{"id":63499,"date":"2026-06-11T18:58:08","date_gmt":"2026-06-11T10:58:08","guid":{"rendered":"https:\/\/blog-admin.thethinkacademy.com\/?p=63499"},"modified":"2026-06-11T18:58:11","modified_gmt":"2026-06-11T10:58:11","slug":"integer-word-problems-explained","status":"publish","type":"post","link":"https:\/\/blog-admin.thethinkacademy.com\/blog\/2026\/06\/11\/integer-word-problems-explained\/","title":{"rendered":"Integer Word Problems: Every Type Explained with Worked Examples (Grade 7\u20139 and Pascal Contest)"},"content":{"rendered":"\n

Integers \u2014 positive numbers, negative numbers, and zero \u2014 appear in real-world contexts that students encounter from Grade 6 onward: temperature, elevation, debt, sea level, profit and loss. Integer word problems are where the rules of integer operations get tested in context, and they are where students who learned the rules procedurally start to make mistakes. This guide covers every type of integer word problem that appears in Ontario Grade 7\u20139 math and the Pascal Contest<\/a>, with clear methods, fully worked examples, and practice problems with solutions.<\/p>\n\n\n\n

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Integer word problems: What are integers?<\/h2>\n\n\n\n

Integers are the set of whole numbers and their negatives, including zero:<\/p>\n\n\n\n

… \u22124, \u22123, \u22122, \u22121, 0, 1, 2, 3, 4 …<\/strong><\/p>\n\n\n\n

They do not include fractions or decimals. In the Ontario curriculum, integers are introduced formally in Grade 6 (ordering and comparing), extended to addition and subtraction in Grade 7, and extended to multiplication and division in Grade 8.<\/p>\n\n\n\n

Integer word problems require students to:<\/p>\n\n\n\n

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  1. Identify whether a quantity is positive or negative from the context<\/li>\n\n\n\n
  2. Choose the correct operation<\/li>\n\n\n\n
  3. Apply the integer operation rules correctly<\/li>\n\n\n\n
  4. Interpret the result in context<\/li>\n<\/ol>\n\n\n\n

    Steps 1 and 4 are where most marks are lost in word problems \u2014 not in the arithmetic itself.<\/p>\n\n\n\n

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    Integer operation rules: a quick reference<\/h2>\n\n\n\n

    Before working through problem types, these are the rules students need to know fluently.<\/p>\n\n\n\n

    Addition and subtraction<\/h3>\n\n\n\n
    Situation<\/th>Rule<\/th>Example<\/th><\/tr><\/thead>
    Adding two positives<\/td>Result is positive<\/td>5 + 3 = 8<\/td><\/tr>
    Adding two negatives<\/td>Result is negative<\/td>\u22125 + (\u22123) = \u22128<\/td><\/tr>
    Adding a positive and a negative<\/td>Subtract the smaller absolute value from the larger; sign follows the larger<\/td>\u22128 + 3 = \u22125<\/td><\/tr>
    Subtracting an integer<\/td>Add the opposite<\/td>7 \u2212 (\u22123) = 7 + 3 = 10<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n

    Multiplication and division<\/h3>\n\n\n\n
    Situation<\/th>Rule<\/th>Example<\/th><\/tr><\/thead>
    Same signs<\/td>Result is positive<\/td>(\u22124) \u00d7 (\u22123) = 12<\/td><\/tr>
    Different signs<\/td>Result is negative<\/td>(\u22124) \u00d7 3 = \u221212<\/td><\/tr>
    Division follows the same sign rules<\/td><\/td>(\u221212) \u00f7 (\u22124) = 3<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n
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    The four types of integer word problems<\/h2>\n\n\n\n
    Type<\/th>Context<\/th>Key language<\/th><\/tr><\/thead>
    Temperature and change<\/td>Rising and falling temperatures<\/td>‘dropped’, ‘rose’, ‘below zero’, ‘above zero’<\/td><\/tr>
    Elevation and depth<\/td>Above and below sea level<\/td>‘below sea level’, ‘altitude’, ‘descended’, ‘climbed’<\/td><\/tr>
    Financial<\/td>Profit, loss, debt, deposits, withdrawals<\/td>‘earned’, ‘spent’, ‘debt’, ‘profit’, ‘loss’, ‘withdrew’<\/td><\/tr>
    Multi-step and mixed<\/td>Combining multiple integer operations<\/td>‘overall change’, ‘net result’, ‘total after’<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n
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    Integer word problems – Type 1: Temperature problems<\/h2>\n\n\n\n

