# NCERT Solutions for Class 7 Maths Chapter 1 Integers (EX 1.3) Exercise 1.3

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Mathematicians work with a variety of numbers. Even though each number is unique, they have certain things in common. These numbers are divided into many groups based on their qualities in order to be understood. The following are a few of the systems groups:

- Natural numbers
- Whole numbers
- Integers
- Rationals
- Real numbers

The NCERT Solutions For Class 7 Maths Chapter 1 Exercise 1.3 are the solutions to Chapter 1 of Class 7 Mathematics and this chapter deals with Integers.

**What are Integers? **

The collection of whole numbers and negative numbers is known as an Integer in Mathematics. Integers, like whole numbers, do not include the fractional portion. Integers can therefore be defined as numbers that can be positive, negative, or zero, but not as fractions. On Integers, students can carry out all arithmetic operations, including addition, subtraction, multiplication, and division. Integer examples include 1, 2, 5, 8, -9, -12, etc. “Z” stands for an Integer.

What is the meaning of Integers –

The Latin term “Integer,” which implies entire or intact, is where the word “Integer” first appeared. Zero, positive numbers, and negative numbers make up the particular set of numbers known as Integers.

Integer examples include – 1, -12, and 6.

Symbol – The letter “Z” stands for an Integer.

Unknown or unidentified Integers are denoted in mathematical equations by lowercase, italicised letters from the “late middle” of the alphabet. P, q, r, and s are the most prevalent.

Denumerable sets include the set Z. Denumerability is the property that even if a set may include an endless number of members, any element in the set may be represented by a list that implies its identity.

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There are three different types of Integers –

- Zero
- Positive Numbers
- Negative Numbers

Students can understand all these numbers with the help of tools like the NCERT Solutions For Class 7 Maths Chapter 1 Exercise 1.3 and more tools that are available on the Extramarks website.

- Positive Numbers

Positive numbers are ones that have a plus symbol (+) before them. Positive numbers are typically shown as plain numbers without the addition sign (+). Every positive number is greater than 0, as are the numbers to its left as well as the negative numbers. Positive numbers are shown to the right of the origin on a number line ( zero).

Examples are 1, 5, 500, 66, 89, etc.

- Negative Numbers

The numbers that are preceded by a minus sign are considered negative numbers (-). Mentioning the sign of negative Integers is required. On a number line, negative numbers are shown to the left of the origin (zero). Examples are -8, -10, -1000, and -1919.

- Zero

Zero cannot be positive or negative, i.e., it lacks the +ve or -ve sign, making it a neutral number. If both of the Integers have the same sign, adding them results in a result that has the same sign as both of the values.

Rules:

- (+) + (+) = +
- (-) + (-) = –

To add two Integers with different signs: Subtraction will occur if one of the numbers has a different sign, and the result will reflect the sign of the larger number. Students can calculate the difference between the absolute values and assign it the same sign as the absolute value of the Integer with the highest difference.

**NCERT Solutions for Class 7 Maths Chapter 1 Integers (EX 1.3) Exercise 1.3 (include PDF) **

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**Access NCERT Solutions for Class 7 Maths Chapter 1 – Integers **

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**Exercise 1.3 **

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**Chapter 1 Integers **

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**Introduction **

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**Properties of Addition and Subtraction of Integers **

Students will study how to add and subtract numbers with the same sign and various signs when learning about the addition and subtraction of Integers. The number line can be used to add and subtract signed Integers. When performing operations on numbers, specific guidelines must be followed.

In contrast to adding two negative Integers, which produces a sum with a negative sign, adding two positive Integers produces positive Integers. However, adding two different signed Integers will only produce subtraction, with the sign of the result matching that of the larger number. Here are a few examples:

- 2+2 = 4
- 2 + (-2) = 0
- -2 + (-2) = -4
- -2 – (-2) = 0

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Subtraction and Addition –

The two main arithmetic operations in Mathematics are addition and subtraction. In addition to these two operations, Multiplication and Division are the other two fundamental operations we learn in elementary Mathematics.

