{"id":61858,"date":"2026-05-06T22:20:34","date_gmt":"2026-05-06T14:20:34","guid":{"rendered":"https:\/\/blog-admin.thethinkacademy.com\/?p=61858"},"modified":"2026-05-07T00:37:12","modified_gmt":"2026-05-06T16:37:12","slug":"algebra-worksheets","status":"publish","type":"post","link":"https:\/\/blog-admin.thethinkacademy.com\/blog\/2026\/05\/06\/algebra-worksheets\/","title":{"rendered":"Grade 8 Algebra Worksheets: Practice Problems for the Gauss Contest"},"content":{"rendered":"\n

Algebra is one of the most consistently tested topic areas on the Gauss math contest, appearing across every difficulty level from the accessible Part A questions through to the challenging Part C problems. These grade 8 algebra worksheets cover everything a student needs to handle the algebraic content on the Gauss contest confidently \u2014 simplifying expressions, solving linear equations, working with patterns and sequences, substitution, and multi-step algebraic word problems. Every problem comes with a full worked solution so students understand not just the answer but the reasoning behind it.<\/p>\n\n\n\n

\n\n\n
\n
\"\"<\/a><\/figure>\n<\/div>\n\n\n
\n\n\n\n

Algebra in Grade 8 and the Gauss contest<\/strong><\/h2>\n\n\n\n

Grade 8 algebra covers the transition from arithmetic thinking to algebraic thinking \u2014 the shift from working with specific numbers to working with unknown quantities and general relationships. This transition is exactly what the Gauss contest tests most heavily at the Grade 8 level.<\/p>\n\n\n\n

The Ontario Grade 8 mathematics curriculum covers expressions, equations, linear relations, and patterning. The Gauss contest draws from all of these but presents them in non-routine ways \u2014 problems that require setting up algebraic relationships from a word problem context, recognising patterns that are not immediately obvious, or solving equations that arise as part of a larger geometry or number theory problem.<\/p>\n\n\n\n

Students who have worked through structured grade 8 algebra worksheets \u2014 building from basic expression simplification through to algebraic word problems \u2014 are significantly better prepared for the Gauss than students who have only covered these topics in class.<\/p>\n\n\n\n

What algebraic skills does the Gauss contest test?<\/strong><\/h3>\n\n\n\n
Skill<\/th>How it appears on the Gauss<\/th><\/tr><\/thead>
Simplifying expressions<\/td>Direct simplification, substitution, evaluating expressions<\/td><\/tr>
Solving linear equations<\/td>One-step, two-step, and multi-step equations<\/td><\/tr>
Word problems<\/td>Setting up equations from context<\/td><\/tr>
Patterns and sequences<\/td>Finding the nth term, identifying rules<\/td><\/tr>
Substitution<\/td>Evaluating expressions for given variable values<\/td><\/tr>
Simultaneous reasoning<\/td>Problems requiring two algebraic relationships<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n

Find out more about the Gauss competition here at: Gauss Math Contest: The Complete Guide for Canadian Students and Parents<\/a>.<\/p>\n\n\n\n

Before competing, make sure to check out: Gauss Contest: What Canadian Students Need to Know Before They Compete<\/a>.<\/p>\n\n\n\n

\n\n\n\n

Grade 8 algebra worksheets \u2014 expressions and substitution<\/strong><\/h2>\n\n\n\n

These problems cover simplifying algebraic expressions and evaluating them for given variable values. They are equivalent in difficulty to Part A questions on the Gauss contest.<\/p>\n\n\n\n

