Advanced Algebra: Variables, Expressions, and Patterns — Challenge Set

1.

Word Problem Evaluate the expression 4x + 3y - 2xy for x = 3 and y = 5. Show your working.

3 pts

Working:

Answer:

Worked Solution:

Step 1: Substitute x = 3 and y = 5 into 4x + 3y - 2xy

Step 2: Compute 4(3) + 3(5) - 2(3)(5) = 12 + 15 - 30

Step 3: Simplify 27 - 30 = -3

Answer: -3

2.

Word Problem A rectangular garden has length L = 3a + 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.

3 pts

Working:

Answer:

Worked Solution:

Step 1: Perimeter P = 2(L + W) = 2((3a + 2) + (a - 1))

Step 2: Inside the parentheses: 3a + 2 + a - 1 = 4a + 1

Step 3: Multiply by 2: P = 2(4a + 1) = 8a + 2

Step 4: Substitute a = 4: P = 8(4) + 2 = 34

Answer: Perimeter expression 8a + 2; when a = 4, P = 34 meters

3.

Word Problem 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.

3 pts

Working:

Answer:

Worked Solution:

Step 1: Discounted price per notebook = p - 3

Step 2: Total cost = 7(p - 3)

Step 3: Substitute p = 18: 7(18 - 3) = 7(15) = 105

Answer: 7(p - 3); when p = 18, total cost = 105

4.

Word Problem Mira is 4 years less than twice Rehan’s age. In 5 years, the sum of their ages will be 54. Find their current ages. Show your algebraic steps.

3 pts

Working:

Answer:

Worked Solution:

Step 1: Let Rehan = r; then Mira = 2r - 4

Step 2: In 5 years: (r + 5) + (2r - 4 + 5) = 54

Step 3: Simplify: r + 5 + 2r + 1 = 54 → 3r + 6 = 54

Step 4: Solve: 3r = 48 → r = 16; Mira = 2(16) - 4 = 28

Answer: Rehan is 16 years; Mira is 28 years

5.

Word Problem Three consecutive even integers have a sum of 114. Find the integers. Show your working.

3 pts

Working:

Answer:

Worked Solution:

Step 1: Let integers be x, x + 2, x + 4

Step 2: Sum: x + (x + 2) + (x + 4) = 114 → 3x + 6 = 114

Step 3: Solve: 3x = 108 → x = 36

Step 4: Integers: 36, 38, 40

Answer: 36, 38, and 40

6.

Word Problem 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.

3 pts

Working:

Answer:

Worked Solution:

Step 1: Deposits: x, x+20, x+40, x+60, x+80, x+100

Step 2: Sum = 6x + (20 + 40 + 60 + 80 + 100) = 6x + 300

Step 3: Substitute x = 100: 6(100) + 300 = 600 + 300 = 900

Answer: 6x + 300; when x = 100, total = 900

7.

Word Problem 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.

3 pts

Working:

Answer:

Worked Solution:

Step 1: First square uses 4 sticks; each of the remaining (n−1) adds 3

Step 2: Sticks S(n) = 4 + 3(n − 1) = 3n + 1

Step 3: For n = 12: S(12) = 3(12) + 1 = 36 + 1 = 37

Answer: S(n) = 3n + 1; for n = 12, 37 sticks

8.

Word Problem A jar has 2x + 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.

3 pts

Working:

Answer:

Worked Solution:

Step 1: Total marbles M = 2x + 7; with x = 8 → M = 2(8) + 7 = 23

Step 2: Each bag holds 5 → 23 ÷ 5 = 4 remainder 3

Step 3: So 4 full bags and 3 marbles left

Answer: 4 full bags; 3 marbles remaining

9.

Word Problem A car travels at (v + 5) km/h for (t − 0.5) hours. Write the distance as an expression in v and t, then evaluate for v = 40 and t = 3. Show your working.

3 pts

Working:

Answer:

Worked Solution:

Step 1: Distance D = speed × time = (v + 5)(t − 0.5)

Step 2: Substitute v = 40, t = 3: D = (45)(2.5)

Step 3: Compute 45 × 2.5 = 112.5

Answer: D = (v + 5)(t − 0.5); for v = 40, t = 3, D = 112.5 km

10.

