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Magazine Chapter 7: Problem 66
Solve each proportion. $$ \frac{n}{2}=\frac{5}{n+3} $$
Short Answer
Expert verified
The solutions to the proportion are \( n = 2 \) and \( n = -5 \).
Step by step solution
01
Identify the Cross Products
To solve the proportion \( \frac{n}{2} = \frac{5}{n+3} \), we first identify the cross products. This involves multiplying the numerator of one ratio by the denominator of the other: \( n(n+3) \) and \( 5 \times 2 \).
02
Set the Cross Products Equal
Equate the cross products identified in Step 1: \[ n(n+3) = 5 \times 2 \]. This simplifies to \[ n^2 + 3n = 10 \].
03
Rearrange to Form a Quadratic Equation
Rearrange the equation \( n^2 + 3n = 10 \) to get everything on one side: \[ n^2 + 3n - 10 = 0 \]. This forms a standard quadratic equation.
04
Solve the Quadratic Equation
To solve the quadratic \( n^2 + 3n - 10 = 0 \), use the quadratic formula \( n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 1 \), \( b = 3 \), and \( c = -10 \). Calculate the discriminant: \[ b^2 - 4ac = 3^2 - 4(1)(-10) = 9 + 40 = 49 \], which is a perfect square.
05
Apply the Quadratic Formula
Substitute the values into the quadratic formula: \[ n = \frac{-3 \pm \sqrt{49}}{2} \]. This simplifies to \[ n = \frac{-3 \pm 7}{2} \], which gives two potential solutions: \( n = 2 \) or \( n = -5 \).
06
Verify the Solutions
Substitute each potential solution back into the original proportion to ensure both sides are equal. For \( n = 2 \): \( \frac{2}{2} = \frac{5}{2+3} \) simplifies to \( 1 = 1 \), which is true. For \( n = -5 \): \( \frac{-5}{2} = \frac{5}{-5+3} \) simplifies to \( -2.5 = -2.5 \), which is also true. Hence, both solutions verify correctly.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Cross Multiplication
Cross multiplication is a fundamental technique used to solve proportions. A proportion is an equation stating that two ratios are equal, like the problem given: \( \frac{n}{2} = \frac{5}{n+3} \). To apply cross multiplication, you multiply the numerator of one fraction by the denominator of the other and vice versa. For this example, the cross products are \( n(n+3) \) and \( 5 \times 2 \).
- Step 1: Identify Cross Products. Multiply across the equal sign: numerator to opposite denominator.
- Step 2: Set Equal. Once you have the products, equate them: \( n(n+3) = 5 \times 2 \).
Quadratic Equation
A quadratic equation is any equation that can be rewritten in the form \( ax^2 + bx + c = 0 \). In our exercise, after using cross multiplication, we derived the equation \( n^2 + 3n = 10 \), which simplifies to \( n^2 + 3n - 10 = 0 \). This is a standard quadratic equation.
- Structure: The quadratic equation has three parts - the square term \( n^2 \), the linear term \( 3n \), and the constant term \(-10\).
- Goal: Solving a quadratic equation means finding the value(s) of \( n \) that make the equation true.
Quadratic Formula
The quadratic formula is a reliable method to find solutions (roots) of quadratic equations. When faced with an equation like \( n^2 + 3n - 10 = 0 \), the quadratic formula \( n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) can be used. Here, \( a = 1 \), \( b = 3 \), and \( c = -10 \).
- Discriminant: Calculate \( b^2 - 4ac \). A positive discriminant means two real solutions.
- Substitute and Solve: Put \( a \), \( b \), and \( c \) into the formula to find \( n \).
Verifying Solutions
After finding potential solutions for a quadratic equation, verifying them is essential. Substituting these solutions back into the original proportion gives you confidence that they are correct. For example, for \( n = 2 \) and \( n = -5 \), check if they satisfy the initial equation \( \frac{n}{2} = \frac{5}{n+3} \).
- Check Solution 1: For \( n = 2 \), you find \( \frac{2}{2} = \frac{5}{2+3} \), which confirms as \( 1 = 1 \).
- Check Solution 2: For \( n = -5 \), verify \( \frac{-5}{2} = \frac{5}{-5+3} \), which holds true as well.