\n\n\n\n

Problem 1 \u2014 simplifying expressions<\/strong><\/p>\n\n\n\n

Simplify: 3x + 5 – 2x + 7<\/p>\n\n\n\n

Solution:<\/strong> Collect like terms: 3x – 2x = x <\/p>\n\n\n\n

5 + 7 = 12 <\/p>\n\n\n\n

Simplified: x + 12<\/p>\n\n\n\n

Answer: x + 12<\/strong><\/p>\n\n\n\n

\n\n\n\n

Problem 2 \u2014 simplifying with brackets<\/strong><\/p>\n\n\n\n

\"gauss<\/figure>\n\n\n\n

Simplify: 4(2x + 3) – 2(x – 5)<\/p>\n\n\n\n

Solution:<\/strong> Expand brackets: 4(2x + 3) = 8x + 12 <\/p>\n\n\n\n

2(x – 5) = 2x – 10<\/p>\n\n\n\n

Subtract: 8x + 12 – 2x + 10 = 6x + 22<\/p>\n\n\n\n

Answer: 6x + 22<\/strong><\/p>\n\n\n\n

Key insight:<\/strong> When subtracting a bracket, every term inside changes sign. 2(x – 5) subtracted becomes -2x + 10, not -2x – 10. This is the most common error in bracket expansion.<\/p>\n\n\n\n

\n\n\n\n

Problem 3 \u2014 substitution<\/strong><\/p>\n\n\n\n

If x = 3 and y = -2, find the value of 2x\u00b2 – 3y + 1.<\/p>\n\n\n\n

Solution:<\/strong> Substitute x = 3 and y = -2: 2(3)\u00b2 – 3(-2) + 1 = 2(9) + 6 + 1 = 18 + 6 + 1 = 25<\/p>\n\n\n\n

Answer: 25<\/strong><\/p>\n\n\n\n

Key insight:<\/strong> Always substitute into the expression before evaluating \u2014 do not try to simplify the expression with the numbers in your head simultaneously. Writing out the substitution step clearly prevents sign errors.<\/p>\n\n\n\n

\n\n\n\n

Problem 4 \u2014 evaluating with fractions<\/strong><\/p>\n\n\n\n

If a = 1\/2 and b = 4, find the value of 2a + b\/a.<\/p>\n\n\n\n

Solution:<\/strong> 2a = 2 x 1\/2 = 1 b\/a = 4 \u00f7 1\/2 = 4 x 2 = 8 2a + b\/a = 1 + 8 = 9<\/p>\n\n\n\n

Answer: 9<\/strong><\/p>\n\n\n\n

\n\n\n\n

Problem 5 \u2014 Gauss style expression problem<\/strong><\/p>\n\n\n\n

If 3x + 7 = 22, what is the value of 6x + 5?<\/p>\n\n\n\n

Solution:<\/strong> From 3x + 7 = 22: 3x = 15 x = 5<\/p>\n\n\n\n

6x + 5 = 6(5) + 5 = 30 + 5 = 35<\/p>\n\n\n\n

Alternatively, without finding x: 6x + 5 = 2(3x + 7) – 9 = 2(22) – 9 = 44 – 9 = 35.<\/p>\n\n\n\n

Answer: 35<\/strong><\/p>\n\n\n\n

Key insight:<\/strong> The faster solution notices that 6x + 5 can be written in terms of 3x + 7 without finding x. This kind of algebraic shortcut appears regularly on the Gauss contest \u2014 recognising when a target expression is a multiple or transformation of a given expression saves significant time.<\/p>\n\n\n\n

\n\n\n\n

Grade 8 algebra worksheets \u2014 solving linear equations<\/strong><\/h2>\n\n\n\n

These algebra worksheets grade 8 problems cover solving equations of increasing complexity. They range from Part A to Part B difficulty on the Gauss contest.<\/p>\n\n\n\n

\n\n\n\n

Problem 6 \u2014 one-step equation<\/strong><\/p>\n\n\n\n

Solve: x + 13 = 28<\/p>\n\n\n\n

Solution:<\/strong> x = 28 – 13 x = 15<\/p>\n\n\n\n

Answer: x = 15<\/strong><\/p>\n\n\n\n

\n\n\n\n

Problem 7 \u2014 two-step equation<\/strong><\/p>\n\n\n\n

Solve: 3x – 8 = 16<\/p>\n\n\n\n

Solution:<\/strong> 3x = 16 + 8 3x = 24 x = 8<\/p>\n\n\n\n

Answer: x = 8<\/strong><\/p>\n\n\n\n

\n\n\n\n

Problem 8 \u2014 equation with brackets<\/strong><\/p>\n\n\n

\n
\"algebra<\/figure>\n<\/div>\n\n\n

Solve: 2(x + 5) = 3x – 4<\/p>\n\n\n\n

Solution:<\/strong> Expand: 2x + 10 = 3x – 4 10 + 4 = 3x – 2x 14 = x<\/p>\n\n\n\n

