{
  "Id": "709eb28e-7214-405b-8270-f739534346bc",
  "Title": "Advanced Algebra: Variables, Expressions, and Patterns \u2014 Challenge Set",
  "Subject": "Mathematics",
  "Topic": "Variables, Algebraic Expressions, Properties, and Pattern Rules",
  "Instructions": "Read each question carefully and answer to the best of your ability. Show all working for word problems.",
  "EstimatedMinutes": 35,
  "StudentProfileId": "",
  "SourceFileName": "algebra.pdf",
  "SessionMaterialId": "728a742e-a3d5-49b9-bfb9-a467d0a1562f",
  "GeneratedAt": "2026-07-25T12:27:18.330302Z",
  "Questions": [
    {
      "Id": 1,
      "Type": "word_problem",
      "Text": "Evaluate the expression 4x \u002B 3y - 2xy for x = 3 and y = 5. Show your working.",
      "Options": [],
      "Answer": "Step 1: Substitute x = 3 and y = 5 into 4x \u002B 3y - 2xy\nStep 2: Compute 4(3) \u002B 3(5) - 2(3)(5) = 12 \u002B 15 - 30\nStep 3: Simplify 27 - 30 = -3\nAnswer: -3",
      "Points": 3
    },
    {
      "Id": 2,
      "Type": "word_problem",
      "Text": "A rectangular garden has length L = 3a \u002B 2 meters and width W = a - 1 meters. Derive a simplified expression for its perimeter in terms of a, then find the perimeter when a = 4. Show your working.",
      "Options": [],
      "Answer": "Step 1: Perimeter P = 2(L \u002B W) = 2((3a \u002B 2) \u002B (a - 1))\nStep 2: Inside the parentheses: 3a \u002B 2 \u002B a - 1 = 4a \u002B 1\nStep 3: Multiply by 2: P = 2(4a \u002B 1) = 8a \u002B 2\nStep 4: Substitute a = 4: P = 8(4) \u002B 2 = 34\nAnswer: Perimeter expression 8a \u002B 2; when a = 4, P = 34 meters",
      "Points": 3
    },
    {
      "Id": 3,
      "Type": "word_problem",
      "Text": "A student buys 7 identical notebooks. Each notebook has a marked price of p, but there is a discount of 3 on each. Write an expression for the total cost and evaluate it when p = 18. Show your working.",
      "Options": [],
      "Answer": "Step 1: Discounted price per notebook = p - 3\nStep 2: Total cost = 7(p - 3)\nStep 3: Substitute p = 18: 7(18 - 3) = 7(15) = 105\nAnswer: 7(p - 3); when p = 18, total cost = 105",
      "Points": 3
    },
    {
      "Id": 4,
      "Type": "word_problem",
      "Text": "Mira is 4 years less than twice Rehan\u2019s age. In 5 years, the sum of their ages will be 54. Find their current ages. Show your algebraic steps.",
      "Options": [],
      "Answer": "Step 1: Let Rehan = r; then Mira = 2r - 4\nStep 2: In 5 years: (r \u002B 5) \u002B (2r - 4 \u002B 5) = 54\nStep 3: Simplify: r \u002B 5 \u002B 2r \u002B 1 = 54 \u2192 3r \u002B 6 = 54\nStep 4: Solve: 3r = 48 \u2192 r = 16; Mira = 2(16) - 4 = 28\nAnswer: Rehan is 16 years; Mira is 28 years",
      "Points": 3
    },
    {
      "Id": 5,
      "Type": "word_problem",
      "Text": "Three consecutive even integers have a sum of 114. Find the integers. Show your working.",
      "Options": [],
      "Answer": "Step 1: Let integers be x, x \u002B 2, x \u002B 4\nStep 2: Sum: x \u002B (x \u002B 2) \u002B (x \u002B 4) = 114 \u2192 3x \u002B 6 = 114\nStep 3: Solve: 3x = 108 \u2192 x = 36\nStep 4: Integers: 36, 38, 40\nAnswer: 36, 38, and 40",
      "Points": 3
    },
    {
      "Id": 6,
