Table of Contents

Grade :8

Subject : Maths

Topic: Numbers and operations

Time: Two week and three days

Objectives

The students will be able to

· Define numbers and describe various uses of numbers

· Classify numbers and identify various types of numbers (natural numbers, integers, rational numbers etc.)

· Describe zero and negative numbers; and their significance

· Solve problems related to negative numbers

· Represent numbers on a number line and show the basic operations (addition, subtraction and multiplication) on a number line

· Define rational number. Distinguish between rational and irrational numbers

· Describe terminating, recurring, non-terminating and nonrecurring decimals

· Identify decimals/fraction as a rational or an irrational number

· Discuss the properties of addition and multiplication of rational numbers

· Write rational numbers in ascending and descending order

· Apply arithmetic operations on rational numbers (fractions)

· Define real numbers as the union of set of rational and irrational number

· Solve word problems on rational numbers

Introduction

Let us recall the numbers that you learnt in your previous classes. An overview of the number system is as follow.

Natural Number

A natural number is a number that occurs commonly and obviously in nature. As such, it is a whole, non-negative number. Natural numbers are also called "counting numbers" because they are used for counting. For example, if you are timing something in seconds, you would use natural numbers (usually starting with 1). When written, natural numbers do not have a decimal point (since they are integers), but large natural numbers may include commas, e.g. 1,000 and 234,567,890. Natural numbers will never include a minus symbol (-) because they cannot be negative. The set of natural numbers, denoted N and mathematically it can be written as N = (1, 2, 3, 4, ...}. Natural numbers are represented on number line by

Whole Number

Whole numbers are almost identical to natural numbers except they include 0. As the name implies, a whole number is not a fraction. It also cannot be negative. The set of Whole numbers, denoted W and mathematically it can be written as W = (1, 2, 3, 4, ...}. They can be represented on number line by

Zero

The idea of zero, though natural to us now, was not natural to early humans ... if there is nothing to count, how can we count it?

Example: we can count dogs, but we can't count an empty space:


Two Dogs		Zero Dogs? Zero Cats?

An empty patch of grass is just an empty patch of grass!

Placeholder

But about 3,000 years ago people needed to tell the difference between numbers like 4 and 40. Without the zero they look the same!

So they used a "placeholder", a space or special symbol, to show "there are no digits here"

The idea of zero had begun, but it wasn't for another thousand years or so that people started thinking of it as an actual number.

But now we can think

"I had 3 oranges, then I ate the 3 oranges, now I have zero oranges...!"

Integers

An integer (pronounced IN-tuh-jer) is a whole number (not a fractional number) that can be positive, negative, or zero. Integers are like whole numbers, but they also include negative numbers.

So, integers can be negative {−1, −2,−3, −4, ... }, positive {1, 2, 3, 4, ... }, or zero {0}

We can put that all together like this:

Integers are denoted by Z . The notation Z came from the first letter of the German word Zahl, which means number. The German terminology for integer numbers specifically is ganze Zahlen, which literally means whole numbers. Z = { ..., −4, −3, −2, −1, 0, 1, 2, 3, 4, ... }

(But numbers like ½, 1.1 and 3.5 are not integers)

Number line

Writing numbers down on a Number Line makes it easy to tell which numbers are greater or lesser

A number on the left is less than a number on the right.

Examples:

5 is less than 8
−1 is less than 1
−8 is less than −5

A number on the right is greater than a number on the left.

Examples:

8 is greater than 5
1 is greater than −1
−5 is greater than −8

Rational Numbers

Any number that can be written as a fraction is called a Rational Number.

So, if "p" and "q" are integers (remember we talked about integers), then p/q is a rational number. Example: If p is 3 and q is 2, then: p/q = 3/2 = 1.5 is a rational number

The only time this doesn't work is when q is zero, because dividing by zero is undefined.

Rational Numbers: {p/q : p and q are integers, q is not zero} So half (½) is a rational number.

