1 Concept Brief ​Mission Briefing: Welcome to the Galactic Command, Commander. A distress signal has been intercepted from the Starship Apollo, stranded in the outer rings of Sector 7. To execute this space rescue mission, you must manage our ship's limited resources. Your primary mathematical tool for this mission is Fractions Addition. ​Before we launch, review the core mechanics of your ship's fraction computers: ​The Anatomy of a Fraction: A fraction represents a part of a whole. ​Numerator (Top Number): Represents the parts of the resource we currently have. ​Denominator (Bottom Number): Represents the total equal parts that make up one whole unit. ​Rule 1: Adding Like Fractions (Same Denominators) When the denominators are the same, the resource parts are identical in size. Simply add the numerators together and keep the denominator exactly the same. Example: \frac{1}{5} + \frac{3}{5} = \frac{4}{5} ​Rule 2: Adding Unlike Fractions (Different Denominators) You cannot directly add fractions with different denominators because the parts are of different sizes. You must first find a Common Denominator. ​Step A: Find the Least Common Multiple (LCM) of the denominators. ​Step B: Convert each fraction into an equivalent fraction using the LCM. Whatever you multiply the bottom by to get the LCM, you must also multiply the top by. ​Step C: Add the new numerators and keep the common denominator. ​Rule 3: Simplification Always reduce your final fraction to its simplest form by dividing the numerator and denominator by their Greatest Common Factor (GCF). This ensures our ship's databanks process the most efficient numbers. ​2 Dynamic Challenge Set ​Mission Objectives: Operation Stellar Rescue ​The rescue sequence has been initiated. Solve the following sequence of terminal alerts to stabilize the ship and rescue the stranded crew. Do not proceed to the solutions until you have locked in your answers. ​Challenge 1: Hull Integrity (Direct Numerical - Like Denominators) Micrometeorites have damaged the starboard wing. The repair droids have sealed \frac{3}{8} of the breached area, while the engineering team has manually sealed another \frac{4}{8} of the area. What total fraction of the breached area has been sealed so far? ​Challenge 2: Navigation Glitch (Conceptual Reasoning) The autopilot system just attempted to merge two navigation coordinates to plot our jump. It calculated: \frac{1}{4} + \frac{1}{4} = \frac{2}{8} The jump is failing! Explain mathematically why the ship's computer is wrong, and provide the correct, simplified fraction to fix the navigation system. ​Challenge 3: Thruster Fuel Synthesis (Direct Numerical - Unlike Denominators) To break out of the sector's gravitational pull, we need to synthesize hyper-fuel. You pump in \frac{1}{3} of a tank of Liquid Trillium and \frac{2}{5} of a tank of Plasma Gel. What is the total fraction of the thruster fuel tank filled by these two components? ​Challenge 4: Life Support Reserves (Application-Based - Mixed Numbers) You reach the Starship Apollo and need to transfer oxygen. Portable Life Support Module Alpha contains 1\frac{1}{2} hours of oxygen. Module Beta contains \frac{3}{4} of an hour of oxygen. What is the total amount of oxygen available in hours for the stranded crew? ​Challenge 5: Power Grid Alignment (Logical Variation - Missing Addend) The rescue pod requires exactly 1 whole gigawatt of power to launch safely back to the mothership. The solar arrays are currently generating \frac{7}{12} of a gigawatt. What fraction of a gigawatt must you manually reroute from the auxiliary battery to reach exactly 1 whole gigawatt for the launch? ​3 Detailed Solutions and Explanation ​Solution 1: Hull Integrity ​Problem: \frac{3}{8} + \frac{4}{8} ​Step 1: Check the denominators. Both are 8. Since they are like fractions, we can add them directly. ​Step 2: Add the numerators: 3 + 4 = 7. ​Step 3: Keep the denominator the same: 8. ​Step 4: Check if it can be simplified. The only common factor of 7 and 8 is 1. ​Final Answer: \frac{7}{8} of the breached area has been sealed. ​Solution 2: Navigation Glitch ​Explanation of Error: The computer incorrectly added the denominators together (4 + 4 = 8). When adding fractions, the denominator represents the size of the parts, which does not change. You only add the numerators (the quantity of those parts). ​Step 1: Correct the calculation. \frac{1}{4} + \frac{1}{4}. The denominators are the same. ​Step 2: Add the numerators (1 + 1 = 2) and keep the denominator (4). This gives \frac{2}{4}. ​Step 3: Simplify the fraction. Both 2 and 4 are divisible by their Greatest Common Factor, which is 2. ​Step 4: Divide top and bottom by 2: \frac{2 \div 2}{4 \div 2} = \frac{1}{2}. ​Final Answer: The computer added the denominators instead of keeping them the same. The correct navigation coordinate is \frac{1}{2}. ​Solution 3: Thruster Fuel Synthesis ​Problem: \frac{1}{3} + \frac{2}{5} ​Step 1: Check the denominators (3 and 5). They are unlike, so we need a common denominator. ​Step 2: Find the LCM of 3 and 5. The multiples of 3 are 3, 6, 9, 12, 15, \dots The multiples of 5 are 5, 10, 15, \dots The LCM is 15. ​Step 3: Convert the first fraction. To turn thirds into fifteenths, multiply the numerator and denominator by 5: \frac{1 \times 5}{3 \times 5} = \frac{5}{15}. ​Step 4: Convert the second fraction. To turn fifths into fifteenths, multiply the numerator and denominator by 3: \frac{2 \times 3}{5 \times 3} = \frac{6}{15}. ​Step 5: Add the equivalent fractions: \frac{5}{15} + \frac{6}{15} = \frac{11}{15}. ​Step 6: Check for simplification. 11 is a prime number and not a factor of 15. It is in simplest form. ​Final Answer: The tank is \frac{11}{15} full. ​Solution 4: Life Support Reserves ​Problem: 1\frac{1}{2} + \frac{3}{4} ​Step 1: Convert the mixed number into an improper fraction. Multiply the whole number by the denominator and add the numerator: (1 \times 2) + 1 = 3. Place this over the original denominator: \frac{3}{2}. ​Step 2: The new problem is \frac{3}{2} + \frac{3}{4}. The denominators (2 and 4) are unlike. ​Step 3: Find the LCM of 2 and 4. The LCM is 4. ​Step 4: Convert \frac{3}{2} into fourths by multiplying top and bottom by 2: \frac{3 \times 2}{2 \times 2} = \frac{6}{4}. ​Step 5: The second fraction (\frac{3}{4}) already has the common denominator. ​Step 6: Add the numerators: \frac{6}{4} + \frac{3}{4} = \frac{9}{4}. ​Step 7: Convert the improper fraction back into a mixed number. Divide 9 by 4. 9 \div 4 = 2 with a remainder of 1. The mixed number is 2\frac{1}{4}. ​Final Answer: There are 2\frac{1}{4} hours of oxygen available. ​Solution 5: Power Grid Alignment ​Problem: You have \frac{7}{12} and need to reach 1. We must find the difference: 1 - \frac{7}{12}. ​Step 1: Convert the whole number 1 into a fraction with the same denominator as the fraction we currently have. Since we are working with twelfths, 1 whole is equal to \frac{12}{12}. ​Step 2: Set up the equation: \frac{12}{12} - \frac{7}{12}. ​(Note: Alternatively, think of it as an addition equation: \frac{7}{12} + \frac{?}{12} = \frac{12}{12}. What plus 7 equals 12?) ​Step 3: The denominators are the same, so subtract the numerators: 12 - 7 = 5. ​Step 4: Keep the denominator: \frac{5}{12}. ​Step 5: Check for simplification. 5 is prime and does not divide evenly into 12. It is in simplest form. ​Final Answer: You must reroute \frac{5}{12} of a gigawatt to reach exactly 1 whole gigawatt.