Common Core Math

Dividing Decimals is much similar to dividing Whole Numbers, except the way we handle the decimal point. Refer to the Dividing Decimal by a Whole Number Step by Step, Solved Examples, etc. Get a good hold of the concept and know How to Divide a Decimal by a Whole Number. Learn the entire procedure used to Divide Decimals by a Whole Number and solve related problems with ease.

Also, Read: Multiplying Decimal by a Decimal Number 

Dividing Decimals – Definition

The process of Dividing Decimals is much similar to the normal division. All you need to keep in mind is to place the decimal point correctly in the quotient. To divide a decimal by a whole number, the division is performed in the same way as in whole numbers ignoring the decimal point. Place the decimal point in the quotient in the same position as in the dividend.

How to Divide a Decimal by a Whole Number?

Follow the simple steps provided below to get acquainted with the Division of a Decimal with a Whole Number. They are along the lines

• Write the division in standard form and divide the whole number part of the decimal number with the divisor.
• Here dividend is the decimal number and the divisor is the whole number.
• Place the decimal point in the quotient above the decimal point of the dividend. Get the tenths digit down.
• Divide the dividend with the divisor.
• Try adding Zeros in the dividend till you get a Zero Remainder.

Solved Examples on Division of a Decimal by Whole Number

1. Solve 112.340 ÷ 5?

Solution:

Decimal Number 112.340 is the dividend and 5 is the whole number. Place the decimal point in the quotient above the decimal point of the dividend 112.340.

Now, we are going to bring down the 3. But, because it follows the decimal point, we have to place a decimal point in the quotient. Later we can bring down the next number.

Therefore, 112.340 ÷ 5 = 22.468

2. Solve 215.8 ÷ 3?

Solution:

Decimal Number 215.8 is the dividend and 3 is the whole number. Place the decimal point in the quotient above the decimal point of the dividend 215.8

Now, we are going to bring down the 8. But, because it follows the decimal point, we have to place a decimal point in the quotient. Later we can bring down the next number.

Therefore, 215.8 ÷ 3 = 71.93

3. Find 142.82 ÷ 4?

Solution:

Decimal Number 142.82 is the dividend and 4 is the whole number. Place the decimal point in the quotient above the decimal point of the dividend 142.82

Now, we are going to bring down the 8. But, because it follows the decimal point, we have to place a decimal point in the quotient. Later we can bring down the next number.

Therefore, 142.82÷ 4 = 35.705

In Multiplication of Decimals, you will learn how to multiply a decimal by decimal. While Multiplying Decimals firstly ignore the decimal points and place the decimal point in the product in a way that decimal places in the product are equal to the sum of decimal places in the given numbers. Refer to the complete article to be well versed with details like Procedure for Multiplying Decimals, Solved Examples on Decimal Multiplication explained step by step.

Also, Read: Multiplying Decimal by a Whole Number

How to Multiply a Decimal by Decimal?

Follow the below-listed guidelines on how or multiply a decimal by decimal. They are along the lines

• Multiply both the numbers as if they are whole numbers and don’t consider the decimal points.
• Place the decimal point after leaving digits equal to the total number of decimal places in both the numbers.
• Remember to count the decimal places from the unit’s place of the product.

Solved Examples on Multiplying Decimal by a Decimal

1. Find the product of 1.3 × 1.3

Solution:

First while performing the multiplication of decimals ignore the decimal points and perform the multiplication as if they are whole numbers

13

x 13

——–––

39

130

(+)

——–––

169

——–––

Count the total number of decimal places i.e. both in the multiplicant and multiplier together. Now, place the decimal point with as many decimal places are in the given numbers.

Since 2 decimal places are there place the decimal point counting from the unit’s place of the product.

