Algebra/Rational & Irrational Numbers Unit test: prove that the sum of 3/4 and 10 is irrational. Show your work, and in 1 sentences, explain why the sum is an irrational number.

Answer 1

To determine whether the sum of \( \frac{3}{4} \) and \( 10 \) is rational or irrational, we first perform the addition:

1. Convert \( 10 \) to a fraction with the same denominator as \( \frac{3}{4} \). Since \( 10 \) can be written as \( \frac{10}{1} \), we convert it to have a denominator of 4: \[ 10 = \frac{10 \times 4}{1 \times 4} = \frac{40}{4} \]

2. Now, we can sum \( \frac{3}{4} \) and \( \frac{40}{4} \): \[ \frac{3}{4} + \frac{40}{4} = \frac{3 + 40}{4} = \frac{43}{4} \]

3. Next, we check if \( \frac{43}{4} \) is a rational number. A number is defined as rational if it can be expressed as the quotient of two integers, where the denominator is not zero.

   In this case:

   • The numerator \( 43 \) is an integer.
   • The denominator \( 4 \) is also an integer (and not zero).

Thus, \( \frac{43}{4} \) is a rational number.

Conclusion:

The sum of \( \frac{3}{4} \) and \( 10 \) is not irrational—it is actually a rational number, \( \frac{43}{4} \). Therefore, it cannot be proven that their sum is irrational; rather, the sum is rational.