And 2 is a rational number also, because we could write it as 2/1

So, Rational Numbers include:

all the integers
and all fractions.

Even a number like 13.3168980325 is a Rational Number.

13.3168980325 = 133,168,980,325 / 10,000,000,000

That seems to include all possible numbers, right?

A rational number is a number that can be written in the form of a numerator upon a denominator. Here the denominator should not be equal to 0. The numerator and the denominator will be integers. A rational number is of the form

p = numerator, q= denominator, where p and q are integers and q ≠0

Examples: 35, −310, 11−15. Here we can see that all the numerators and denominators are integers and even the denominators should be non-zero.

Positive and Negative Rational Numbers

Any rational number can be called as the positive rational number if both the numerator and denominator have like signs. A rational number which has either the numerator negative or the denominator negative is called the negative rational number.

Identify the Rational Numbers

2/7: Here 2 is an integer, 7 is an integer so yes it is a rational number.
0/0: Here there is 0 in the denominator too. So it is not a rational number.
-9: Here -9 can be written −91. So it is a rational number.
0: 0 is a rational number.

Properties of Rational Number

1. A rational number remains unchanged when a non zero integer m is multiplied to both numerator and denominator. p×mq×m

Suppose we take the number 25 and multiply both numerator and denominator by 3 then, 2×35×3 the result that we get is 615. Now this 615 is the standard form. If we express it in its simplest form we get it as 25.

2. A rational number remains unchanged when a non zero same integer m is divided to both numerator and denominator.

p÷mq÷m

Suppose we take the number 615 and divide both numerator and denominator by 3 then, 6÷315÷3 the result that we get is 25.

Standard Form of Rational Number

Now 2436 is a rational number. But when this number is expressed in its simplest form, it is 23. A rational number is in its standard form if it has no common factors other than 1 between the numerator and denominator and the denominator is positive.

Solved Examples for You

Question 1: What fraction lies exactly halfway between 23 and 34 ?

Answer : The correct option is “E”. Comsider 3 × 4 = 12. So,
23 = 812
34 = 912
Multiplying the numerator and denominatoe by 2
1624 = 1824
The midpint is 1724

Question 2: If we divide a positive integer by another positive integer, what is the resulting number?

Always a natural number
Always an integer
A rational number
An irrational number

Answer : The correct option is “C”. If we divide a positive integer by another positive integer, the resulting number is always a rational number. Though it can be a natural number and an integer only if the denominator is 1.

Question 3: Give a simple definition of rational number?

Answer: A rational number refers to a number that one can write as a fraction. Rational numbers happen to be real numbers. Moreover, these numbers can be positive or negative.

Question 4: How can you identify a rational number?

Answer: A rational number is a number whose writing can take place as a ratio. That means it is possible to write it as a fraction such that both the numerator and the denominator are whole numbers. The number 8 is a rational number because one can write it as the fraction f8/1.

Question 5: Can we say that 3.5 is a rational number?

Answer: Yes, we can say that 3.5 is a rational number. Furthermore, 3.5 has a decimal because of which it is not a whole number. A rational number refers to any value which has equivalence to the ratio two integers. 3.5 is equivalent to the ratio of 7 and 2, thus it is a rational number.

Question 6: Can we say that 0.25 is a rational number?

Answer: Decimal 0.25 is certainly a rational number. This is because it shows the ratio or fraction 25/100 and both 25 and 100 are integers.

Watch this video for more understanding https://www.youtube.com/watch?v=9yvtLN_24G0

Converting Recurring Decimal

A recurring decimal exists when decimal numbers repeat forever. For example, $0. \dot{3}$ means 0.333333... - the decimal never ends.

Dot notation is used with recurring decimals. The dot above the number shows which numbers recur, for example $0.5 \dot{7}$ is equal to 0.5777777... and $0. \dot{2} \dot{7}$ is equal to 0.27272727...

