Step-by-Step Integer Word Problem Solutions for Students
For solving integer word problems students need the right base of knowledge on integers. Proper practice is required for solving integer questions correctly. In this article, we will help students develop the base required for answering word problems on integers for Class 7.
Let’s start with the fact that an arithmetic operation is an elementary branch of mathematics. Arithmetic operations are subtraction, addition, division, and multiplication. These arithmetic operations are used for solving integer problems. There are also other types of numbers that can be solved with the help of arithmetic operations.
It should also be noted that integers are a special group that does not contain any decimal or fractional part. Integers include positive numbers, negative numbers, and zero. Also, arithmetic operations on integers are similar to whole numbers.
It also makes it a little confusing to solve word problems on integers for class 6 and word problems on integers for class 7 pdf because there are both positive and negative numbers. This is also why integers are different from whole numbers.
Integers can also be plotted on a number line. A number line might also be used by students when learning how to solve integers questions. These types of questions are more common when it comes to integers word problems Class 7. If you have never seen a number line, then an image of a number line is attached below.
Rules of Integers
There are several rules that students need for learning how to solve integer word problems. Some of those rules are mentioned below.
The sum of any two positive integers will result in an integer.
The sum of any two negative integers is an integer.
The product of two positive integers will give an integer.
The product of two negative integers will be given an integer.
The sum of any integer and its inverse will be equal to zero.
The product of an integer and its reciprocal will be equal to 1.
Now, let’s look at addition, multiplication, subtraction, and division of signed integer numbers. This will help students to work on story problems with integers answer key.
The Addition of Signed Integer Numbers
As mentioned above, if we add two integers with the same sign, then we have to add the absolute value along with the sign that was provided with the number. For example, (+4) + (+7) = +11 and (-6) + (-4) = -10.
Also, if we add two integers with different signs, then we have to subtract the absolute values and write down the difference. This should be done with the sign of the number that has the largest absolute value. For example, (-4) + (+2) = -2 and (+6) + (-4) = +2.
Subtraction of Signed Integer Numbers
If a student wants to solve integer example problems, then he or she needs to know that while subtracting two integers, we have to change the sign of the second number. The second number should be subtracted and the rules of addition should also be followed. For example, (-7) - (+4) = (-7) + (-4) = -11 and (+8) - (+3) = (+8) + (-3) = +5.
Multiplication and Division of Signed Integer Numbers
When it comes to working on integer word problems with solutions of multiplying and dividing two integer numbers, then the rules are quite straightforward. If both the integers have the same sign, then the final results are positive. If the integers have different signs, then the final result is negative. For example, (+2) x (+3) = +6, (+3) x (-4) = -12, (+6) / (+2) = +3, and (-16) / (+4) = -4.
Properties of Integers
Students should also be familiar with the properties of integers if they want to work on integer word problems grade 6 with solutions. Some of those properties of integers are:
Closure property
Associative property
Commutative property
Distributive property
Additive inverse property
Multiplicative inverse property
Identity property
We also need to look at these properties in detail for solving integer problems in 6th grade. Let’s move on to that discussion.
Closure Property:
According to the closure property of integers, if two integers are added or multiplied together, the final result will be an integer only. This means that if a and b are integers, then:
a + b = integer
a x b = integer
For example, 2 + 5 = 7, which is an integer, and 2 x 5 = 10, which is also an integer.
Commutative Property:
According to this property, if a and b are two integers, then a + b = b + a and a x b = b x a. For example, 3 + 8 = 8 x 3 = 24 and 3 + 8 = 8 + 3 = 11. It should be noted that this property is not followed in the case of subtraction and division.
According to the associative property, if a, b, and c are integers, then:
a + (b + c) = (a + b) + c
a x (b x c) = (a x b) x c
For example, 2 + (3 + 4) = (2 + 3) + 4 = 9 and 2 x (3 x 4) = (2 x 3) x 4 = 24.
This property is only valid when it comes to addition and multiplication.
Distributive Property:
The distributive property states that if a, b, and c are integers, then a x (b + c) = a x b + a x c. For example, if we have to prove that 3 x (5 +1) = 3 x 5 + 3 x 1, then we should be start by finding:
LHS = 3 x (5 + 1) = 3 x 6 = 18
RHS = 3 x 5 + 3 x 1 = 15 + 3 = 18
Since, LHS = RHS
This proves our example.
Additive Inverse Property:
This additive inverse property states that if a is an integer, then a + (-a) = 0. This means that-a is the additive inverse of integer a.
Multiplicative Inverse Property:
The multiplicative inverse property states that if a is an integer, then a x (1 / a) = 1. This means that 1 / a is the multiplicative inverse of integer a.
Identity Property of Integers:
The identity property of integers states that a + 0 = a and a x 1 = a. For example, 4 + 0 = 4 and 4 x 1 = 4.
Types of Integers
Earlier, we mentioned that there are three types of integers. In this section, we will look at these types of integers in more depth. The list of those types of integers is mentioned below.
Zero:
Zero can be characterized as neither a positive nor a negative integer. It can be best defined as a neutral number. This also refers to the fact that zero has no sign (+ or -).
