Chapter 3: Problem 26
What number can you add to the numerator and denominator of \(\frac{7}{9}\) to get \(\frac{1}{2} ?\) F. \(-11\) G. \(-5\) H. \(-2 \frac{1}{2}\) J. \(-1 \frac{2}{3}\) K. 5
Short Answer
Expert verified
Answer: G: -5
Step by step solution
01
Set up the equation for the numerator
First, we will create an equation for the numerator. Since we are adding the same unknown number to both the numerator and denominator, let's call this unknown number "x."
In the numerator, we have the expression 7 + x, and we want this to equal half of the modified denominator (since \(\frac{1}{2}\) means one out of two parts). So we can set up an equation as follows:
\(7 + x = \frac{1}{2}(9 + x)\)
02
Solve the equation for x
Now, we will solve the equation for x:
1. Distribute the \(\frac{1}{2}\) on the right side of the equation:
\: \(7 + x = \frac{1}{2}(9 + x)\)
\: \(7 + x = \frac{9}{2} + \frac{1}{2}x\)
2. Get all the x's on one side of the equation by subtracting \(\frac{1}{2}x\) from both sides:
\: \(7 + x - \frac{1}{2}x = \frac{9}{2}\)
\: \(7 + \frac{1}{2}x = \frac{9}{2}\)
3. Get all the numbers on the other side of the equation by subtracting 7 from both sides:
\: \(\frac{1}{2}x = \frac{9}{2} - 7\)
4. Turn 7 into \(\frac{14}{2}\) so we can subtract it from \(\frac{9}{2}\):
\: \(\frac{1}{2}x = \frac{9}{2} - \frac{14}{2}\)
5. Subtract \(\frac{14}{2}\) from \(\frac{9}{2}\):
\: \(\frac{1}{2}x = -\frac{5}{2}\)
6. Solve for x by multiplying both sides by 2:
\: \(x = -\frac{5}{2} \cdot 2 = -5\)
Looking at the given options, we can see that we get Answer G: \(-5\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Fraction Addition
Understanding how to add fractions is essential for solving many algebraic problems, including those found on the ACT Math Practice. Fraction addition involves combining fractions with either the same or different denominators in such a way that we obtain a single fraction that represents the sum.
For fractions with the same denominator, the addition is straightforward: we simply add the numerators and keep the same denominator. However, when the denominators are different, we first need to find a common denominator. This involves finding the least common multiple of the denominators and adjusting the numerators accordingly before adding them together.
The ability to competently manipulate and combine fractions is fundamental to solving equations that involve fractions. When working with equations, both sides must be balanced, so any fraction addition must maintain equality. In practice problems like the one from the ACT Math section, understanding the concept of fraction addition and how to manage different denominators becomes particularly important in terms of setting up the correct equation to solve for the unknown variable.
For fractions with the same denominator, the addition is straightforward: we simply add the numerators and keep the same denominator. However, when the denominators are different, we first need to find a common denominator. This involves finding the least common multiple of the denominators and adjusting the numerators accordingly before adding them together.
The ability to competently manipulate and combine fractions is fundamental to solving equations that involve fractions. When working with equations, both sides must be balanced, so any fraction addition must maintain equality. In practice problems like the one from the ACT Math section, understanding the concept of fraction addition and how to manage different denominators becomes particularly important in terms of setting up the correct equation to solve for the unknown variable.
Equation Solving
Equation solving is a critical skill in mathematics, forming the foundation of algebra and higher-level math subjects. To solve an equation, we must find the value of the variable that makes the equation true. In the context of the ACT Math Practice, solving equations allows students to find unknown quantities, which can be applied to various real-world problems.
There are several methods to solve equations, including adding or subtracting the same value from both sides, multiplying or dividing both sides by the same number, and using the distributive property to simplify expressions. The goal is always to isolate the variable on one side of the equation to find its value. For the problem in question, the equation required a series of operations, including distribution and collecting like terms, to solve for 'x'.
Students should be comfortable with these operations, and with transforming an equation step by step to get to the solution. Practice in equation solving involves recognizing which operation to apply at what stage of the problem, something that becomes intuitive with time and experience.
There are several methods to solve equations, including adding or subtracting the same value from both sides, multiplying or dividing both sides by the same number, and using the distributive property to simplify expressions. The goal is always to isolate the variable on one side of the equation to find its value. For the problem in question, the equation required a series of operations, including distribution and collecting like terms, to solve for 'x'.
Students should be comfortable with these operations, and with transforming an equation step by step to get to the solution. Practice in equation solving involves recognizing which operation to apply at what stage of the problem, something that becomes intuitive with time and experience.
Mathematical Operations
Mathematical operations are the building blocks for solving mathematical problems and include addition, subtraction, multiplication, and division. In more complex problems, these fundamental operations are combined and applied to numbers, fractions, and variables to manipulate and solve equations.
It's crucial for students to understand how these operations work independently and together, especially on exams like the ACT. For example, multiplication and division are inverse operations, meaning one undoes the effect of the other, which is essential when solving for a variable in an equation. Similarly, addition and subtraction are inverse operations commonly used to move terms from one side of an equation to the other.
In the practice problem, we see these operations at play. To solve the equation, we distributed the multiplication of a fraction, combined like terms through addition and subtraction, and then used multiplication to isolate the variable 'x'. Each of these operations moves us closer to the solution, demonstrating the importance of a solid understanding of basic mathematical operations.
It's crucial for students to understand how these operations work independently and together, especially on exams like the ACT. For example, multiplication and division are inverse operations, meaning one undoes the effect of the other, which is essential when solving for a variable in an equation. Similarly, addition and subtraction are inverse operations commonly used to move terms from one side of an equation to the other.
In the practice problem, we see these operations at play. To solve the equation, we distributed the multiplication of a fraction, combined like terms through addition and subtraction, and then used multiplication to isolate the variable 'x'. Each of these operations moves us closer to the solution, demonstrating the importance of a solid understanding of basic mathematical operations.
