Integer Word Problems

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Tackling word problems with positive and negative numbers can feel tricky, but it's like solving a real-world puzzle! This guide will show you how to decode the clues, set up the right equation, and confidently find the correct answer every time.

What Are Integer Word Problems?

Integer word problems are real-life scenarios that use positive numbers, negative numbers, and zero to ask a question. Integers are the set of all whole numbers and their opposites. This means they include numbers like \{-3, -2, -1, 0, 1, 2, 3\} and so on, extending infinitely in both positive and negative directions. They do not include fractions or decimals.

Think about a number line. Zero is in the middle. The numbers to the right are positive integers ( $1, 2, 3, . . .$ ), and the numbers to the left are negative integers ( $. . . - 3, - 2, - 1$ ). Integer word problems take these numbers off the line and put them into situations you might actually encounter. For example:

Temperature: A thermometer showing degrees above or below zero. A temperature of $10$ degrees below zero is written as $- 10^{\circ} F$ .
Elevation: The height of land relative to sea level. A diver who is $50$ feet below the surface is at an elevation of $- 50$ feet.
Money: Bank accounts can have a positive balance (money you have) or a negative balance (money you owe), which is called debt.

Solving these problems is all about translating the words into a mathematical equation using integers.

How Do You Find Clues in Word Problems?

The key to solving integer word problems is to identify keywords that tell you which mathematical operation to use. Words can signal whether you need to add, subtract, multiply, or divide. They also give you clues about whether a number is positive or negative. Here is a table to help you become a math detective!

Operation	Keywords That Often Mean...
Addition (+)	increase, rise, gain, deposit, ascend, earn, more than, total, sum, combine
Subtraction (-)	decrease, fall, drop, loss, withdraw, spend, descend, less than, difference, take away
Multiplication (×)	per, each, every, rate of, times, product of, twice, triple
Division (÷)	split equally, share, distribute, average, quotient of, out of, ratio

Remember that the context is everything. A 'loss' of $5$ yards in football means you should use the integer $- 5$ . A 'deposit' of $$ 20$ means you should use $+ 20$ . Paying close attention to these clues is the first step toward finding the right answer.

What Is a Foolproof Method for Solving Integer Word Problems?

When you face a word problem, it can look like a big wall of text. The best way to tackle it is with a clear plan. By following these steps, you can break any problem down into manageable pieces.

Read and Understand: Read the entire problem carefully, maybe even two or three times. Don't rush! Your goal is to understand the story. What is happening? What information are you given? What is the question asking you to find?
Identify the Integers: Find all the numbers in the problem. Decide if each number is positive or negative based on the keywords and the context. For example, if a submarine descends $100$ feet, the integer is $- 100$ .
Find the Operation: Use the keyword table and the story to decide if you need to add, subtract,multiply, or divide. Sometimes a problem might require more than one operation.
Write the Equation: Translate the words and numbers into a mathematical sentence. For example, if you start with $$ 15$ and spend $$ 5$ , your equation would be $15 - 5$ or $15 + (- 5)$ .
Solve the Equation: Now it's time to do the math. Use the rules for operating with integers to find the value of your unknown. Be careful with your signs!
Write Your Answer with Units: The answer to a word problem isn't just a number. It needs context. Make sure you answer the question that was asked and include the correct units (like dollars, feet, degrees, etc.). Does your answer make sense in the context of the story?

Following these steps will help you stay organized and avoid getting lost in the details.

Can We See Some Addition and Subtraction Examples?

Let's put the step-by-step method into action. The rules for adding and subtracting integers are your most important tool here.

Integer Addition Rules:
Same Signs: Add the numbers and keep the sign. (e.g.,

- 3 + (- 4) = - 7

)
Different Signs: Subtract the smaller absolute value from the larger, and keep the sign of the number with the larger absolute value. (e.g.,

- 8 + 5 = - 3

)

Integer Subtraction Rule:
To subtract an integer, add its opposite. (Keep-Change-Change)
Example:

5 - (- 2) = 5 + (+ 2) = 7

Example 1

At sunrise, the temperature in Anchorage, Alaska was $- 8^{\circ} F$ . By noon, the temperature had risen by $15^{\circ} F$ . By sunset, it had dropped $6^{\circ} F$ from the noon temperature. What was the temperature at sunset?

Step 1 & 2: We have a starting temperature of $- 8$ . A rise of $15$ is $+ 15$ . A drop of $6$ is $- 6$ .

Step 3 & 4: This is a two-part problem. First, we find the noon temperature, then the sunset temperature. The equation is: $(- 8) + 15 - 6$ .

Step 5: Let's solve it step-by-step. First, the morning to noon change: $- 8 + 15 = 7$ The signs are different, so we subtract $15 - 8 = 7$ , and keep the sign of the larger absolute value ( $15$ ), which is positive. So at noon, it was $7^{\circ} F$ .
Next, the noon to sunset change: $7 - 6 = 1$ .

Step 6: The temperature at sunset was $1^{\circ} F$ .

Example 2

Maria has $$ 50$ in her checking account. She uses her debit card to buy a video game that costs $$ 65$ . What is the new balance in her account?

Step 1 & 2: Maria starts with $$ 50$ , which is $+ 50$ . Buying something for $$ 65$ means she is spending, so we can represent this as $- 65$ .

Step 3 & 4: We need to find the new balance, so we start with her initial amount and subtract the cost. The equation is: $50 - 65$ .