Word Problem A rectangle has width w = l − 3 and perimeter P = 50 meters. Use P = 2(l + w) to find the length l. Show your algebraic steps.

3 pts

Working:

Answer:

Worked Solution:

Step 1: Substitute w into P: 50 = 2(l + (l − 3))

Step 2: Inside parentheses: l + l − 3 = 2l − 3

Step 3: 50 = 2(2l − 3) = 4l − 6

Step 4: 4l = 56 → l = 14

Answer: l = 14 meters

11.

Word Problem 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.

3 pts

Working:

Answer:

Worked Solution:

Step 1: Sandwiches S = 2n; Juices J = n

Step 2: For n = 7: S = 2(7) = 14; J = 7

Answer: 14 sandwiches and 7 juices

12.

Word Problem A rectangle’s area is A(x) = (x + 3)(x − 1). If x increases by 2, find the increase in area and evaluate it when x = 5. Show your working.

3 pts

Working:

Answer:

Worked Solution:

Step 1: A(x) = (x + 3)(x − 1) = x^2 + 2x − 3

Step 2: A(x + 2) = (x + 5)(x + 1) = x^2 + 6x + 5

Step 3: Increase ΔA = A(x + 2) − A(x) = (x^2 + 6x + 5) − (x^2 + 2x − 3) = 4x + 8

Step 4: For x = 5: ΔA = 4(5) + 8 = 28

Answer: Increase is 4x + 8; when x = 5, increase = 28 square units

13.

Word Problem Adult tickets cost a each; child tickets cost (a − 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.

3 pts

Working:

Answer:

Worked Solution:

Step 1: Total cost T = 3a + 2(a − 40)

Step 2: Simplify: T = 3a + 2a − 80 = 5a − 80

Step 3: Substitute a = 150: T = 5(150) − 80 = 750 − 80 = 670

Answer: 5a − 80; when a = 150, total = 670

14.

Word Problem Evaluate (x − 2)(x + 3) − (x − 1)^2 at x = 6 by simplifying the expression first. Show all steps.

3 pts

Working:

Answer:

Worked Solution:

Step 1: Expand (x − 2)(x + 3) = x^2 + x − 6

Step 2: Expand (x − 1)^2 = x^2 − 2x + 1

Step 3: Subtract: (x^2 + x − 6) − (x^2 − 2x + 1) = 3x − 7

Step 4: Substitute x = 6: 3(6) − 7 = 18 − 7 = 11

Answer: 11

15.

Word Problem 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.

3 pts

Working:

Answer:

Worked Solution:

Step 1: Arithmetic sequence with first term a1 = 7 and difference d = 5

Step 2: General term a_n = a1 + (n − 1)d = 7 + (n − 1)5

Step 3: For n = 20: a_20 = 7 + 19×5 = 7 + 95 = 102

Answer: 102 flowers on day 20

16.

Word Problem A recipe mixes sugar and flour in a 2:3 ratio. If the total mass is 5k + 10 grams, find expressions for the sugar and flour masses, then compute both when k = 8. Show your steps.

3 pts

Working:

Answer:

Worked Solution:

Step 1: Total parts = 2 + 3 = 5

Step 2: Sugar = (2/5)(5k + 10) = 2k + 4; Flour = (3/5)(5k + 10) = 3k + 6

Step 3: For k = 8: Sugar = 2(8) + 4 = 20; Flour = 3(8) + 6 = 30

Answer: Sugar = 2k + 4 (20 g when k = 8); Flour = 3k + 6 (30 g when k = 8)

17.

Word Problem A regular n-gon has side length s(n) = 0.5n + 1 meters. Write the perimeter as a function of n and evaluate it when n = 10. Show your working.

3 pts

Working:

Answer:

Worked Solution:

Step 1: Perimeter P(n) = n × s(n) = n(0.5n + 1)

Step 2: For n = 10: P(10) = 10(0.5×10 + 1) = 10(5 + 1) = 60

Answer: P(n) = n(0.5n + 1); when n = 10, perimeter = 60 meters

18.

Word Problem 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.

