make the expression true, as seen with finding the value of \( N \) in our step-by-step solution.
By learning to work with algebraic expressions, students are equipped to handle a wide array of problems, from simple equations to complex functions.
In this exercise, \( 8x^2y \) is an algebraic expression that forms the numerator of the first fraction. It's made up of variables and coefficients. The coefficient \( 8 \) is a constant that multiplies the variables \( x^2 \) and \( y \), which are raised to powers, indicating repeated multiplication. Algebraic expressions can take various forms, be manipulated using algebraic rules, and are used to find the values of variables that make the expression true, as seen with finding the value of \( N \) in our step-by-step solution.
By learning to work with algebraic expressions, students are equipped to handle a wide array of problems, from simple equations to complex functions.
Other exercises in this chapter
Problem 32
For the following problems, show that the fractions are equivalent. $$ \frac{-2}{7} \text { and }-\frac{2}{7} $$
View solution Problem 32
For the following problems, perform the multiplications and divisions. $$ \frac{x+10}{x-4} \cdot \frac{x-4}{x-1} $$
View solution Problem 32
For the following problems, add or subtract the rational expressions. $$ \frac{y-1}{y+4}+\frac{y+7}{y+4} $$
View solution Problem 32
For the following problems, reduce each rational expression to lowest terms. $$ \frac{(y-2)^{6}(y-1)^{4}}{(y-2)^{3}(y-1)^{2}} $$
View solution