      "Type": "word_problem",
      "Text": "A saver deposits x this month and increases the deposit by 20 each subsequent month for a total of 6 months. Find a simplified expression for the 6-month total and evaluate it for x = 100. Show your working.",
      "Options": [],
      "Answer": "Step 1: Deposits: x, x\u002B20, x\u002B40, x\u002B60, x\u002B80, x\u002B100\nStep 2: Sum = 6x \u002B (20 \u002B 40 \u002B 60 \u002B 80 \u002B 100) = 6x \u002B 300\nStep 3: Substitute x = 100: 6(100) \u002B 300 = 600 \u002B 300 = 900\nAnswer: 6x \u002B 300; when x = 100, total = 900",
      "Points": 3
    },
    {
      "Id": 7,
      "Type": "word_problem",
      "Text": "Matchstick squares in a row share sides: the first square needs 4 sticks; each additional square adds 3 sticks. Derive a formula for the number of sticks needed for n squares in a row and compute it for n = 12. Show your reasoning.",
      "Options": [],
      "Answer": "Step 1: First square uses 4 sticks; each of the remaining (n\u22121) adds 3\nStep 2: Sticks S(n) = 4 \u002B 3(n \u2212 1) = 3n \u002B 1\nStep 3: For n = 12: S(12) = 3(12) \u002B 1 = 36 \u002B 1 = 37\nAnswer: S(n) = 3n \u002B 1; for n = 12, 37 sticks",
      "Points": 3
    },
    {
      "Id": 8,
      "Type": "word_problem",
      "Text": "A jar has 2x \u002B 7 marbles. They are packed into bags that hold 5 marbles each. For x = 8, find the number of completely filled bags and how many marbles remain. Show your steps.",
      "Options": [],
      "Answer": "Step 1: Total marbles M = 2x \u002B 7; with x = 8 \u2192 M = 2(8) \u002B 7 = 23\nStep 2: Each bag holds 5 \u2192 23 \u00F7 5 = 4 remainder 3\nStep 3: So 4 full bags and 3 marbles left\nAnswer: 4 full bags; 3 marbles remaining",
      "Points": 3
    },
    {
      "Id": 9,
      "Type": "word_problem",
      "Text": "A car travels at (v \u002B 5) km/h for (t \u2212 0.5) hours. Write the distance as an expression in v and t, then evaluate for v = 40 and t = 3. Show your working.",
      "Options": [],
      "Answer": "Step 1: Distance D = speed \u00D7 time = (v \u002B 5)(t \u2212 0.5)\nStep 2: Substitute v = 40, t = 3: D = (45)(2.5)\nStep 3: Compute 45 \u00D7 2.5 = 112.5\nAnswer: D = (v \u002B 5)(t \u2212 0.5); for v = 40, t = 3, D = 112.5 km",
      "Points": 3
    },
    {
      "Id": 10,
      "Type": "word_problem",
      "Text": "A rectangle has width w = l \u2212 3 and perimeter P = 50 meters. Use P = 2(l \u002B w) to find the length l. Show your algebraic steps.",
      "Options": [],
      "Answer": "Step 1: Substitute w into P: 50 = 2(l \u002B (l \u2212 3))\nStep 2: Inside parentheses: l \u002B l \u2212 3 = 2l \u2212 3\nStep 3: 50 = 2(2l \u2212 3) = 4l \u2212 6\nStep 4: 4l = 56 \u2192 l = 14\nAnswer: l = 14 meters",
      "Points": 3
    },
    {
      "Id": 11,
      "Type": "word_problem",
      "Text": "A cafe sells a combo of 2 sandwiches and 1 juice. If a group orders n combos, express the total number of sandwiches and juices, then find both totals for n = 7. Show your working.",
      "Options": [],
      "Answer": "Step 1: Sandwiches S = 2n; Juices J = n\nStep 2: For n = 7: S = 2(7) = 14; J = 7\nAnswer: 14 sandwiches and 7 juices",