The “addition” stands for the values that have been added to the original value. For instance, if students add more than two balls to a basket, it will contain a total of four balls. Similar to this, subtracting two balls from a basket containing four balls results in the basket having just two balls, which demonstrates subtraction.

In addition to Integers, rational and irrational numbers can also be added to and subtracted from. Both operations are therefore applicable to all real and complex numbers. Additionally, when conducting algebraic operations, the addition and subtraction of algebraic expressions are done according to the same laws.

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**Multiplication of Integers **

Multiplication is the repeated addition of numbers, according to its definition. However, the guidelines for multiplying Integers differ from those for addition. There are three options available. As follows:

- Multiplication of a positive number by a negative number,
- between two positive numbers,
- and between a positive and a negative number.

When two Integers with comparable sign numbers are multiplied, the result is always positive. So, whether two positive numbers or two negative numbers are added together, the result will always be positive. While an Integer with different signs that is both positive and negative will always result in a negative number.

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**Division of Integers **

The division is the distribution of numbers, whereas the multiplication is the adding up of numbers. The reverse of multiplication is the operation of dividing numbers. Integer division rules, however, are identical to multiplication rules. However, it need not always be the case that the quotient is an Integer.

Rule 1: A pair of positive Integers will always have a positive quotient.

Rule 2: A pair of negative Integers will always have a positive quotient.

Rule 3: A positive Integer and a negative Integer will never have a positive quotient.

Similar to multiplying, students should divide the Integers without the sign, and then add the sign in accordance with the table’s rules. Positive quotient results from the division of two Integers with similar signs, while negative quotient results from the division of two Integers with opposite signs.

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**Secret Tips to Understand the Chapter **

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**NCERT Solutions for Class 7 Maths **

In computer programming and coding, if objects must be on or off, i.e., 1 or 0, Integers are used. Everything that a computer performs can be reduced to a series of ones and zeros; this is referred to as being “binary.” In Mathematics, Integers are crucial. They are useful not just in Mathematics but also in daily life.

In practically every field, Integers are useful for computing the effectiveness of positive or negative numbers.

- Students can determine their position using Integers.
- Calculating how many steps should be taken—more or fewer—to get better results is also helpful.
- The intensity of actual life and intense emotions, however, cannot be quantified.

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**NCERT Solution Class 7 Maths of Chapter 1 All Exercises **

Integers are employed in a variety of everyday applications, including clocks, addresses, thermometers, and money. Maps, altitude measurements, and hockey scores are other applications for Integers.

Whole numbers, often known as positive Integers, are frequently used in everyday life. Roadway speed limits are posted along with highway numbers. To determine the speed to travel on a certain road, one uses Integers. Clock numbers are used to read the time and set alarms. Integers are used in buildings and homes have numerals on them that serve as both address numbers and floor numbers within buildings. Integers are used in maps to show direction and information. Students can understand this concept better when they practice with the help of the NCERT Solutions For Class 7 Maths Chapter 1 Exercise 1.3 and other similar tools made for this particular chapter that are easily available on the Extramarks website.

Negative Integers may be more difficult to employ in everyday situations. The use of positive Integers can be seen, for instance, in bank calculations, hockey game scorekeeping, altitude, and thermometer readings. All of these examples use positive Integers, although they also make use of negative Integers. For temperatures below zero, a thermometer utilises negative Integers. In hockey, when the first team scores, it is a plus; however, when the other team scores on their line, it is a minus. Negative numbers are used to indicate elevation below sea level. In banks and credit unions, debits are represented by negative Integers, and credits are represented by positive Integers.