Answer: x = 14<\/strong><\/p>\n\n\n\n

\n\n\n\n

Problem 9 \u2014 equation with fractions<\/strong><\/p>\n\n\n\n

Solve: x\/3 + 2 = 7<\/p>\n\n\n\n

Solution:<\/strong> x\/3 = 5 x = 15<\/p>\n\n\n\n

Answer: x = 15<\/strong><\/p>\n\n\n\n

\n\n\n\n

Problem 10 \u2014 equation with variables on both sides<\/strong><\/p>\n\n\n\n

Solve: 5x + 3 = 2x + 18<\/p>\n\n\n\n

Solution:<\/strong> 5x – 2x = 18 – 3 3x = 15 x = 5<\/p>\n\n\n\n

Answer: x = 5<\/strong><\/p>\n\n\n\n

\n\n\n\n

Problem 11 \u2014 Gauss style equation problem<\/strong><\/p>\n\n\n\n

The average of three consecutive integers is 25. What is the largest of the three integers?<\/p>\n\n\n\n

Solution:<\/strong> Let the three consecutive integers be n, n+1, and n+2. Average = (n + n+1 + n+2)\/3 = 25 (3n + 3)\/3 = 25 3n + 3 = 75 3n = 72 n = 24<\/p>\n\n\n\n

The three integers are 24, 25, 26. The largest is 26.<\/p>\n\n\n\n

Answer: 26<\/strong><\/p>\n\n\n\n

Key insight:<\/strong> Consecutive integer problems always work cleanly when expressed as n, n+1, n+2. Do not let the word “consecutive” cause hesitation \u2014 translate it directly into algebra before solving.<\/p>\n\n\n\n

\n\n\n\n

Problem 12 \u2014 Gauss style, two equations<\/strong><\/p>\n\n\n\n

Two numbers have a sum of 40 and a difference of 12. What is the larger number?<\/p>\n\n\n\n

Solution:<\/strong> Let the numbers be a and b where a > b. a + b = 40 a – b = 12<\/p>\n\n\n\n

Add the equations: 2a = 52, so a = 26. Check: b = 40 – 26 = 14. Difference: 26 – 14 = 12 \u2713<\/p>\n\n\n\n

Answer: 26<\/strong><\/p>\n\n\n\n

Key insight:<\/strong> Adding two equations to eliminate a variable is faster than substitution for this problem type. Recognising when to add or subtract equations is a key algebraic skill tested on the Gauss.<\/p>\n\n\n\n

\n\n\n\n

Grade 8 algebra worksheets \u2014 patterns and sequences<\/strong><\/h2>\n\n\n\n

Patterns and sequences appear on every Gauss contest paper. These algebra grade 8 worksheets problems develop the skill of finding rules and generalising sequences algebraically.<\/p>\n\n\n\n

\n\n\n\n

Problem 13 \u2014 finding the rule<\/strong><\/p>\n\n\n\n

A pattern produces the sequence 5, 9, 13, 17, 21…<\/p>\n\n\n\n

What is the rule for the nth term?<\/p>\n\n\n\n

Solution:<\/strong> The sequence increases by 4 each time \u2014 it is an arithmetic sequence with common difference 4.<\/p>\n\n\n\n

First term = 5. nth term = first term + (n-1) x common difference = 5 + (n-1) x 4 = 5 + 4n – 4 = 4n + 1<\/p>\n\n\n\n

Check:<\/strong> n=1: 4(1) + 1 = 5 \u2713. n=3: 4(3) + 1 = 13 \u2713<\/p>\n\n\n\n

Answer: nth term = 4n + 1<\/strong><\/p>\n\n\n\n

\n\n\n\n

Problem 14 \u2014 using the rule to find a specific term<\/strong><\/p>\n\n\n\n

Using the sequence from Problem 13, what is the 50th term?<\/p>\n\n\n\n

Solution:<\/strong> 50th term = 4(50) + 1 = 200 + 1 = 201<\/p>\n\n\n\n

Answer: 201<\/strong><\/p>\n\n\n\n

\n\n\n\n

Problem 15 \u2014 finding position from value<\/strong><\/p>\n\n\n\n

Using the same sequence (nth term = 4n + 1), which term in the sequence equals 97?<\/p>\n\n\n\n