      "Points": 3
    },
    {
      "Id": 12,
      "Type": "word_problem",
      "Text": "A rectangle\u2019s area is A(x) = (x \u002B 3)(x \u2212 1). If x increases by 2, find the increase in area and evaluate it when x = 5. Show your working.",
      "Options": [],
      "Answer": "Step 1: A(x) = (x \u002B 3)(x \u2212 1) = x^2 \u002B 2x \u2212 3\nStep 2: A(x \u002B 2) = (x \u002B 5)(x \u002B 1) = x^2 \u002B 6x \u002B 5\nStep 3: Increase \u0394A = A(x \u002B 2) \u2212 A(x) = (x^2 \u002B 6x \u002B 5) \u2212 (x^2 \u002B 2x \u2212 3) = 4x \u002B 8\nStep 4: For x = 5: \u0394A = 4(5) \u002B 8 = 28\nAnswer: Increase is 4x \u002B 8; when x = 5, increase = 28 square units",
      "Points": 3
    },
    {
      "Id": 13,
      "Type": "word_problem",
      "Text": "Adult tickets cost a each; child tickets cost (a \u2212 40). A family buys 3 adult and 2 child tickets. Write the total cost in terms of a and evaluate it when a = 150. Show your working.",
      "Options": [],
      "Answer": "Step 1: Total cost T = 3a \u002B 2(a \u2212 40)\nStep 2: Simplify: T = 3a \u002B 2a \u2212 80 = 5a \u2212 80\nStep 3: Substitute a = 150: T = 5(150) \u2212 80 = 750 \u2212 80 = 670\nAnswer: 5a \u2212 80; when a = 150, total = 670",
      "Points": 3
    },
    {
      "Id": 14,
      "Type": "word_problem",
      "Text": "Evaluate (x \u2212 2)(x \u002B 3) \u2212 (x \u2212 1)^2 at x = 6 by simplifying the expression first. Show all steps.",
      "Options": [],
      "Answer": "Step 1: Expand (x \u2212 2)(x \u002B 3) = x^2 \u002B x \u2212 6\nStep 2: Expand (x \u2212 1)^2 = x^2 \u2212 2x \u002B 1\nStep 3: Subtract: (x^2 \u002B x \u2212 6) \u2212 (x^2 \u2212 2x \u002B 1) = 3x \u2212 7\nStep 4: Substitute x = 6: 3(6) \u2212 7 = 18 \u2212 7 = 11\nAnswer: 11",
      "Points": 3
    },
    {
      "Id": 15,
      "Type": "word_problem",
      "Text": "On day 1 a gardener plants 7 flowers; each day after that, 5 more flowers than the previous day. How many flowers are planted on day 20? Show your working.",
      "Options": [],
      "Answer": "Step 1: Arithmetic sequence with first term a1 = 7 and difference d = 5\nStep 2: General term a_n = a1 \u002B (n \u2212 1)d = 7 \u002B (n \u2212 1)5\nStep 3: For n = 20: a_20 = 7 \u002B 19\u00D75 = 7 \u002B 95 = 102\nAnswer: 102 flowers on day 20",
      "Points": 3
    },
    {
      "Id": 16,
      "Type": "word_problem",
      "Text": "A recipe mixes sugar and flour in a 2:3 ratio. If the total mass is 5k \u002B 10 grams, find expressions for the sugar and flour masses, then compute both when k = 8. Show your steps.",
      "Options": [],
      "Answer": "Step 1: Total parts = 2 \u002B 3 = 5\nStep 2: Sugar = (2/5)(5k \u002B 10) = 2k \u002B 4; Flour = (3/5)(5k \u002B 10) = 3k \u002B 6\nStep 3: For k = 8: Sugar = 2(8) \u002B 4 = 20; Flour = 3(8) \u002B 6 = 30\nAnswer: Sugar = 2k \u002B 4 (20 g when k = 8); Flour = 3k \u002B 6 (30 g when k = 8)",
      "Points": 3
    },
    {
      "Id": 17,
      "Type": "word_problem",
      "Text": "A regular n-gon has side length s(n) = 0.5n \u002B 1 meters. Write the perimeter as a function of n and evaluate it when n = 10. Show your working.",
      "Options": [],