Positive Integers:
As the name indicates, positive integers are those numbers that are positive in their nature. These numbers are represented by a positive or plus (+) sign. The positive integers lie on the right side of the zero on the number line. This also means that all positive integers are greater than zero. For example, 122, 54, and 9087268292.
Negative Integers:
Negative integers, on the other hand, are numbers that are represented by a minus (-) or negative sign. These numbers are present on the left side of the zero on a number line. For example, -182, -8292, and -2927225.
Fun Facts About Integers
The word integer comes from the Latin word “integer” which literally means whole.
You might find it interesting to note that integers are not just simple numbers on paper. Instead, these numbers have real-life applications! Both positive and negative integers are used to symbolize two contradicting situations in the real world. For example, if the temperature is above zero, then positive integers are used for denoting the temperature. But if the temperature is less than zero, then negative integers are used for denoting the temperature.
Integers can also help an individual in comparing and measuring two things like how small or big or few or more things are. These integers help in quantifying things. For example, in games like cricket and soccer, integers are used for keeping a track of scores. Movies and songs can also be rated by using integers!
FAQs on Word Problems on Integers: Practice with Answers
1. What are integers and why are they important for solving real-world problems?
Integers are the set of all whole numbers, including zero, positive numbers, and negative numbers (e.g., -3, 0, 42). They do not include fractions or decimals. Their importance lies in representing concepts that have opposite values or direction. For example, they are used to describe:
Temperature above or below zero.
Elevation as sea level (0), above sea level (+), or below sea level (-).
Financial transactions like a profit (+) or a loss (-).
2. What is the best first step to solving a word problem on integers?
The most crucial first step is to read and understand the problem carefully. Before performing any calculation, you should identify the keywords and the quantities involved. A good approach is to break it down:
Identify keywords: Look for words that indicate positive values (e.g., 'rise', 'deposit', 'gain') and negative values (e.g., 'fall', 'withdraw', 'loss').
Assign integers: Translate these quantities into positive and negative integers.
Determine the operation: Decide if the problem requires addition, subtraction, multiplication, or division based on the context.
3. How can you identify whether to add, subtract, multiply, or divide integers in a word problem?
You can identify the correct operation by looking for specific keywords and understanding the context:
Addition: Indicated by words like 'sum', 'total', 'increase', 'ascend', 'deposit', and 'altogether'.
Subtraction: Indicated by 'difference', 'decrease', 'deduct', 'descend', 'how much more', or 'how much is left'.
Multiplication: Used for repeated addition, indicated by 'product', 'times', or rates like 'loses 3 points per game'.
Division: Used for sharing or grouping, indicated by 'distribute equally', 'average', or 'share per person'.
4. What are some common examples of word problems involving integers?
Common examples of integer word problems often mirror real-life situations. These include:
Temperature Changes: Calculating the final temperature after it rises by a few degrees and then drops significantly.
Elevation and Depth: Finding the distance between a bird flying above sea level and a submarine diving below it.
Financial Scenarios: Calculating the final balance in an account after a series of deposits and withdrawals.
Game Scores: Determining a final score when a player earns points for correct answers and loses points for incorrect ones.
5. How does a number line help in visualising and solving integer word problems?
A number line is an excellent visual tool for understanding integer operations. It makes the abstract concept of negative numbers concrete. When solving a word problem, you can:
Start at the initial value.
Move to the right for adding a positive integer (representing a gain, rise, or increase).
Move to the left for subtracting a positive integer or adding a negative integer (representing a loss, fall, or decrease).
The final position on the number line gives the answer to the problem.
6. What is the difference between an integer and a decimal, and why can't integers have fractional parts?
The fundamental difference is that integers represent whole units, while decimals represent parts of a unit. An integer is a whole number like -5, 0, or 20. A decimal, like 2.5, includes a fractional part. Integers cannot have fractional parts by definition because they are used to count complete, indivisible items. For instance, you can have 3 apples (an integer), but if you have half an apple, you need a fraction (1/2) or a decimal (0.5) to represent it.
7. Why is correctly identifying positive and negative keywords (like 'profit' vs 'loss') crucial in integer word problems?
Correctly identifying these keywords is crucial because they determine the sign (+ or -) of the integer you write in your equation. Assigning the wrong sign is the most common source of error. For example, interpreting a 'loss of ₹500' as +500 instead of -500 will completely reverse the operation and lead to a wildly incorrect answer. The sign is the mathematical translation of the real-world direction or value, so getting it right is the foundation for solving the problem correctly.
8. What is the most common mistake students make when setting up an equation from a word problem with integers?
A very common mistake occurs in subtraction problems involving two integers, especially when one or both are negative. For example, when asked to find the 'difference in temperature between 8°C and -3°C', many students incorrectly calculate 8 - 3 = 5. The correct method is to subtract the second value from the first: 8 - (-3). Here, subtracting a negative becomes addition, so the correct calculation is 8 + 3 = 11. Confusing the order of subtraction ('subtract 5 from -2' should be -2 - 5) is another frequent error.