Solution:<\/strong> 4n + 1 = 97 4n = 96 n = 24<\/p>\n\n\n\n

Answer: The 24th term<\/strong><\/p>\n\n\n\n

\n\n\n\n

Problem 16 \u2014 Gauss style pattern problem<\/strong><\/p>\n\n\n\n

A pattern of dots is arranged in a sequence. In the first figure there are 3 dots, in the second there are 7, in the third there are 11. If this pattern continues, how many dots are in the 20th figure?<\/p>\n\n\n\n

Solution:<\/strong> Sequence: 3, 7, 11… Common difference = 4. nth term = 3 + (n-1) x 4 = 3 + 4n – 4 = 4n – 1<\/p>\n\n\n\n

20th term: 4(20) – 1 = 80 – 1 = 79<\/p>\n\n\n\n

Answer: 79<\/strong><\/p>\n\n\n\n

\n\n\n\n

Problem 17 \u2014 Gauss style, finding a pattern rule<\/strong><\/p>\n\n\n\n

A sequence begins 2, 6, 18, 54… What is the 7th term?<\/p>\n\n\n\n

Solution:<\/strong> Each term is multiplied by 3 \u2014 this is a geometric sequence with ratio 3.<\/p>\n\n\n\n

nth term = 2 x 3^(n-1)<\/p>\n\n\n\n

7th term = 2 x 3^6 = 2 x 729 = 1458<\/p>\n\n\n\n

Answer: 1458<\/strong><\/p>\n\n\n\n

Key insight:<\/strong> Not all sequences on the Gauss are arithmetic. When the difference between consecutive terms is not constant, check whether the ratio between consecutive terms is constant instead \u2014 if so it is a geometric sequence.<\/p>\n\n\n\n

\n\n\n\n

Grade 8 algebra worksheets \u2014 word problems<\/strong><\/h2>\n\n\n\n

Algebraic word problems require setting up an equation from a written context. This is where many students lose marks on the Gauss contest \u2014 not because they cannot solve the equation once it is set up, but because they struggle to translate the words into algebra.<\/p>\n\n\n\n

A method for every word problem<\/strong><\/h3>\n\n\n\n

Step one \u2014 identify the unknown and assign it a variable. Write “Let x = …” before doing anything else.<\/p>\n\n\n\n

Step two \u2014 write an equation that expresses the relationship described in the problem.<\/p>\n\n\n\n

Step three \u2014 solve the equation.<\/p>\n\n\n\n

Step four \u2014 check the answer makes sense in the original context.<\/p>\n\n\n\n

\n\n\n\n

Problem 18 \u2014 basic word problem<\/strong><\/p>\n\n\n

\n
\"grade<\/figure>\n<\/div>\n\n\n

A number is multiplied by 5 and then 8 is added. The result is 43. What is the number?<\/p>\n\n\n\n

Solution:<\/strong> Let x = the number. 5x + 8 = 43 5x = 35 x = 7<\/p>\n\n\n\n

Check:<\/strong> 5(7) + 8 = 35 + 8 = 43 \u2713<\/p>\n\n\n\n

Answer: 7<\/strong><\/p>\n\n\n\n

\n\n\n\n

Problem 19 \u2014 age problem<\/strong><\/p>\n\n\n\n

Sam is 4 years older than twice his sister’s age. The sum of their ages is 25. How old is Sam?<\/p>\n\n\n\n

Solution:<\/strong> Let sister’s age = x. Sam’s age = 2x + 4.<\/p>\n\n\n\n

x + (2x + 4) = 25 3x + 4 = 25 3x = 21 x = 7<\/p>\n\n\n\n

Sam’s age = 2(7) + 4 = 18.<\/p>\n\n\n\n

Answer: Sam is 18<\/strong><\/p>\n\n\n\n

\n\n\n\n

Problem 20 \u2014 perimeter word problem<\/strong><\/p>\n\n\n\n

A rectangle has a length that is 3 more than twice its width. The perimeter is 48. Find the dimensions.<\/p>\n\n\n\n