Thus the product becomes 1.69

Therefore, the Product of 1.3 by 1.3 gives 1.69

2. Find the product of 3.5 × 0.06?

Solution:

First while performing the multiplication of decimals ignore the decimal points and perform the multiplication as if they are whole numbers

35

x 6

——–––

210

——–––

Count the total number of decimal points both in multiplicand and multiplier together. Place a decimal point as many decimal places are there in the given numbers.

Since there are 3 decimal places all together in given numbers place a decimal point counting from the unit’s place of the product.

Thus, the product becomes 0.210

Therefore, the product of 3.5 by 0.006 gives 0.210

3. Multiply 118.12 by 3.5?

Solution:

Before performing the decimal multiplication multiply as if they are whole numbers and ignore the decimal points.

11812

x     35

——–––––––

59060

35436

(+)

——–––––––

413420

——–––––––

Count the total number of decimal points both in multiplicand and multiplier together. Place a decimal point as many decimal places are there in the given numbers.

Since there are 3 decimal places all together in given numbers place a decimal point counting from the unit’s place of the product.

Thus, the product becomes 413.420

Therefore, the product of118.12 by 3.5 gives 413.420

Do you wish to learn Multiplication of Decimal with a Whole Number? Then this is the right place where you will get complete knowledge on Step by Step Procedure for Multiplication of Decimal with a Whole Number. Check out the Definition, Solved Examples listed here to get a grip on the concept. Learn the approach used here so that it becomes easy for you during your math calculations.

Also, See:

• Adding Decimals
• Subtracting Decimals

How to Multiply a Decimal by a Whole Number?

To Multiply a Decimal with a Whole Number follow the simple procedure listed below. They are along the lines

• Multiply the decimal as you would do with the whole number.
• Count the number of decimal places in the factors.
• Now, mark the decimal point in the result obtained from right to left as per the number of decimal places in the given decimal number.

Worked Out Problems on Multiplication of Decimals with a Whole Number

1. Find the Product

6.36 × 7

Solution:

Firstly, ignore the decimal places and multiply as if it is they are whole numbers.

Count the number of decimal places in the given decimal number and place the decimal point in the result obtained after multiplication. Rewrite the product with 2 decimal places as the decimal 6.36 has 2 decimal places.

Thus, the product of 6.36 × 7 gives 44.52

2. The length and breadth of a rectangle are 16.82 m and 6 m. Find the area of the rectangle?

Solution:

Length of a Rectangle = 16.82 m

Breadth of a Rectangle = 6m

Area of Rectangle = l*b

= 16.82*6

Ignore the decimal places and multiply as if it is they are whole numbers.

=

Count the number of decimal places in the given decimal number and place the decimal point in the result obtained after multiplication. Rewrite the product with 2 decimal places as the decimal 16.82 has 2 decimal places.

Thus, the product of 16.82*6 results in 100.92

3. Find the Product 8.54×3?

Solution:

Firstly, ignore the decimal places and multiply as if it is they are whole numbers.

Count the number of decimal places in the given decimal number and place the decimal point in the result obtained after multiplication. Rewrite the product with 2 decimal places as the decimal 8.54 has 2 decimal places.

Place the decimal point in the product with 2 decimal places i.e. 25.62

4.  The length and breadth of a rectangle are 11.82 m and 3 m. Find the area of the rectangle?

Solution:

Length of the Rectangle = 11.82m

Breadth of Rectangle = 3m

Area of Rectangle = l*b

= 11.82*3

Count the number of decimal places in the given decimal number and place the decimal point in the result obtained after multiplication. Rewrite the product with 2 decimal places as the decimal 11.82 has 2 decimal places.

Place the decimal point in the product with 2 decimal places i.e. 35.46

A multiple is the product of one number with another number. Also, we can define a multiple as the result that is obtained by multiplying a number by an integer. But it is not a function. The multiples of the whole numbers are found by doing the product of the counting numbers and that of whole numbers. For example, multiples of 5 can be obtained when we multiply 5 by 1, 5 by 2, 5 by 3, and so on.