If two dots are used, they show the beginning and end of the recurring group of numbers: $0. \dot{3} 1 \dot{2}$ is equal to 0.312312312...

Example

How is the number 0.57575757... written using dot notation?

In this case, the recurring numbers are the 5 and the 7, so the answer is $0. \dot{5} \dot{7}$ .

Example

Convert $\frac{5}{6}$ to a recurring decimal.

Divide 5 by 6. 5 divided by 6 is 0, remainder 5, so carry the 5 to the tenths column.

50 divided by 6 is 8, remainder 2.

20 divided by 6 is 3 remainder 2.

Because the remainder is 2 again, the digit 3 is going to recur:

$\frac{5}{6}$ = 0.8333…= $0.8 \dot{3}$

Algebra can be used to convert recurring decimals into fractions.

Example

Convert $0. \dot{1}$ to a fraction.

$0. \dot{1}$ has 1 digit recurring.

Firstly, write out $0. \dot{1}$ as a number, using a few iterations (repeats) of the decimal.

0.111111111…

Call this number . Written as an equation $x = 0.1111111 \dotsc$

If this number is multiplied by 10 it will give a different number with the same digit recurring.

So if:

$x = 0.11111111 \dotsc$ then:

$10x = 1.11111111 \dotsc$

Notice that after the decimal points the recurring digits match up. So subtracting these equations gives:

$10x - x = 1.11111 \dotsc - 0.11111 \dotsc$

So:

Dividing both sides by 9 gives:

$x = \frac{1}{9}$

So, $0. \dot{1}$ = $\frac{1}{9}$ .

Worksheet on Repeating Decimals

Top of Form

Bottom of Form

Practice the questions given in the worksheet on repeating decimals or recurring decimals. The questions are based in expressing the decimal form.

1. Identify pure recurring decimals and mixed recurring decimals.

(a) 1.4

(b) 0.752

(c) 5.27

(d) 11.011

(e) 1.25

2. Identify whether the following are terminating or non-terminating:

(a) 3/5

(b) 3/4

(c) 81/25

(d) 1/9

(e) 57/6

(f) 19/4

(g) 29/7

3. Convert the pure recurring decimals into vulgar fractions:

(a) 0.8

(b) 2.4

(c) 0.89

(d) 5.381

(e) 0.45

(f ) 1.23

(g) 0.7

(h) 0.473

4. Express each of the infinite repeating decimal or mixed recurring decimals in the form of a/b also a/b should be positive integers.

(a) 0.8526
(b) 0.4321
(c) 3.012
(d) 1.1234
(e) 0.154
(f) 1.234
(g) 0.389
(h) 0.26
(i) 6.78
5. Find the value of the infinite repeating decimals:

(a) 3.2 + 2.3
(b) 0.9 - 0.4
(c) 2.12 + 1.21
(d) 4.13 - 2.05
Answers for the worksheet on repeating decimals or recurring decimals are given below to check the exact answers of the above decimal form.

Answers:

1. (a) Pure recurring decimals

(b) Mixed recurring decimals

(c) Pure recurring decimals

(d) Mixed recurring decimals

(e) Pure recurring decimals

2. (a) Terminating

(b) Terminating

(c) Terminating

(d) Non-terminating

(e) Terminating

(f) Terminating

(g) Non-terminating

3. (a) 8/9

(b) 22/9

(c) 89/99

(d) 1792/333

(e) 5/11

(f) 122/99

(g) 7/9

(h) 473/999

4. (a) 8441/9900

(b) 713/1650

(c) 2711/900

(d)5561/4950

(e) 17/110

(f) 611/495

(g) 193/495

(h) 4/15

(i) 611/90

5. (a) 5.5

(b) 0.5

(c) 3.3

(d) 2.07

But There Is More

People didn't stop asking the questions ...and here is one that caused a lot of fuss during the time of Pythagoras:

square root 2

If we draw a square (of size "1"), what is the distance across the diagonal?