Solution:<\/strong> Let width = w. Length = 2w + 3.<\/p>\n\n\n\n

Perimeter = 2(l + w) = 48 l + w = 24 (2w + 3) + w = 24 3w + 3 = 24 3w = 21 w = 7<\/p>\n\n\n\n

Length = 2(7) + 3 = 17.<\/p>\n\n\n\n

Check:<\/strong> Perimeter = 2(17 + 7) = 2(24) = 48 \u2713<\/p>\n\n\n\n

Answer: Width = 7, Length = 17<\/strong><\/p>\n\n\n\n

Try more area and perimeter problems at: Area and Perimeter Worksheets: How to Solve Every AMC 8 Geometry Problem<\/a>.<\/p>\n\n\n\n

\n\n\n\n

Problem 21 \u2014 Gauss style word problem<\/strong><\/p>\n\n\n\n

\"gauss<\/figure>\n\n\n\n

A shop sells pens for $2 each and notebooks for $5 each. A customer buys a total of 12 items and spends $39. How many pens did they buy?<\/p>\n\n\n\n

Solution:<\/strong> Let pens = p. Notebooks = 12 – p.<\/p>\n\n\n\n

2p + 5(12 – p) = 39 <\/p>\n\n\n\n

2p + 60 – 5p = 39 <\/p>\n\n\n\n

-3p = -21 <\/p>\n\n\n\n

p = 7<\/p>\n\n\n\n

Check:<\/strong> 7 pens + 5 notebooks = 12 items. 7(2) + 5(5) = 14 + 25 = 39 \u2713<\/p>\n\n\n\n

Answer: 7 pens<\/strong><\/p>\n\n\n\n

\n\n\n\n

Problem 22 \u2014 Gauss style, harder word problem<\/strong><\/p>\n\n\n\n

Three friends share some money. The second friend receives twice as much as the first. The third receives $10 more than the second. Together they receive $90. How much does the first friend receive?<\/p>\n\n\n\n

Solution:<\/strong> Let first friend’s share = x. Second = 2x. Third = 2x + 10.<\/p>\n\n\n\n

x + 2x + (2x + 10) = 90 <\/p>\n\n\n\n

5x + 10 = 90 <\/p>\n\n\n\n

5x = 80 x = 16<\/p>\n\n\n\n

Check:<\/strong> 16 + 32 + 42 = 90 \u2713<\/p>\n\n\n\n

Answer: $16<\/strong><\/p>\n\n\n\n

\n\n\n\n

Algebra grade 8 worksheets \u2014 Gauss style harder problems<\/strong><\/h2>\n\n\n\n

These problems combine algebra with other mathematical ideas in the way the Gauss Part B and Part C questions do.<\/p>\n\n\n\n

\n\n\n\n

Problem 23<\/strong><\/p>\n\n\n\n

If 2^x = 8, what is the value of 2^(x+3)?<\/p>\n\n\n\n

Solution:<\/strong> 2^x = 8 = 2^3, so x = 3. 2^(x+3) = 2^6 = 64.<\/p>\n\n\n\n

Alternatively: 2^(x+3) = 2^x x 2^3 = 8 x 8 = 64.<\/p>\n\n\n\n

Answer: 64<\/strong><\/p>\n\n\n\n

Key insight:<\/strong> The second approach is faster and does not require finding x explicitly. Recognising that 2^(x+3) = 2^x x 2^3 uses the exponent law a^(m+n) = a^m x a^n.<\/p>\n\n\n\n

\n\n\n\n

Problem 24<\/strong><\/p>\n\n\n\n

The sum of five consecutive even integers is 120. What is the largest of the five integers?<\/p>\n\n\n\n

Solution:<\/strong> Consecutive even integers differ by 2. Let them be n, n+2, n+4, n+6, n+8.<\/p>\n\n\n\n

n + (n+2) + (n+4) + (n+6) + (n+8) = 120 <\/p>\n\n\n\n

5n + 20 = 120 <\/p>\n\n\n\n

5n = 100 n = 20<\/p>\n\n\n\n

Largest = 20 + 8 = 28.<\/p>\n\n\n\n

Answer: 28<\/strong><\/p>\n\n\n\n

\n\n\n\n

Problem 25<\/strong><\/p>\n\n\n\n

A rectangle has area 60. Its length is 3 more than its width. What is the perimeter?<\/p>\n\n\n\n