Step 5: We are subtracting a larger number from a smaller one. $50 - 65 = - 15$ . We can also think of this as $50 + (- 65)$ . The signs are different, so we subtract their absolute values: $65 - 50 = 15$ . We keep the sign of the number with the larger absolute value, which is $- 65$ . So the result is $- 15$ .

Step 6: The new balance in Maria's account is $- $ 15$ . This means she has a debt of $$ 15$ to the bank.

What About Multiplication and Division Problems?

Multiplication and division problems often involve a rate of change over time or splitting a quantity into equal groups. The rules for signs are simpler to remember here.

Integer Multiplication & Division Rules:
Same Signs (Positive × Positive or Negative × Negative) = Positive Answer
Different Signs (Positive × Negative or Negative × Positive) = Negative Answer

Example 3

A submarine starts at the surface of the ocean. It begins to descend at a rate of $20$ meters per minute. What is its depth after $7$ minutes?

Step 1 & 2: Descending at a rate of $20$ meters per minute can be written as the integer $- 20$ . The time is $7$ minutes.

Step 3 & 4: We have a constant rate of change over time, which signals multiplication. The equation is: $- 20 \times 7$ .

Step 5: We are multiplying a negative number by a positive number, so the result will be negative. $- 20 \times 7 = - 140$

Step 6: The submarine's depth after $7$ minutes is $- 140$ meters, or $140$ meters below the surface.

Example 4

Four friends go out for lunch and their total bill comes to $$ 92$ . If they decide to split the cost equally, what is the change in each person's amount of money?

Step 1 & 2: The total cost is $$ 92$ . Since this is money being spent, we can represent it as $- 92$ . There are $4$ friends.

Step 3 & 4: The phrase "split the cost equally" tells us to use division. We want to divide the total cost by the number of people. The equation is: $- 92 \div 4$ .

Step 5: We are dividing a negative number by a positive number, so the result will be negative. $- 92 \div 4 = - 23$

Step 6: The change in each person's amount of money is $- $ 23$ . This means each friend spent $$ 23$ .

What Are Common Mistakes to Avoid?

Working with integers can be tricky, and a small mistake can lead to a wrong answer. Here are some common pitfalls to watch out for:

Sign Confusion: The most common error is mixing up the rules for signs. Forgetting that subtracting a negative is the same as adding a positive is a classic mistake. For example, $8 - (- 3)$ is $11$ , not $5$ .
Misinterpreting Keywords: A student might see the word "total" and automatically add, even when the context requires something else. Always read the whole problem to understand the situation before choosing an operation.
Ignoring Zero: Remember that zero is an integer! It represents a neutral point, like sea level or having no money and no debt.
Calculation Errors: Double-check your basic arithmetic. It's easy to make a small error like $7 \times 8 = 54$ when you're focused on the integer rules.
Forgetting the Units: Your final answer should always refer back to the story. An answer of $- 50$ is incomplete. Is it $- 50$ dollars, $- 50$ feet, or $- 50$ degrees? Always label your answer.

By being aware of these common mistakes, you can learn to check your work more effectively and build confidence in your problem-solving skills.

Quick Summary: Your Integer Cheat Sheet

Feeling overwhelmed? Keep this quick summary handy as a reference. These are the most important ideas for conquering integer word problems.

The 6-Step Plan:

Read to understand the story.
Identify the positive and negative integers.
Find the operation (add, subtract, multiply, divide).
Write the mathematical equation.
Solve the equation using integer rules.
Answer the question with proper units.

The Integer Operation Rules:

Addition: Same signs, add and keep the sign. Different signs, subtract and keep the sign of the 'bigger' number (larger absolute value).
Subtraction: Keep-Change-Change. To subtract, add the opposite. $a - b = a + (- b)$ .
Multiplication/Division: Same signs = POSITIVE answer. Different signs = NEGATIVE answer.

Practice is the best way to become an expert. The more problems you solve, the more these rules and steps will become second nature!

Frequently Asked Questions

What's the difference between a whole number and an integer?

Whole numbers are the set of positive numbers and zero, like $0, 1, 2, 3, . . .$ . Integers include all the whole numbers plus their negative opposites, like $. . ., - 3, - 2, - 1, 0, 1, 2, 3, . . .$ .

Why is zero an integer?

Zero is an integer because it is a whole number that is neither positive nor negative. It represents a starting point or a balance, like sea level or an account with no money and no debt.

How do I know if the final answer should be positive or negative?

The sign of your answer depends on the operation and the numbers involved. For addition and subtraction, it's about the relationship between the numbers. For multiplication and division, if the two numbers have the same sign, the answer is positive; if they have different signs, the answer is negative.

What's the trick for subtracting a negative number?

The best trick is called 'Keep-Change-Change.' You keep the first number the same, change the subtraction sign to an addition sign, and change the sign of the second number. So, $5 - (- 4)$ becomes $5 + (+ 4)$ , which equals $9$ .

Do the rules for multiplying integers also apply to division?

Yes, they do! The sign rules for multiplication and division are identical. If the signs of the two numbers are the same, the result is positive. If the signs are different, the result is negative.

What if a word problem has more than one step?

For multi-step problems, break it down into smaller parts. Solve the first part of the problem, then use that answer to solve the second part. Following the order of operations (PEMDAS/BODMAS) is very important in these cases.

Are there any keywords that are always for a specific operation?

While keywords are great clues, context is always the most important thing. Words like 'sum' or 'product' are very specific to addition and multiplication. However, a word like 'left' could mean subtraction, but you need to read the whole sentence to be sure.