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**NCERT Solutions for Class 7 **

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**Q.1 **

$\begin{array}{l}\mathrm{Find}\text{}\mathrm{each}\text{}\mathrm{of}\text{}\mathrm{the}\text{}\mathrm{following}\text{}\mathrm{products}:\\ \left(\mathrm{a}\right)\text{}3\times (\u20131)\\ \left(\mathrm{b}\right)\text{}(\u20131)\times 225\\ \left(\mathrm{c}\right)\text{}(\u201321)\times (\u201330)\\ \left(\mathrm{d}\right)\text{}(\u2013316)\times (\u20131)\\ \left(\mathrm{e}\right)\text{}(\u201315)\times 0\times (\u201318)\\ \left(\mathrm{f}\right)\text{}(\u201312)\times (\u201311)\times \left(10\right)\\ \left(\mathrm{g}\right)9\times (\u20133)\times (\u20136)\\ \left(\mathrm{h}\right)(\u201318)\times (\u20135)\times (\u20134)\\ \left(\mathrm{i}\right)(\u20131)\times (\u20132)\times (\u20133)\times 4\\ \left(\mathrm{j}\right)\text{}(\u20133)\times (\u20136)\times (\u20132)\times (\u20131)\end{array}$

**Ans.**

\begin{array}{l}\left(\text{a}\right)\text{3}\times \text{}\left(-\text{1}\right)=\overline{)-3}\\ \left(\text{b}\right)\text{}\left(-\text{1}\right)\text{}\times \text{225}=\text{}\overline{)-225}\\ \left(\text{c}\right)\text{}\left(-\text{21}\right)\text{}\times \text{}\left(-\text{3}0\right)=\overline{)630}\\ \left(\text{d}\right)\text{}\left(-\text{316}\right)\text{}\times \text{}\left(-\text{1}\right)=\overline{)316}\\ \left(\text{e}\right)\text{}\left(-\text{15}\right)\text{}\times \text{}0\text{}\times \text{}\left(-\text{18}\right)=\overline{)0}\\ \left(\text{f}\right)\text{}\left(-\text{12}\right)\text{}\times \text{}\left(-\text{11}\right)\text{}\times \text{}\left(\text{1}0\right)=\overline{)1320}\\ \left(\text{g}\right)\text{9}\times \text{}\left(-\text{3}\right)\text{}\times \text{}\left(-\text{6}\right)=\overline{)162}\\ \left(\text{h}\right)\text{}\left(-\text{18}\right)\text{}\times \text{}\left(-\text{5}\right)\text{}\times \text{}\left(-\text{4}\right)=\overline{)-360}\\ \left(\text{i}\right)\text{}\left(-\text{1}\right)\text{}\times \text{}\left(-\text{2}\right)\text{}\times \text{}\left(-\text{3}\right)\text{}\times \text{4}=\overline{)-24}\\ \left(\text{j}\right)\text{}\left(-\text{3}\right)\text{}\times \text{}\left(-\text{6}\right)\text{}\times \text{}\left(-\text{2}\right)\text{}\times \text{}\left(-\text{1}\right)=\overline{)36}\end{array}

**Q.2 **

$\begin{array}{l}\mathrm{Verify}\text{}\mathrm{the}\text{}\mathrm{following}:\\ \left(\mathrm{a}\right)18\times [7+(\u20133)]=[18\times 7]+[18\times (\u20133)]\\ \left(\mathrm{b}\right)(\u201321)\times [(\u20134)+(\u20136)]=[(\u201321)\times (\u20134)]+[(\u201321)\times (\u20136)]\end{array}$