Solution:<\/strong> Let width = w. Length = w + 3.<\/p>\n\n\n\n

Perimeter = 2(l + w) = 54 <\/p>\n\n\n\n

l + w = 27 (w + 3) + w = 27 <\/p>\n\n\n\n

2w + 3 = 27 <\/p>\n\n\n\n

2w = 24 <\/p>\n\n\n\n

w = 12<\/p>\n\n\n\n

Length = 12 + 3 = 15. Area = 12 x 15 = 180.<\/p>\n\n\n\n

Check:<\/strong> Perimeter = 2(15 + 12) = 2(27) = 54 \u2713<\/p>\n\n\n\n

Answer: 180<\/strong><\/p>\n\n\n\n

Key insight:<\/strong> This problem gives perimeter and asks for area \u2014 the reverse of what many students expect. Setting up the perimeter equation finds the dimensions, then area follows directly. Gauss word problems often ask for a secondary quantity (area here) that requires a two-step process: find the dimensions first, then calculate what the question actually asks for.<\/p>\n\n\n\n

\n\n\n\n

Problem 26 \u2014 hardest<\/strong><\/p>\n\n\n\n

A function is defined as f(x) = 3x – 2. If f(a) = 13 and f(b) = 7, what is f(a + b)?<\/p>\n\n\n\n

Solution:<\/strong> f(a) = 3a – 2 = 13 \u2192 3a = 15 \u2192 a = 5. <\/p>\n\n\n\n

f(b) = 3b – 2 = 7 \u2192 3b = 9 \u2192 b = 3.<\/p>\n\n\n\n

a + b = 8. <\/p>\n\n\n\n

f(8) = 3(8) – 2 = 22.<\/p>\n\n\n\n

Answer: 22<\/strong><\/p>\n\n\n\n

Key insight:<\/strong> Function notation is introduced in Grade 8 and appears on the Gauss. f(x) simply means “the value of the expression when x is substituted.” Treating f(x) as a machine that takes a number in and gives a number out \u2014 rather than trying to manipulate f algebraically \u2014 is the clearest way to approach these problems.<\/p>\n\n\n

\n
\"\"<\/a><\/figure>\n<\/div>\n\n\n
\n\n\n\n

Grade 8 algebra reference sheet<\/strong><\/h2>\n\n\n\n

Use this as a quick reference alongside the practice problems above.<\/p>\n\n\n\n

Key rules for expressions<\/strong><\/p>\n\n\n\n

Operation<\/th>Rule<\/th>Example<\/th><\/tr><\/thead>
Collecting like terms<\/td>Add coefficients of same variable<\/td>3x + 5x = 8x<\/td><\/tr>
Expanding brackets<\/td>Multiply each term inside<\/td>3(x + 4) = 3x + 12<\/td><\/tr>
Subtracting brackets<\/td>All signs inside change<\/td>-(x – 3) = -x + 3<\/td><\/tr>
Substitution<\/td>Replace variable with number<\/td>x = 5 in 3x + 2 \u2192 17<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n

Solving equations \u2014 steps in order<\/strong><\/p>\n\n\n\n

    \n
  1. Expand all brackets<\/li>\n\n\n\n
  2. Collect variable terms on one side<\/li>\n\n\n\n
  3. Collect number terms on the other side<\/li>\n\n\n\n
  4. Divide both sides by the coefficient<\/li>\n\n\n\n
  5. Check by substituting back into the original equation<\/li>\n<\/ol>\n\n\n\n

    nth term of a sequence<\/strong><\/p>\n\n\n\n

    Sequence type<\/th>nth term formula<\/th><\/tr><\/thead>
    Arithmetic (common difference d, first term a)<\/td>a + (n-1)d<\/td><\/tr>
    Geometric (common ratio r, first term a)<\/td>a x r^(n-1)<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n

    Setting up word problem equations<\/strong><\/p>\n\n\n\n