    Temperature is the most natural context for integers. Temperatures above zero are positive; temperatures below zero are negative.<\/p>\n\n\n\n

    Worked example 1: Temperature drop<\/h3>\n\n\n\n

    The temperature in Winnipeg was \u22128\u00b0C in the morning. By evening it had dropped a further 11\u00b0C. What was the evening temperature?<\/p>\n\n\n\n

    Setup:<\/strong> Start at \u22128. Drop 11 more degrees means subtract 11. \u22128 \u2212 11 = \u22128 + (\u221211) = \u221219\u00b0C<\/strong><\/p>\n\n\n\n

    Worked example 2: Temperature rise<\/h3>\n\n\n\n

    The overnight low in Ottawa was \u221214\u00b0C. By midday the temperature had risen 23\u00b0C. What was the midday temperature?<\/p>\n\n\n\n

    Setup:<\/strong> Start at \u221214. Rise of 23 means add 23. \u221214 + 23 = 9\u00b0C<\/strong><\/p>\n\n\n\n

    Worked example 3: Temperature difference<\/h3>\n\n\n\n

    Toronto recorded a high of 6\u00b0C and a low of \u22129\u00b0C on the same day. What was the difference between the high and low temperatures?<\/p>\n\n\n\n

    Setup:<\/strong> Difference = high \u2212 low = 6 \u2212 (\u22129) = 6 + 9 = 15\u00b0C<\/strong><\/p>\n\n\n\n

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    Type 2: Elevation and depth problems<\/h2>\n\n\n\n

    Elevation above sea level is positive; depth below sea level is negative.<\/p>\n\n\n\n

    Worked example 4: Descent<\/h3>\n\n\n\n

    A submarine is at an elevation of \u221245 m (45 m below sea level). It descends a further 78 m. What is its new elevation?<\/p>\n\n\n\n

    Setup:<\/strong> \u221245 \u2212 78 = \u221245 + (\u221278) = \u2212123 m<\/strong><\/p>\n\n\n\n

    Worked example 5: Change in elevation<\/h3>\n\n\n\n

    A hiker starts at an elevation of 340 m above sea level and descends to a point 80 m below sea level. What is the total change in elevation?<\/p>\n\n\n\n

    Setup:<\/strong> Change = final \u2212 initial = \u221280 \u2212 340 = \u2212420 m<\/strong><\/p>\n\n\n\n

    The total change is \u2212420 m, meaning the hiker descended 420 m overall.<\/p>\n\n\n\n

    Worked example 6: Multi-step elevation<\/h3>\n\n\n\n

    A drone starts at ground level (0 m), rises 85 m, then descends 120 m. What is its final elevation?<\/p>\n\n\n\n

    Setup:<\/strong> 0 + 85 \u2212 120 = 85 \u2212 120 = \u221235 m<\/strong><\/p>\n\n\n\n

    The drone is 35 m below ground level \u2014 this would typically indicate it has landed somewhere lower than its starting point, or is underground.<\/p>\n\n\n\n

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    Type 3: Financial problems<\/h2>\n\n\n\n

    Positive integers represent money received or gained; negative integers represent money spent or lost.<\/p>\n\n\n\n

    Worked example 7: Bank account<\/h3>\n\n\n\n

    A bank account has a balance of $215. Three withdrawals of $80, $65, and $120 are made. What is the new balance?<\/p>\n\n\n\n

    Setup:<\/strong> 215 \u2212 80 \u2212 65 \u2212 120 = 215 \u2212 265 = \u2212$50<\/strong><\/p>\n\n\n\n

    The account is overdrawn by $50.<\/p>\n\n\n\n

    Worked example 8: Profit and loss<\/h3>\n\n\n\n

    A small business made a profit of $1,200 in January, a loss of $450 in February, and a profit of $780 in March. What was the overall result for the three months?<\/p>\n\n\n\n

    Setup:<\/strong> 1200 + (\u2212450) + 780 = 1200 \u2212 450 + 780 = $1,530 profit<\/strong><\/p>\n\n\n\n

    Worked example 9: Multiplication in financial context<\/h3>\n\n\n\n

    A company loses $340 per day for 6 days. What is the total change in the company’s finances?<\/p>\n\n\n\n