The answer is the square root of 2, which is 1.4142135623730950...(etc)

But it is not a number like 3, or five-thirds, or anything like that ...... in fact we cannot answer that question using a ratio of two integers square root of 2 ≠ p/q ... and so it is not a rational number (read more here)

Irrational Numbers

So, the square root of 2 (√2) is an irrational number. It is called irrational because it is not rational (can't be made using a simple ratio of integers). It isn't crazy or anything, just not rational. And we know there are many more irrational numbers. Pi (π) is a famous one.

So irrational numbers are useful. We need them to

find the diagonal distance across some squares,
to work out lots of calculations with circles (using π),
and more,

Real Numbers

Real Numbers include:

the rational numbers, and
the irrational numbers

Real Numbers: {x : x is a rational or an irrational number}

https://study.com/academy/practice/quiz-worksheet-classification-of-numbers.html

Worksheets

Practice the questions given in the worksheet on addition and subtraction of rational number. The questions are based on adding and subtracting rational numbers with same denominators and adding and subtracting rational numbers with different denominators.

1. Add the following rational numbers:

(i) -5/7 and 3/7

(ii) -15/4 and 7/4

(iii) -8/11 and -4/11

(iv) 6/13 and -9/13

2. Add the following rational numbers:

(i) 3/4 and -3/5

(ii) -3 and 3/5

(iii) -7/27 and 11/18

(iv) 31/-4 and -5/8

3. Verify the following:

(i) (3/4 + -2/5) + -7/10 = 3/4 + (-2/5 + -7/10)

(ii) (-7/11 + 2/-5) + -13/22 = -7/11 + (2/-5 + -13/22)

(iii) -1 + (-2/3 + -3/4) = (-1 + -2/3) + -3/4

4. Subtract the following rational numbers:

(i) 3/4 from 1/3

(ii) -5/6 from 1/3

(iii) -8/9 from -3/5

(iv) -9/7 from -1

(v) -18/11 from 1

(vi) -13/9 from 0

(vii) -32/13 from -6/5

(viii) -7 from -4/7

5. The sum of two rational numbers is -2. If one of the numbers is -14/5, find the other.

6. The sum of two rational numbers is -1/2. If one of the numbers is 5/6 find the other.

7. What number should be added to -5/8 so as to get -3/2?

8. What number should be added to -1 so as to get 5/7?

9. What number should be subtracted from -2/3 to get -1/6?

10. Is the difference of two rational numbers a rational number?

11. What is the negative of a negative rational number?

Answers for the worksheet on addition and subtraction of rational number are given below to check the exact answers of the above questions on addition and subtraction.

Answers:

(i) -2/7

(ii) -2

(iii) -12/11

(iv) -3/13

(i) 3/20

(ii) -12/5

(iii) 9/54

(iv) -67/6

. (i) -5/12

(ii) 7/6

(iii) 13/45

(iv) 2/7

(v) 29/11

(vi) 13/9

(vii) 82/65

(viii) 45/7

5. 4/5

6. -4/3

7. -7/8

8. 12/7

9. -1/2

10. Yes

11. The number itself.

Practice the questions given in the worksheet on operations on rational expressions. The questions are related to the operations of addition, subtraction, multiplication and division on rational numbers.

1. Simplify the following rational numbers:

(i) (25/8 × 2/5) – (3/5 × -10/9)

(ii) (1/2 × ¼) + (1/2 × 6)

(iii) (-5 × 2/15) – (-6 × 2/9)

(iv) (-9/4 × 5/3) + (13/2 × 5/6)

2. Simplify the rational expressions:

(i) (3/2 × 1/6) + (5/3 × 7/2) – (13/8 × 4/3)

(ii) (1/4 × 2/7) – (5/14 × -2/3) + (3/7 × 9/2)

(iii) (13/9 × -15/2) + (7/3 × 8/5) + (3/5 × ½)