**Ans.**

\begin{array}{l}\left(\text{a}\right)\\ \text{18}\times \text{}\left[\text{7}+\text{}\left(\u2013\text{3}\right)\right]=18\times \left[4\right]=72\\ \text{and}\\ \left[\text{18}\times \text{7}\right]\text{}+\text{}\left[\text{18}\times \text{}\left(\u2013\text{3}\right)\right]=\left[126\right]+\left[-56\right]=70\\ \text{So},\overline{)\text{18}\times \text{}\left[\text{7}+\text{}\left(\u2013\text{3}\right)\right]\ne \left[\text{18}\times \text{7}\right]\text{}+\text{}\left[\text{18}\times \text{}\left(\u2013\text{3}\right)\right]}\\ \left(\text{b}\right)\\ \left(-\text{21}\right)\text{}\times \text{}\left[\left(-\text{4}\right)\text{}+\text{}\left(-\text{6}\right)\right]=\left(-21\right)\times \left[-10\right]=210\\ \text{and}\\ \left[\left(-\text{21}\right)\text{}\times \text{}\left(-\text{4}\right)\right]\text{}+\text{}\left[\left(-\text{21}\right)\text{}\times \text{}\left(-\text{6}\right)\right]=\left[84\right]+\left[126\right]=210\\ \text{So},\\ \overline{)\left(\u2013\text{21}\right)\text{}\times \text{}\left[\left(\u2013\text{4}\right)\text{}+\text{}\left(\u2013\text{6}\right)\right]=\left[\left(\u2013\text{21}\right)\text{}\times \text{}\left(\u2013\text{4}\right)\right]\text{}+\text{}\left[\left(\u2013\text{21}\right)\text{}\times \text{}\left(\u2013\text{6}\right)\right]}\end{array}

**Q.3 **

$\begin{array}{l}\left(\mathrm{i}\right)\mathrm{For}\mathrm{any}\mathrm{integera},\mathrm{what}\mathrm{is}(\u20131)\times \mathrm{a}\mathrm{equal}\mathrm{to}?\\ \left(\mathrm{ii}\right)\mathrm{Determine}\mathrm{the}\mathrm{integer}\mathrm{whose}\mathrm{product}\mathrm{with}(\u20131)\mathrm{is}\\ \left(\mathrm{a}\right)\u201322\left(\mathrm{b}\right)\mathrm{}37\left(\mathrm{c}\right)0\end{array}$

**Ans.**

\begin{array}{l}\left(\text{i}\right)\text{\hspace{0.17em}}\left(-\text{1}\right)\times \text{a=}-\text{a}\\ \text{(ii) (a)}\overline{)22}\times \left(-1\right)=-22\\ \text{(b)}\overline{)-\text{37}}\times \left(-1\right)=37\\ \text{(c)}\overline{)\text{0}}\times \left(-1\right)=0\end{array}

**Q.4 **

$\begin{array}{l}\mathrm{Starting}\mathrm{from}(\u20131)\times 5,\mathrm{write}\mathrm{various}\mathrm{products}\mathrm{showing}\mathrm{some}\\ \mathrm{pattern}\mathrm{to}\mathrm{show}(\u20131)\times (\u20131)=1\end{array}$

**Ans.**

\begin{array}{l}-1\times 5=-5\\ -1\times 4=-4=-5+1\\ -1\times 3=-3=-4+1\\ -1\times 2=-2=-3+1\\ -1\times 1=-1=-2+1\\ -1\times 0=0=-1+1\\ \text{Thus,}\overline{)-\text{1}\times \left(-1\right)=0+1=1}\end{array}

**Q.5 **

\begin{array}{l}Findtheproduct,usingsuitableproperties:\\ \left(a\right)26\times \left(\u201348\right)+\left(\u201348\right)\times \left(\u201336\right)\left(b\right)8\times 53\times \left(\u2013125\right)\\ \left(c\right)15\times \left(\u201325\right)\times \left(\u20134\right)\times \left(\u201310\right)\text{}\left(d\right)\left(\u201341\right)\times 102\\ \left(e\right)625\times \left(\u201335\right)+\left(\u2013625\right)\times 65\left(f\right)7\times \left(50\u20132\right)\\ \left(g\right)\left(\u201317\right)\times \left(\u201329\right)\text{}\text{}\text{}\text{}\left(h\right)\left(\u201357\right)\times \left(\u201319\right)+57\end{array}