      "Answer": "Step 1: Perimeter P(n) = n \u00D7 s(n) = n(0.5n \u002B 1)\nStep 2: For n = 10: P(10) = 10(0.5\u00D710 \u002B 1) = 10(5 \u002B 1) = 60\nAnswer: P(n) = n(0.5n \u002B 1); when n = 10, perimeter = 60 meters",
      "Points": 3
    },
    {
      "Id": 18,
      "Type": "word_problem",
      "Text": "The product of a number and 6 decreased by the sum of the number and 4 equals 38. Find the number. Show your algebraic steps.",
      "Options": [],
      "Answer": "Step 1: Let the number be x; equation: 6x \u2212 (x \u002B 4) = 38\nStep 2: Simplify: 6x \u2212 x \u2212 4 = 38 \u2192 5x \u2212 4 = 38\nStep 3: Solve: 5x = 42 \u2192 x = 42/5 = 8.4\nAnswer: 8.4",
      "Points": 3
    },
    {
      "Id": 19,
      "Type": "word_problem",
      "Text": "Notebooks cost 25 each; if you buy at least 5, a 10% discount applies to notebooks only. Pens cost 12 each with no discount. With a budget of 200, how much change remains after buying 5 notebooks and 3 pens? Show your working.",
      "Options": [],
      "Answer": "Step 1: Notebook cost with discount: 5 \u00D7 25 \u00D7 0.9 = 112.5\nStep 2: Pen cost: 3 \u00D7 12 = 36\nStep 3: Total spent: 112.5 \u002B 36 = 148.5\nStep 4: Change: 200 \u2212 148.5 = 51.5\nAnswer: 51.5",
      "Points": 3
    },
    {
      "Id": 20,
      "Type": "word_problem",
      "Text": "Two numbers have a sum of 37. The difference (larger \u2212 smaller) equals twice the smaller minus 5. Find both numbers. Show your working.",
      "Options": [],
      "Answer": "Step 1: Let larger = L, smaller = S; L \u002B S = 37\nStep 2: Given L \u2212 S = 2S \u2212 5 \u2192 L = 3S \u2212 5\nStep 3: Substitute into sum: (3S \u2212 5) \u002B S = 37 \u2192 4S \u2212 5 = 37\nStep 4: 4S = 42 \u2192 S = 10.5; L = 37 \u2212 10.5 = 26.5\nAnswer: Smaller = 10.5, Larger = 26.5",
      "Points": 3
    },
    {
      "Id": 21,
      "Type": "fill_blank",
      "Text": "The simplified form of 3(x \u002B 4) \u2212 2(2x \u2212 5) is _____.",
      "Options": [],
      "Answer": "-x \u002B 22",
      "Points": 3
    },
    {
      "Id": 22,
      "Type": "fill_blank",
      "Text": "If P = 2l \u002B 2w and w = 3l \u2212 5, then P expressed only in terms of l is _____.",
      "Options": [],
      "Answer": "8l - 10",
      "Points": 3
    },
    {
      "Id": 23,
      "Type": "fill_blank",
      "Text": "If two consecutive odd integers are n and n \u002B 2, then their product minus their sum simplifies to _____.",
      "Options": [],
      "Answer": "n^2 - 2",
      "Points": 3
    },
    {
      "Id": 24,
      "Type": "fill_blank",
      "Text": "If a = 2b \u2212 3 and b = 5, then the value of 3a \u2212 b is _____.",
      "Options": [],
      "Answer": "16",
      "Points": 3
    },
    {
      "Id": 25,
      "Type": "fill_blank",
      "Text": "The perimeter of a regular 11-gon with side length s is _____.",
      "Options": [],
      "Answer": "11s",
      "Points": 3
    },
    {
      "Id": 26,
      "Type": "fill_blank",
      "Text": "The expression equivalent to (x \u002B y)^2 \u2212 (x \u2212 y)^2 is _____.",
      "Options": [],
      "Answer": "4xy",
      "Points": 3
    },
    {
      "Id": 27,
