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Chapter 1: Problem 31

Find each sum without the use of a number line. $$\frac{9}{10}+\left(-\frac{3}{5}\right)$$

Short Answer

Expert verified
The sum is \( \frac{3}{10} \).

Step by step solution

01

Find the Least Common Denominator

The fractions \( \frac{9}{10} \) and \( -\frac{3}{5} \) do not have the same denominator. To add or subtract fractions, they must share the same denominator as this represents the same dividing of the whole. In this case, the least common denominator or least common multiple (LCM) between 10 and 5 is 10.
02

Convert the Fractions

Change the fraction \( -\frac{3}{5} \) to have the denominator 10. Multiply the numerator and the denominator by 2 to keep the ratio and overall value unchanged. This gives us \( -\frac{6}{10} \). So now we have \( \frac{9}{10} + -\frac{6}{10} \).
03

Perform the Addition

\Once we have the same denominator, we just add the numerators. Here, \( 9 + -6 = 3 \), so the final result is \( \frac{3}{10} \).

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Key Concepts

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

Understanding the Least Common Denominator (LCD)
When adding fractions, it鈥檚 crucial that they have the same denominator before performing the addition. The denominator represents how many equal parts the whole is divided into. Therefore, fractions with different denominators cannot be directly added or subtracted. This is exactly where the Least Common Denominator (LCD) comes in. It is the smallest number that both denominators can divide into evenly.

For the fractions \( \frac{9}{10} \) and \( -\frac{3}{5} \), the denominators are 10 and 5, respectively. To find the LCD, we identify the least common multiple of these numbers. The number 10 is already a multiple of 5 (since \( 5 \times 2 = 10 \)), making 10 the LCD. By using the LCD, we can ensure that both fractions refer to parts of the same whole, allowing us to accurately add them together.
The Art of Fraction Conversion
Once we have determined the least common denominator, the next step is to convert the fractions so that they each have this denominator. Fraction conversion is simply adjusting the numerator and denominator in a way that the value of the fraction remains unchanged. This is achieved by multiplying both the numerator and denominator by the same number.

In the exercise, \( -\frac{3}{5} \) needs to be converted to have a denominator of 10. Since 10 is twice 5, we multiply the numerator \(-3\) and the denominator \(5\) by 2, resulting in \( -\frac{6}{10} \). This conversion ensures that both fractions \( \frac{9}{10} \) and \( -\frac{6}{10} \) are now compatible and ready for addition.
Performing Addition of Fractions
After ensuring both fractions have the same denominator through conversion, adding them becomes straightforward. The denominators are now equal, so the focus shifts to the numerators. Addition of fractions with like denominators is simplified to just performing the operation on the numerators, while the common denominator remains unchanged.

For \( \frac{9}{10} + -\frac{6}{10} \), we add the numerators: \(9 + (-6)\). This results in 3. Therefore, the fraction becomes \( \frac{3}{10} \). The common denominator, 10, stays the same throughout this process. Thus, the final answer to the fraction addition is \( \frac{3}{10} \). This exemplifies the importance of having a common denominator when adding fractions.

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Most popular questions from this chapter

Without a calculator, you can add numbers using a number line, using absolute value, or using gains and losses. Which method do you find most helpful? Why is this so?

In Exercises \(109-116\), write a numerical expression for each phrase. Then simplify the numerical expression by performing the given operations. The difference between \(-11\) and the quotient of 20 and \(-5\)

In Exercises \(109-116\), write a numerical expression for each phrase. Then simplify the numerical expression by performing the given operations. 8 added to the product of 4 and \(-10\)

Use a calculator to find a decimal approximation for each irrational number, correct to three decimal places. Between which two integers should you graph each of these numbers on the number line? $$\sqrt{3}$$

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I evaluated the formula \(d=\sqrt{1.5 h}\) for a value of \(h\) that resulted in a rational number for \(d\).
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