3 pts

Working:

Answer:

Worked Solution:

Step 1: Let the number be x; equation: 6x − (x + 4) = 38

Step 2: Simplify: 6x − x − 4 = 38 → 5x − 4 = 38

Step 3: Solve: 5x = 42 → x = 42/5 = 8.4

Answer: 8.4

19.

Word Problem 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.

3 pts

Working:

Answer:

Worked Solution:

Step 1: Notebook cost with discount: 5 × 25 × 0.9 = 112.5

Step 2: Pen cost: 3 × 12 = 36

Step 3: Total spent: 112.5 + 36 = 148.5

Step 4: Change: 200 − 148.5 = 51.5

Answer: 51.5

20.

Word Problem Two numbers have a sum of 37. The difference (larger − smaller) equals twice the smaller minus 5. Find both numbers. Show your working.

3 pts

Working:

Answer:

Worked Solution:

Step 1: Let larger = L, smaller = S; L + S = 37

Step 2: Given L − S = 2S − 5 → L = 3S − 5

Step 3: Substitute into sum: (3S − 5) + S = 37 → 4S − 5 = 37

Step 4: 4S = 42 → S = 10.5; L = 37 − 10.5 = 26.5

Answer: Smaller = 10.5, Larger = 26.5

21.

Fill in the Blank The simplified form of 3(x + 4) − 2(2x − 5) is _____.

3 pts

Answer: -x + 22

22.

Fill in the Blank If P = 2l + 2w and w = 3l − 5, then P expressed only in terms of l is _____.

3 pts

Answer: 8l - 10

23.

Fill in the Blank If two consecutive odd integers are n and n + 2, then their product minus their sum simplifies to _____.

3 pts

Answer: n^2 - 2

24.

Fill in the Blank If a = 2b − 3 and b = 5, then the value of 3a − b is _____.

3 pts

Answer: 16

25.

Fill in the Blank The perimeter of a regular 11-gon with side length s is _____.

3 pts

Answer: 11s

26.

Fill in the Blank The expression equivalent to (x + y)^2 − (x − y)^2 is _____.

3 pts

Answer: 4xy

27.

Fill in the Blank If f(n) = 5n − 1 and g(n) = n^2, then f(g(3)) equals _____.

3 pts

Answer: 44

28.

Fill in the Blank In the expansion of (2x − 3)(x + 5), the coefficient of x is _____.

3 pts

Answer: 7

29.

Fill in the Blank For a rectangle with fixed area A and width w, the length expressed in terms of A and w is _____.

3 pts

Answer: A/w

30.

Fill in the Blank 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 _____.

3 pts

Answer: 2n + 1

31.

Fill in the Blank Simplify 4(x − 2y) + 3(2y − x) to _____.

3 pts

Answer: x - 2y

32.

Fill in the Blank If x = 2y + 1 and y = 4, then x^2 − y equals _____.

3 pts

Answer: 77

33.

Fill in the Blank The sum of the first n odd numbers equals _____.

3 pts

Answer: n^2

34.

Fill in the Blank The simplified value of 1/2(6x − 4) − 1/3(3x + 6) is _____.

3 pts

Answer: 2x - 4

35.

Fill in the Blank If a rectangle’s length increases by k and its width decreases by k, the new perimeter in terms of l, w, and k is _____.

3 pts

Answer: 2(l + w)

36.

Fill in the Blank The solution for x in the equation 3(x − 4) = 2x + 5 is _____.

3 pts

Answer: 17

37.

Fill in the Blank If h(t) = t^2 − 4t, then the change h(t + 3) − h(t) simplifies to _____.

3 pts

Answer: 6t - 3

38.

Fill in the Blank The simplified form of (a + b + c) − (2a − b + 3c) is _____.

3 pts

Answer: -a + 2b - 2c

39.

Fill in the Blank If a square has the same perimeter as a rectangle with sides l and w, then the side length of the square is _____.

3 pts

Answer: (l + w)/2

40.

Fill in the Blank For the sequence defined by T1 = 2 and Tn = Tn−1 + 3 (n ≥ 2), the nth term Tn equals _____.

3 pts

Answer: 3n - 1