(iv) (3/11 × 5/6) – (9/12 × 4/3) + (5/13 × 6/15)

3. Find the value and express as a rational number in standard form:

(i) 2/5 ÷ 26/15

(ii) 10/3 ÷ (-35/12)

(iii) -6 ÷ (-8/17)

(iv) 40/98 ÷ (-20)

4. By what number should we multiply -15/28, so that the product may be -5/7?

5. By what number should we multiply -8/13, so that the product may be 24?

6. By what number should -3/4 be multiplied in order to produce 2/3?

7. Find (m + n) ÷ (m – n), if

(i) m = 2/3, n = 3/2

(ii) m = 2/5, n = 1/2

(iii) m = 5/4, y = -1/3

8. The cost of 7 2/3 meters of rope is $12 3/4. Find its cost per meter.

9. The cost of 2 1/3 meters of rope is $75 1/4. Find cost of cloth per meter.

10. Divide the sum of -13/5 and 12/7 by the product of -31/7 and -1/2.

11. Divide the sum of 65/12 and 8/3 by their difference.

12. If 24 trousers of equal size can be prepared in 54 meters of cloth, what length of the cloth is required for each trouser?

13. Divide the sum of 13/5 and -12/7 by the product of -31/7 and 1/-2.

14. Divide the sum of 65/12 and 8/3 by their difference.

Answers for the worksheet on operations on rational expressions are given below to check the exact answers of the above questions on rational numbers.

Answer:

. (i) 23/12

(ii) 25/8

(iii) 2/3

(iv)5/3

(i) 47/12

(ii) 47/21

(iii) -34/5

(iv) -177/286

. (i) 3/13

(ii) -8/7

(iii) 51/4

(iv) -1/49

4. 4/3

5. -39

6. -8/9

. (i) -13/5

(ii) -9

(iii) 11/19

8. $1 61/92

9. $32 1/4

10. -2/5

11. 97/33

12. 9/4 meters

13. 2/5

14. 97/33

OPERATIONS WITH RATIONAL NUMBERS WORKSHEET

Operations with Rational Numbers Worksheet :

Worksheet given in this section will be much useful for the students who would like to practice problems rational numbers.

Operations with Rational Numbers Worksheet - Problems

Problem 1 :

Simplify : 2/5 + 3/5

Problem 2 :

Simplify : 7/5 - 3/5

Problem 3 :

Simplify : 1/8 + 1/3

Problem 4 :

Simplify : 5/12 + 1/20

Problem 5 :

Convert the fraction 17/5 into mixed number.

Problem 6 :

Multiply 2/3 and 4/5.

Problem 7 :

Divide 6 by 2/5.

Problem 8 :

Divide 1/5 by 3/7

Problem 9 :

Lily earned $54 mowing lawns in two days. She worked 2.5 hours yesterday and 4.25 hours today. If Naomi was paid the same amount for every hour she works, how much did she earn per hour ?

Problem 10 :

David traveled from A to B in 3 hours at the rate of 50 miles per hour. Then he traveled from B to C in 2 hours at the rate of 60 miles per hour. What is the average speed of David from A to C ?

Problem 11 :

Each part of a multipart question on a test is worth the same number of points. The whole question is worth 37.5 points. Daniel got 1/2 of the parts of a question correct. How many points did Daniel receive ?

Problem 12 :

The bill for a pizza was $14.50. Charles paid for 3/5 of the bill. How much did he pay ?

Operations with Rational Numbers Worksheet - Solutions

Problem 1 :

Simplify : 2/5 + 3/5

Solution :

Here, for both the fractions, we have the same denominator, we have to take only one denominator and add the numerators.

Then, we get

2/5 + 3/5 = (2+3) / 5 = 5/5 = 1

Problem 2 :

Simplify : 7/5 - 3/5

Solution :

Here, for both the fractions, we have the same denominator, we have to take only one denominator and subtract the numerators.