**Ans.**

\begin{array}{l}\left(\text{a}\right)\text{26}\times \text{}\left(-\text{48}\right)\text{}+\text{}\left(-\text{48}\right)\text{}\times \text{}\left(-\text{36}\right)\\ =\left(-48\right)\times 26+\left(-48\right)\times \left(-36\right)\\ =\left(-48\right)\left[26-36\right]\\ =\left(-48\right)\left[-10\right]\\ =\overline{)480}\\ \left(\text{b}\right)\text{8}\times \text{53}\times \text{}\left(-\text{125}\right)\\ =424\times \left(-125\right)\\ =\overline{)53000}\\ \left(\text{c}\right)\text{15}\times \text{}\left(-\text{25}\right)\text{}\times \text{}\left(-\text{4}\right)\text{}\times \text{}\left(-\text{1}0\right)\\ =\left(15\times -25\right)\times \left(-4\times -10\right)\\ =\left(-375\right)\times 40\\ =\overline{)-15000}\\ \left(\text{d}\right)\text{}\left(-\text{41}\right)\text{}\times \text{1}0\text{2}\\ =\overline{)-\text{4182}}\\ \left(\text{e}\right)\text{625}\times \text{}\left(-\text{35}\right)\text{}+\text{}\left(-\text{625}\right)\text{}\times \text{65}\\ =-21875+\left(-40625\right)\\ =\overline{)-62500}\\ \\ \left(\text{f}\right)\text{7}\times \text{}\left(\text{5}0\text{}-\text{2}\right)\\ =7\times \left(48\right)\\ =\overline{)336}\\ \left(\text{g}\right)\text{}\left(-\text{17}\right)\text{}\times \text{}\left(-\text{29}\right)\\ =\overline{)493}\\ \left(\text{h}\right)\text{}\left(-\text{57}\right)\text{}\times \text{}\left(-\text{19}\right)\text{}+\text{57}\\ =1083+57\\ =\overline{)1140}\end{array}

**Q.6 **

$\begin{array}{l}\left(\text{i}\right)\text{For any integer a, what is}\left(\text{\u20131}\right)\text{\xd7 a equal to?}\\ \left(\text{ii}\right)\text{Determine the integer whose product with}\left(\text{\u20131}\right)\text{is}\\ \left(\text{a}\right)\text{\u201322}\left(\text{b}\right)\text{37}\left(\text{c}\right)\text{0}\end{array}$

**Ans.**

\begin{array}{l}\left(\text{i}\right)\text{\hspace{0.17em}}\left(-\text{1}\right)\times \text{a=}-\text{a}\\ \text{(ii) (a)}\overline{)22}\times \left(-1\right)=-22\\ \text{(b)}\overline{)-\text{37}}\times \left(-1\right)=37\\ \text{(c)}\overline{)\text{0}}\times \left(-1\right)=0\end{array}

**Q.7 **

$\begin{array}{l}\text{Starting from}\left(\text{-1}\right)\text{\xd75, write various products showing some}\\ \text{pattern to show}\left(\text{-1}\right)\text{\xd7}\left(\text{-1}\right)\text{=1}\end{array}$

**Ans.**

\begin{array}{l}-1\times 5=-5\\ -1\times 4=-4=-5+1\\ -1\times 3=-3=-4+1\\ -1\times 2=-2=-3+1\\ -1\times 1=-1=-2+1\\ -1\times 0=0=-1+1\\ \text{Thus ,}\overline{)-\text{1}\times \left(-1\right)=0+1=1}\end{array}