      "Type": "fill_blank",
      "Text": "If f(n) = 5n \u2212 1 and g(n) = n^2, then f(g(3)) equals _____.",
      "Options": [],
      "Answer": "44",
      "Points": 3
    },
    {
      "Id": 28,
      "Type": "fill_blank",
      "Text": "In the expansion of (2x \u2212 3)(x \u002B 5), the coefficient of x is _____.",
      "Options": [],
      "Answer": "7",
      "Points": 3
    },
    {
      "Id": 29,
      "Type": "fill_blank",
      "Text": "For a rectangle with fixed area A and width w, the length expressed in terms of A and w is _____.",
      "Options": [],
      "Answer": "A/w",
      "Points": 3
    },
    {
      "Id": 30,
      "Type": "fill_blank",
      "Text": "For a path of n triangles where the first needs 3 sticks and each additional triangle adds 2, the total sticks needed as a function of n is _____.",
      "Options": [],
      "Answer": "2n \u002B 1",
      "Points": 3
    },
    {
      "Id": 31,
      "Type": "fill_blank",
      "Text": "Simplify 4(x \u2212 2y) \u002B 3(2y \u2212 x) to _____.",
      "Options": [],
      "Answer": "x - 2y",
      "Points": 3
    },
    {
      "Id": 32,
      "Type": "fill_blank",
      "Text": "If x = 2y \u002B 1 and y = 4, then x^2 \u2212 y equals _____.",
      "Options": [],
      "Answer": "77",
      "Points": 3
    },
    {
      "Id": 33,
      "Type": "fill_blank",
      "Text": "The sum of the first n odd numbers equals _____.",
      "Options": [],
      "Answer": "n^2",
      "Points": 3
    },
    {
      "Id": 34,
      "Type": "fill_blank",
      "Text": "The simplified value of 1/2(6x \u2212 4) \u2212 1/3(3x \u002B 6) is _____.",
      "Options": [],
      "Answer": "2x - 4",
      "Points": 3
    },
    {
      "Id": 35,
      "Type": "fill_blank",
      "Text": "If a rectangle\u2019s length increases by k and its width decreases by k, the new perimeter in terms of l, w, and k is _____.",
      "Options": [],
      "Answer": "2(l \u002B w)",
      "Points": 3
    },
    {
      "Id": 36,
      "Type": "fill_blank",
      "Text": "The solution for x in the equation 3(x \u2212 4) = 2x \u002B 5 is _____.",
      "Options": [],
      "Answer": "17",
      "Points": 3
    },
    {
      "Id": 37,
      "Type": "fill_blank",
      "Text": "If h(t) = t^2 \u2212 4t, then the change h(t \u002B 3) \u2212 h(t) simplifies to _____.",
      "Options": [],
      "Answer": "6t - 3",
      "Points": 3
    },
    {
      "Id": 38,
      "Type": "fill_blank",
      "Text": "The simplified form of (a \u002B b \u002B c) \u2212 (2a \u2212 b \u002B 3c) is _____.",
      "Options": [],
      "Answer": "-a \u002B 2b - 2c",
      "Points": 3
    },
    {
      "Id": 39,
      "Type": "fill_blank",
      "Text": "If a square has the same perimeter as a rectangle with sides l and w, then the side length of the square is _____.",
      "Options": [],
      "Answer": "(l \u002B w)/2",
      "Points": 3
    },
    {
      "Id": 40,
      "Type": "fill_blank",
      "Text": "For the sequence defined by T1 = 2 and Tn = Tn\u22121 \u002B 3 (n \u2265 2), the nth term Tn equals _____.",
      "Options": [],
      "Answer": "3n - 1",
      "Points": 3
    }
  ],
  "TotalPoints": 120
}