Then, we get

7/5 - 3/5 = (7-3) / 5 = 4/5

Problem 3 :

Simplify : 1/8 + 1/3

Solution :

In the given two fractions, denominators are 8 and 3.

For 8 and 3, there is no common divisor other than 1.

So 8 and 3 are co-prime.

Here we have to apply cross-multiplication method to add the two fractions 1/8 and 1/3 as given below.

So,

1/8 + 1/3 = 11/24

Problem 4 :

Simplify : 5/12 + 1/20

Solution :

In the given two fractions, denominators are 12 and 20.

For 12 and 20, if there is at least one common divisor other than 1, then 12 and 20 are not co-prime.

For 12 & 20, we have the following common divisors other than 1.

2 & 4

So 12 and 20 are not co-prime.

In the next step, we have to find the L.C.M (Least common multiple) of 12 and 20.

12 = 2² x 3

20 = 2² x 5

When we decompose 12 and 20 in to prime numbers, we find 2, 3 and 5 as prime factors for 12 and 20.

To get L.C.M of 12 and 20, we have to take 2, 3 and 5 with maximum powers found above.

So, L.C.M of 12 and 20 = 2² x 3 x 5

= 4 x 3 x 5

= 60

Now we have to make the denominators of both the fractions to be 60 and add the two fractions 5/12 and 1/20 as given below.

So,

5/12 + 1/20 = 7/15

Problem 5 :

Convert the fraction 17/5 into mixed number.

Solution :

The picture given below clearly illustrates, how to convert the fraction 17/5 into mixed number.

So,

17/5 = 3 2/5

Problem 6 :

Multiply 2/3 and 4/5.

Solution :

To multiply a proper or improper fraction by another proper or improper fraction, we have to multiply the numerators and denominators.

That is,

2/3 x 4/5 = 8/15

Problem 7 :

Divide 6 by 2/5.

Solution :

To divide a whole number by any fraction, multiply that whole number by the reciprocal of that fraction.

That is,

6 ÷ 2/5 = 6 x 5/2 = 30/2 = 15

Problem 8 :

Divide 1/5 by 3/7

Solution :

To divide a fraction by another fraction, multiply the first fraction by the reciprocal of the second fraction.

That is,

1/5 ÷ 3/7 = 1/5 x 7/3 = 7/15

Problem 9 :

Lily earned $54 mowing lawns in two days. She worked 2.5 hours yesterday and 4.25 hours today. If Naomi was paid the same amount for every hour she works, how much did she earn per hour ?

Solution :

Analyze Information :

Identify the important information.

• Naomi made $54 mowing lawns.

• Naomi worked 2.5 hours yesterday and 4.25 hours today.

• We are asked to find how much she earned per hour

Formulate a plan :

• The total amount she earned divided by the total hours she worked gives the amount she earns per hour.

• Use the expression 54 ÷ (2.5 + 4.25) to find the amount she earned per hour.

Solve :

Follow the order of operations.

(2.5 + 4.25) = 6.75 ---- > (Add inside parentheses)

54 ÷ 6.75 = 8 ---- > (Divide)

Lily earned $8 per hour mowing lawns.

Problem 10 :

David traveled from A to B in 3 hours at the rate of 50 miles per hour. Then he traveled from B to C in 2 hours at the rate of 60 miles per hour. What is the average speed of David from A to C ?

Solution :

Analyze Information :

Identify the important information.

• David traveled from A to B in 3 hours @ 50 mph.

• David traveled from B to C in 2 hours @ 60 mph.

• We are asked to find the average speed from A to C.

Formulate a plan :

• The total distance covered from A to C divided by total time taken gives the average speed from A to C.

• Use the expression (3 x 50) + (2 x 60) to find the total distance from A to C.

That is, 270 miles

• Use the expression (3 + 2) to find the total time taken from A to C.

That is, 5 hours