    Setup:<\/strong> (\u2212340) \u00d7 6 = \u2212$2,040<\/strong><\/p>\n\n\n\n

    The company’s finances changed by \u2212$2,040 (a loss of $2,040).<\/p>\n\n\n\n

    Worked example 10: Division in financial context<\/h3>\n\n\n\n

    A business incurred a total loss of $2,400 over 8 months. If the loss was equal each month, what was the monthly loss?<\/p>\n\n\n\n

    Setup:<\/strong> \u22122400 \u00f7 8 = \u2212$300 per month<\/strong><\/p>\n\n\n\n

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    Type 4: Multi-step and mixed problems<\/h2>\n\n\n\n

    These combine multiple operations and often require students to work through several steps before reaching the final answer.<\/p>\n\n\n\n

    Worked example 11: Average temperature<\/h3>\n\n\n\n

    The temperatures recorded over five days were: \u22126, 3, \u22122, 8, \u22123. What was the average daily temperature?<\/p>\n\n\n\n

    Setup:<\/strong> Sum = \u22126 + 3 + (\u22122) + 8 + (\u22123) = 0 Average = 0 \u00f7 5 = 0\u00b0C<\/strong><\/p>\n\n\n\n

    Worked example 12: Net position<\/h3>\n\n\n\n

    A submarine starts at \u221230 m, rises 45 m, descends 60 m, then rises 25 m. What is its final position?<\/p>\n\n\n\n

    Setup:<\/strong> \u221230 + 45 \u2212 60 + 25 = \u221230 + 45 = 15 = 15 \u2212 60 = \u221245 = \u221245 + 25 = \u221220 m<\/strong><\/p>\n\n\n\n

    Worked example 13: Pascal-style problem<\/h3>\n\n\n\n

    The temperature at the top of a mountain is \u221212\u00b0C. The temperature decreases by 3\u00b0C for every 200 m of altitude gained. If the summit is 800 m higher than the current position, what is the temperature at the summit?<\/p>\n\n\n\n

    Setup:<\/strong> Altitude gain = 800 m. Number of 200 m steps = 800 \u00f7 200 = 4. Temperature change = 4 \u00d7 (\u22123) = \u221212\u00b0C Summit temperature = \u221212 + (\u221212) = \u221224\u00b0C<\/strong><\/p>\n\n\n\n

    This type of multi-step integer problem \u2014 where multiplication and addition of integers are combined in a real-world context \u2014 is typical of Pascal Contest Part B problems.<\/p>\n\n\n\n

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    Integer word problems in the Ontario curriculum<\/h2>\n\n\n\n

    Integer word problems appear across several grades in Ontario:<\/p>\n\n\n\n

    Grade<\/th>Integer content<\/th><\/tr><\/thead>
    Grade 6<\/td>Ordering and comparing integers; introduction to negative numbers in context<\/td><\/tr>
    Grade 7<\/td>Adding and subtracting integers<\/a>; solving contextual problems with + and \u2212<\/td><\/tr>
    Grade 8<\/td>Multiplying and dividing integers; multi-step integer problems; order of operations with integers<\/td><\/tr>
    Grade 9 (MTH1<\/a>W<\/a>)<\/a><\/td>Integer operations in algebraic contexts; rational numbers extending integers<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n

    The most common contexts in Ontario curriculum word problems are temperature, elevation, and financial scenarios. EQAO Grade 6<\/a> includes simple integer comparison and ordering; by Grade 9<\/a>, integer reasoning is embedded across multiple strands.<\/p>\n\n\n\n

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    Integer word problems and the Pascal Contest<\/h2>\n\n\n\n

    The Pascal Contest is written by Grade 9 and 10 students and tests mathematical reasoning across the curriculum. Integer word problems appear in two main forms:<\/p>\n\n\n\n

    Direct application:<\/strong> Part A problems (worth 5 marks) sometimes include straightforward integer word problems \u2014 temperature changes, net financial positions, or elevation problems \u2014 where the key skill is correct operation selection and sign handling.<\/p>\n\n\n\n

    Embedded integer reasoning:<\/strong> Part B and Part C problems often require integer arithmetic as a step within a larger problem. A student who makes a sign error on an intermediate step will get the final answer wrong even if the rest of their reasoning is correct. Fluency with integer operations prevents avoidable errors on harder problems.<\/p>\n\n\n\n

    The most common integer mistakes in Pascal problems:<\/p>\n\n\n\n