**Q.8 **

$\begin{array}{l}\text{Find the product, using suitable properties:}\\ \text{}\left(\text{a}\right)\text{26 \xd7}\left(\text{\u2013 48}\right)\text{+}\left(\text{\u2013 48}\right)\text{\xd7}\left(\text{\u201336}\right)\text{}\left(\text{b}\right)\text{8 \xd7 53 \xd7}\left(\text{\u2013125}\right)\\ \text{}\left(\text{c}\right)\text{15 \xd7}\left(\text{\u201325}\right)\text{\xd7}\left(\text{\u2013 4}\right)\text{\xd7}\left(\text{\u201310}\right)\text{}\left(\text{d}\right)\text{}\left(\text{\u2013 41}\right)\text{\xd7 102}\\ \text{}\left(\text{e}\right)\text{625 \xd7}\left(\text{\u201335}\right)\text{+}\left(\text{\u2013 625}\right)\text{\xd7 65}\left(\text{f}\right)\text{7 \xd7}\left(\text{50 \u2013 2}\right)\\ \text{}\left(\text{g}\right)\text{}\left(\text{\u201317}\right)\text{\xd7}\left(\text{\u201329}\right)\text{}\left(\text{h}\right)\text{}\left(\text{\u201357}\right)\text{\xd7}\left(\text{\u201319}\right)\text{+ 57}\end{array}$

**Ans.**

\begin{array}{l}\left(\text{a}\right)\text{26}\times \text{}\left(-\text{48}\right)\text{}+\text{}\left(-\text{48}\right)\text{}\times \text{}\left(-\text{36}\right)\\ =\left(-48\right)\times 26+\left(-48\right)\times \left(-36\right)\\ =\left(-48\right)\left[26-36\right]\\ =\left(-48\right)\left[-10\right]\\ =\overline{)480}\\ \left(\text{b}\right)\text{8}\times \text{53}\times \text{}\left(-\text{125}\right)\\ =424\times \left(-125\right)\\ =\overline{)53000}\\ \left(\text{c}\right)\text{15}\times \text{}\left(-\text{25}\right)\text{}\times \text{}\left(-\text{4}\right)\text{}\times \text{}\left(-\text{1}0\right)\\ =\left(15\times -25\right)\times \left(-4\times -10\right)\\ =\left(-375\right)\times 40\\ =\overline{)-15000}\\ \left(\text{d}\right)\text{}\left(-\text{41}\right)\text{}\times \text{1}0\text{2}\\ =\overline{)-\text{4182}}\\ \left(\text{e}\right)\text{625}\times \text{}\left(-\text{35}\right)\text{}+\text{}\left(-\text{625}\right)\text{}\times \text{65}\\ =-21875+\left(-40625\right)\\ =\overline{)-62500}\\ \left(\text{f}\right)\text{7}\times \text{}\left(\text{5}0\text{}-\text{2}\right)\\ =7\times \left(48\right)\\ =\overline{)336}\\ \left(\text{g}\right)\text{}\left(-\text{17}\right)\text{}\times \text{}\left(-\text{29}\right)\\ =\overline{)493}\\ \left(\text{h}\right)\text{}\left(-\text{57}\right)\text{}\times \text{}\left(-\text{19}\right)\text{}+\text{57}\\ =1083+57\\ =\overline{)1140}\end{array}

**Q.9 **

$\begin{array}{l}\text{A certain freezing process requires that room temperature}\\ \text{be lowered from 40\xb0C at the rate of 5\xb0C every hour. What will}\\ \text{be the room temperature 10 hours after the process begins?}\end{array}$

**Ans.**

\begin{array}{l}\text{Given initial temprature}=40\xb0C\\ \text{Change in temprature per hour}=-5\xb0C\\ \text{Change in temprature after 10 hours}=-5\xb0C\times 10=-50\xb0C\\ \text{Final temprature}=40\xb0C-50\xb0C=\overline{)-10\xb0C}\end{array}

## FAQs (Frequently Asked Questions)

### 1. Are Integers a part of Algebra?

Algebra has Integers and whole numbers etc but the Integers are not exclusively of use in Algebra Mathematics. With the use of the NCERT Solutions For Class 7 Maths Chapter 1 Exercise 1.3, students can understand this better and practice better too.

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