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Chapter 11: Problem 17

Use algebra to show that if \(\frac{a}{b}=\frac{c}{d},\) then \(\frac{a+b}{b}=\frac{c+d}{d}\)

Short Answer

Expert verified
\( \frac{a+b}{b} = \frac{c+d}{d} \)

Step by step solution

01

Start with the Given Equation

Begin with the equation \(\frac{a}{b} = \frac{c}{d}\).
02

Cross Multiply

Cross multiply the given equation to eliminate the fractions. This gives \(a \times d = b \times c\).
03

Add Terms to Both Sides

Add \(b \times d\) to both sides of the equation to prepare for combining terms: \(a \times d + b \times d = b \times c + b \times d\).
04

Factor Out Common Terms

Factor \(d\) from the left-hand side and \(b\) from the right-hand side: \d(a + b) = b(c + d)\.
05

Divide Both Sides by \(bd\)

Divide both sides of the equation by \(b \times d\) to isolate the fractions: \frac{a+b}{b} = \frac{c+d}{d}\.

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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 an essential tool for solving equations involving fractions. It helps us eliminate the denominators by multiplying both sides of the equation by the denominators of the fractions involved. For example, if we have \(\frac{a}{b} = \frac{c}{d}\), cross multiplication means multiplying both sides by \(b \times d\). This transforms it to \(a \times d = b \times c\). This method is particularly useful because it simplifies the equation, making it easier to handle.
Algebraic Manipulation
Algebraic manipulation involves rearranging and simplifying equations to isolate the variable or expression of interest. In our exercise, after cross multiplication, we have \(a \times d = b \times c\). To move forward, we add \(b \times d\) to both sides of the equation. This gives us \(a \times d + b \times d = b \times c + b \times d\). Through algebraic manipulation, we can factor out the common terms on both sides. On the left, we factor out \(d\), and on the right, we factor out \(b\), leading us to \(d (a + b) = b (c + d)\).
Fraction Equivalence
Understanding fraction equivalence is crucial when working with equations like the one in our exercise. Fractions are equivalent if their cross products are equal, which led us from \(\frac{a}{b} = \frac{c}{d}\) to \(a \times d = b \times c\). Once we have identified this relationship, we can derive more relationships by manipulating the equation. By dividing both sides by \(b \times d\), we isolate the fractions to show the equivalence: \( \frac{a+b}{b} = \frac{c+d}{d}\). Recognizing and proving fraction equivalence is a powerful technique to simplify and solve algebraic problems involving fractions.

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Most popular questions from this chapter

In Exercises \(13-16\), use a proportion to solve the problem. The floor plan of a house is drawn to the scale of \(\frac{1}{4}\) in. \(=1\) ft. The master bedroom measures 3 in. by \(3 \frac{3}{4}\) in on the blueprints. What is the actual size of the room?

A tabloid magazine at a supermarket checkout exclaims, "Scientists Breed 4-Foot-Tall Chicken." A photo shows a giant chicken that supposedly weighs 74 pounds and will solve the world's hunger problem. What do you think about this headline? Assuming an average chicken stands 14 inches tall and weighs 7 pounds, would a 4 -foot chicken weigh 74 pounds? Is it possible for a chicken to be 4 feet tall? Explain your reasoning.

The sequence \(6,15,24,33,42, \ldots\) is an example of an arithmetic sequence - each term is generated by adding a constant, in this case \(9,\) to the previous term. The sequence \(6,12,24,48,96, \ldots\) is an example of a geometric sequence - each term is generated by multiplying the previous term by a constant, in this case 2 . Find the missing terms assuming each pattern is an arithmetic sequence, and then find the missing terms assuming each pattern is a geometric sequence. a. \(10,\text{_}, 40,...\) b. \(2,\text{_}, 50,...\) c. \(4,\text{_}, 36,...\)

In Exercises \(4-12,\) solve the proportion. \(\frac{14}{10}=\frac{x+9}{15}\)

In Exercises \(4-12,\) solve the proportion. \(\frac{20}{13}=\frac{60}{c}\)
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