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Chapter 7: Problem 10

Add or subtract. Write answer in lowest terms. \(\frac{5}{p}+\frac{12}{p}\)

Short Answer

Expert verified
\( \frac{17}{p} \)

Step by step solution

01

Identify the Common Denominator

Both fractions \(\frac{5}{p}\) and \(\frac{12}{p}\) have the same denominator, which is \p\.
02

Combine the Numerators

Since the denominators are the same, add the numerators: 5 + 12. This gives \(\frac{5 + 12}{p}\).
03

Simplify the Result

Calculate the sum of the numerators: 5 + 12 = 17. So the fraction becomes \(\frac{17}{p}\).
04

Express in Lowest Terms

The fraction \(\frac{17}{p}\) is already in its lowest terms since 17 is a prime number and does not share any common factors with \p\.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Common Denominator
A common denominator is a shared multiple of the denominators of two or more fractions. When adding or subtracting fractions, having a common denominator simplifies the process.

In the exercise \(\frac{5}{p} + \frac{12}{p}\), both fractions already share the same denominator, which is \(p\). This means we do not need to find a new common denominator 鈥� a step that often involves calculating the Least Common Multiple (LCM) of different denominators.

Because \(p\) is common to both fractions, we can proceed directly to combining the numerators. This simplifies our calculations significantly.
Numerator
The numerator is the top part of a fraction, representing the number of parts we have. In the fraction \(\frac{5}{p}\), the '5' is the numerator. Understanding the role of the numerator is essential for performing operations like addition and subtraction of fractions.

In the given exercise, after identifying the common denominator, we combine the numerators of the fractions involved. The numerators are 5 and 12. Adding these together, we get \(5 + 12 = 17\). Therefore, the new fraction formed is \(\frac{17}{p}\).

This step focuses on the numerators while keeping the common denominator unchanged.
Lowest Terms
Expressing a fraction in its lowest terms means reducing it to the simplest form, where the numerator and the denominator share no common factors other than 1. This process makes the fraction as simple as possible to understand and work with.

In our exercise, the fraction resulting from combining the numerators was \(\frac{17}{p}\). Since 17 is a prime number, it has no factors other than 1 and itself. Therefore, it does not share any common factors with \(p\) (assuming \(p\) is a different prime number or a composite number that doesn't include 17 as a factor).

Conclusively, \(\frac{17}{p}\) is already in its lowest terms, and no further simplification is required.

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Most popular questions from this chapter

Exercises \(61-82 \text { involve factoring by grouping (Section } 6.1)\) and factoring sums and differences of cubes (Section 6.4). Write each rational expression in lowest terms. $$ \frac{1+p^{3}}{1+p} $$

When solving an equation with variables in denominators, we must determine the values thar cause these denominators to equal 0, so that we can reject these extraneous values if they= appear as potential solutions. Find all values for which at least one denominator is equal to \(0 .\) Write answers using the symbol \(\neq\) Do not solve. See Examples \(3-7\). $$ \frac{4}{x^{2}+8 x-9}+\frac{1}{x^{2}-4}=0 $$

Determine whether each equation represents direct or inverse variation. $$ y=\frac{3}{x} $$

Solve each problem involving direct or inverse variation. If \(z\) varies inversely as \(x,\) and \(z=50\) when \(x=2,\) find \(z\) when \(x=25\)

Solve each problem. An airplane maintaining a constant airspeed takes as long to go \(450 \mathrm{mi}\) with the wind as it does to go \(375 \mathrm{mi}\) against the wind. If the wind is blowing at \(15 \mathrm{mph},\) what is the rate of the plane in still air?
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