Rational and irrational numbers Transitional Reasoning Practice

Questions

1. 1.

   In a mathematics workshop, four students are asked to choose two distinct irrational numbers such that both their sum and their product result in rational numbers. The pairs chosen by the students are Elena ($\sqrt{18}$ and $\sqrt{2}$), Marcus ($3 + \sqrt{5}$ and $1 - \sqrt{5}$), Priya ($4 - \sqrt{7}$ and $4 + \sqrt{7}$), and Lucas ($\sqrt{20}$ and $-\sqrt{5}$). Which student successfully selected a pair that meets both conditions?

   • A) Elena's pair: $\sqrt{18}$ and $\sqrt{2}$
   • B) Marcus's pair: $3 + \sqrt{5}$ and $1 - \sqrt{5}$
   • C) Priya's pair: $4 - \sqrt{7}$ and $4 + \sqrt{7}$
   • D) Lucas's pair: $\sqrt{20}$ and $-\sqrt{5}$
2. 2.

   A digital signal processing engineer must configure an audio filter with a coefficient that is a rational number to eliminate timing drift. Four candidate coefficients are derived from different mathematical expressions. Which of the following coefficients meets the engineer's requirement by representing a rational number?

   • A) $\pi + 1.45$
   • B) $\sqrt{1.\overline{7}} - \sqrt{0.64}$
   • C) $\sqrt{2.5}$
   • D) $\sqrt{8} + 3$
3. 3.

   A telecommunications engineer is evaluating four candidate frequency offsets to prevent periodic harmonic interference with a rational reference clock. An offset avoids periodic interference only if its calculated value in kilohertz is an irrational number. The four candidate frequency offsets are Offset 1: $(\sqrt{7} + 3) - \sqrt{7}$ kHz, Offset 2: $\frac{\sqrt{50}}{\sqrt{2}}$ kHz, Offset 3: $(\sqrt{6})^2 + 1.\overline{6}$ kHz, and Offset 4: $5 - \sqrt{3}$ kHz. Which candidate frequency offset meets the engineer's criterion?

   • A) Offset 1: $(\sqrt{7} + 3) - \sqrt{7}$ kHz
   • B) Offset 2: $\frac{\sqrt{50}}{\sqrt{2}}$ kHz
   • C) Offset 3: $(\sqrt{6})^2 + 1.\overline{6}$ kHz
   • D) Offset 4: $5 - \sqrt{3}$ kHz
4. 4.

   An architect needs to choose a rectangular solar panel whose surface area is a rational number of square meters. Four prototype panels are tested with different length and width dimensions: Panel A is $\sqrt{18}$ meters by 2 meters, Panel B is $\sqrt{5}$ meters by $\sqrt{15}$ meters, Panel C is $\sqrt{12}$ meters by $\sqrt{27}$ meters, and Panel D is $\sqrt{8}$ meters by $\sqrt{6}$ meters. Which panel design meets the architect's requirement?

   • A) Panel A
   • B) Panel B
   • C) Panel C
   • D) Panel D
5. 5.

   A robotics team programs a rover to navigate along four paths on a test grid, each defined by a measured distance: Path 1 is $\sqrt{1.44}$ meters, Path 2 is $3.\overline{18}$ meters, Path 3 is $\sqrt{18}$ meters, and Path 4 is $\frac{22}{7}$ meters. The rover's navigation software can store any distance that can be expressed as a ratio of two integers $\frac{a}{b}$ where $b \neq 0$, but it flags any distance that cannot be written in this form for manual calibration. Which path distance will be flagged by the software as an irrational number?

   • A) Path 1 ($\sqrt{1.44}$ meters)
   • B) Path 2 ($3.\overline{18}$ meters)
   • C) Path 3 ($\sqrt{18}$ meters)
   • D) Path 4 ($\frac{22}{7}$ meters)

Answers and step-by-step solutions appear after you finish the test.