Rational Sum: -2 3/4 + 5/9 Explained!

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Rational Sum: -2 3/4 + 5/9 Explained!

Hey guys! Today, we're diving into a cool math problem: figuring out why the sum of (−234){\left(-2 \frac{3}{4}\right)} and 59{\frac{5}{9}} is a rational number. Let's break it down step by step so it's super clear.

Understanding Rational Numbers

First, let's quickly recap what rational numbers are. A rational number is any number that can be expressed as a fraction pq{\frac{p}{q}}, where p and q are integers, and q is not zero. Basically, if you can write a number as a ratio of two integers, it's rational!

Examples of Rational Numbers:

  • Integers: Like -3, 0, 5 (because you can write them as -3/1, 0/1, 5/1)
  • Fractions: Like 1/2, -3/4, 5/9
  • Terminating decimals: Like 0.25 (which is 1/4)
  • Repeating decimals: Like 0.333... (which is 1/3)

Now that we have that down, let's jump into our specific problem.

Analyzing the Numbers: -2 3/4 and 5/9

We're given two numbers: −234{-2 \frac{3}{4}} and 59{\frac{5}{9}}. To determine if their sum is rational, let's first confirm that each number is rational on its own.

Checking -2 3/4

The number −234{-2 \frac{3}{4}} is a mixed number. We can convert it into an improper fraction:

−234=−(2×4)+34=−8+34=−114{ -2 \frac{3}{4} = -\frac{(2 \times 4) + 3}{4} = -\frac{8 + 3}{4} = -\frac{11}{4} }

Since -11 and 4 are both integers, and 4 is not zero, −114{-\frac{11}{4}} is indeed a rational number. This is because it fits perfectly into our definition of a rational number: it's a ratio of two integers.

Checking 5/9

The number 59{\frac{5}{9}} is already in the form of a fraction, where 5 and 9 are integers, and 9 is not zero. Therefore, 59{\frac{5}{9}} is also a rational number. It's as simple as that!

Summing the Numbers

Now that we know both numbers are rational, let's add them together:

−114+59{ -\frac{11}{4} + \frac{5}{9} }

To add these fractions, we need a common denominator. The least common multiple (LCM) of 4 and 9 is 36. So, we convert both fractions to have a denominator of 36:

−114=−11×94×9=−9936{ -\frac{11}{4} = -\frac{11 \times 9}{4 \times 9} = -\frac{99}{36} }

59=5×49×4=2036{ \frac{5}{9} = \frac{5 \times 4}{9 \times 4} = \frac{20}{36} }

Now we can add them:

−9936+2036=−99+2036=−7936{ -\frac{99}{36} + \frac{20}{36} = \frac{-99 + 20}{36} = \frac{-79}{36} }

So, the sum is −7936{-\frac{79}{36}}.

Determining if the Sum is Rational

We found that the sum of −234{-2 \frac{3}{4}} and 59{\frac{5}{9}} is −7936{-\frac{79}{36}}. To determine if this sum is rational, we check if it can be expressed as a fraction pq{\frac{p}{q}}, where p and q are integers and q is not zero.

In our case, -79 and 36 are both integers, and 36 is not zero. Therefore, −7936{-\frac{79}{36}} fits the definition of a rational number.

Why the Sum of Rational Numbers is Always Rational

A key property of rational numbers is that when you add, subtract, multiply, or divide (except by zero) two rational numbers, the result is always a rational number. This is because performing these operations on fractions results in another fraction, which fits the definition of a rational number.

In our example, we added two rational numbers (−114{-\frac{11}{4}} and 59{\frac{5}{9}}), and the result (−7936{-\frac{79}{36}}) was also a rational number. This illustrates this fundamental property.

Analyzing the Given Options

Now, let's look at the options provided and see which one correctly explains why the sum is rational:

A. The sum is a nonterminating and a non-repeating decimal.

  • This is incorrect. Nonterminating and non-repeating decimals are irrational numbers, not rational numbers.

B. The sum is a fraction.

  • This is the correct answer. Since the sum −7936{-\frac{79}{36}} can be expressed as a fraction where both the numerator and the denominator are integers, it is a rational number.

C. The sum is a terminating and a repeating decimal.

  • This is incorrect. While terminating and repeating decimals can be rational, the primary reason we know the sum is rational is because it's a fraction.

D. The sum is an integer.

  • This is incorrect. −7936{-\frac{79}{36}} is not an integer; it's a fraction between -2 and -3.

Final Answer

The correct answer is B. The sum is a fraction. This is the most direct and accurate explanation of why the sum of (−234){\left(-2 \frac{3}{4}\right)} and 59{\frac{5}{9}} is a rational number.

Converting to Decimal Form (For Extra Understanding)

For those curious, let's convert −7936{-\frac{79}{36}} to decimal form to further illustrate the point:

−7936≈−2.19444...{ -\frac{79}{36} \approx -2.19444... }

The decimal representation is -2.19444..., which is a repeating decimal. Repeating decimals are rational numbers, but identifying it as a fraction is the clearest way to classify it as rational from the start.

Key Takeaways

  • Rational numbers can be expressed as a fraction pq{\frac{p}{q}}, where p and q are integers and q is not zero.
  • The sum of two rational numbers is always a rational number.
  • Identifying a number as a fraction is a clear way to determine if it is rational.

Hope this breakdown helps you understand why the sum of (−234){\left(-2 \frac{3}{4}\right)} and 59{\frac{5}{9}} is a rational number. Keep practicing, and you'll master these concepts in no time! You got this!