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Problem-solving Pack

Ecolint Campus des NationsMathematics
Year 8 · 8.7 Algebra Expressions

Problem-solving Pack

Name: _________________________________
Date: _________________ Class: ___________

These problems are designed to challenge you. Read each question carefully. Show all your reasoning — a correct answer without working receives no credit.

1Problem 1 of 12
Equivalent expressions check. Determine whether each pair of expressions is equivalent, by expanding/simplifying:

(a) 3(x+4)3(x + 4) and 3x+123x + 12.
(b) (x+2)2(x + 2)^2 and x2+4x^2 + 4.
(c) 2(x−3)−(x−5)2(x - 3) - (x - 5) and x−1x - 1.
(d) 6x2\dfrac{6x}{2} and 3x3x.

Working space

2Problem 2 of 12
Substitution with mixed signs. Calculate the value of each expression for x=−3,y=2x = -3, y = 2:

(a) 2x+3y2x + 3y
(b) x2−y2x^2 - y^2
(c) x+yx−y\dfrac{x + y}{x - y}
(d) (2x−y)2(2x - y)^2

Working space

3Problem 3 of 12
Factorise practice. Factorise each expression fully.

(a) 6x+126x + 12
(b) 4x2−8x4x^2 - 8x
(c) 12a2b−18ab212 a^2 b - 18 a b^2
(d) 5(x−1)+y(x−1)5(x - 1) + y(x - 1)

Working space

4Problem 4 of 12
Algebraic fractions. Simplify each:

(a) 8a2b4ab2\dfrac{8a^2 b}{4 a b^2}
(b) 6x2−9x3x\dfrac{6x^2 - 9x}{3x}
(c) x2+2x5\dfrac{x}{2} + \dfrac{2x}{5}
(d) x−34−x+16\dfrac{x - 3}{4} - \dfrac{x + 1}{6}

Working space

5Problem 5 of 12
Songs in terms of xx (CdN Algebra Feb 2022). Pete has three times as many songs as Sabrina. Sabrina has 120 fewer songs than Elena.

(a) If Elena has xx songs, write expressions for (i) Sabrina's songs and (ii) Pete's songs.
(b) Pete has 300 songs. Write and solve an equation to find Elena's count.

Working space

6Problem 6 of 12
Rectangle expressions. A rectangle has dimensions (2x+1)(2x + 1) and (x+3)(x + 3).

(a) Write expressions for perimeter and area.
(b) Expand the area expression.
(c) When x=4x = 4, find numerical values for perimeter and area.

Working space

7Problem 7 of 12
Two-bracket expansion. Expand and simplify each:

(a) (x+5)(x+3)(x + 5)(x + 3)
(b) (x−4)(x+7)(x - 4)(x + 7)
(c) (2x−1)(x+3)(2x - 1)(x + 3)
(d) (x+2)2(x + 2)^2

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8Problem 8 of 12
Pete & Sabrina extended. Pete has 3 times as many songs as Sabrina. Sabrina has dd fewer songs than Elena. If Pete has PP songs, express Elena's number of songs in terms of PP and dd.

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9Problem 9 of 12
Algebra in geometry. A triangle has angles (2x)°(2x)°, (3x+10)°(3x + 10)°, (x−10)°(x - 10)°.

(a) Set up an equation in xx and solve.
(b) Find the three angles.

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10Problem 10 of 12
Patterns in algebraic identities. Verify (by expansion) that for any aa and bb:

(a) (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab + b^2
(b) (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab + b^2
(c) (a+b)(a−b)=a2−b2(a + b)(a - b) = a^2 - b^2
(d) Use (a) to compute 52252^2 from 502=250050^2 = 2500.

Working space

11Problem 11 of 12
Comparing fractions algebraically. Simplify each and decide which is larger when x>0x > 0:

(a) x+62\dfrac{x + 6}{2} vs 2x+63\dfrac{2x + 6}{3}
(b) Find the value of xx where they are equal.

Working space

12Problem 12 of 12
Repeating-decimal investigation. Recurring decimals can be turned into fractions algebraically. Let x=0.ab‾x = 0.\overline{ab} (two-digit repeating block).

(a) Show that 100x−x=ab100x - x = ab and hence x=ab99x = \tfrac{ab}{99}.
(b) Use this to convert 0.36‾0.\overline{36} to a fraction in simplest form.
(c) Convert 0.142857‾0.\overline{142857} — what fraction do you recognise?

Working space