Example 1: Find the multiples of whole number 4?
Firstly, do the multiplication of 4 with other numbers to get multiples of 4.
Multiplication: 4 x 1, 4 x 2, 4 x 3, 4 x 4, 4 x 5, 4 x 6, 4 x 7, 4 x 8
Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32
Solution: The multiples of 4 are 4, 8, 12, 16, 20, 24, 28, 32,……

Example 2: Find the multiples of whole number 6?
Firstly, do the multiplication of 6 with other numbers to get multiples of 6.
Multiplication: 6 x 1, 6 x 2, 6 x 3, 6 x 4, 6 x 5, 6 x 6, 6 x 7, 6 x 8
Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48.
Solution: The multiples of 6 are 6, 12, 18, 24, 30, 36, 42, 48,……

Also, Check: Common Multiples

Properties of Multiples

Here we have given some important Properties of Multiples. Check out the properties and get a grip on them to make your learning easy.
(i) Every number is a multiple of itself.
For example, the first multiple of 5 is 5 because 5 × 1 = 5.
(ii) The multiples of a number are infinite.
We know that numbers are infinite. Therefore, the multiples of a number also infinite. If you take the example of multiples of 2, we begin with 2, 4, 6, 8, 10, 12, 14,…. and so on.
(iii) The multiple of a number is greater than or equal to the number itself.
For example, if we take the multiples of 3: 3, 6, 9, 12, 15, .… and so on. We can see that: The 1st multiple of 3 is equal to 3: 3 × 1 = 3. The 2nd multiple, the 3rd multiple, and the following multiples of 3 are all greater than 3 (6 > 3, 9 > 3, ….)
(iv) 0 is a multiple of every number.

Common Multiples

Multiples that are common to any given two numbers are known as common multiples of those numbers. Check out the example for better understanding.
Consider two numbers– 2 and 3. Multiples of 2 and 3 are –
Multiples of 2 = 2, 4, 6, 8, 10, 12, ….
Multiples of 3 = 3, 6, 9, 12, 15, 18,……….
We observe that 6 and 12 are the first two common multiples of 2 and 3. But what can be the real-life use of common multiples?
Suppose Arun and Anil are cycling on a circular track. They start from the same point but Arun takes 30 seconds to cover a lap while Anil takes 45 seconds to cover the lap. So when will be the first time they meet again at the starting point?
This can be deduced from the list of common multiples. Arun and Anil will meet again after 90 minutes.

First Ten Multiples of the Numbers

Find out the first ten multiples of the numbers from the below figure.

Multiples of Different Numbers

When two numbers are multiplied the result is called the product of the multiple of given numbers. If the number 6 is multiplied with other numbers, then you get different multiples. Also, if the number 7 is multiplied with other numbers, then you get different multiples.

Solved Examples on Multiples

1. Find the first three multiples of 7.

Solution:
The given number is 7.
To find the first three multiples of 7, you need to multiply 7 with 1, 2, 3.
7 × 1 = 7
7 × 2 = 14
7 × 3 = 21

So, 7, 14, 21 are the first 3 multiples of 7.

2. Four friends Alex, Ram, Vijay, and Venu decided to pluck flowers from the garden in the order of the first four multiples of 5. Can you list the number of flowers that each of them plucked as a series of the first four multiples of 5?

Solution:
Given that four friends Alex, Ram, Vijay, and Venu decided to pluck flowers from the garden in the order of the first four multiples of 5.
To find the first four multiples of 5, you need to multiply 5 with 1, 2, 3, and 4.
5 × 1 = 5
5 × 2 = 10
5 × 3 = 15
5 × 4 = 20
The first four multiples of 5 are (5 × 1) = 5, (5 × 2) =10, (5 × 3) = 15, and (5 × 4) = 20.

Hence, Alex plucked 5 flowers, Ram plucked 10 flowers, Vijay plucked 15 flowers and Venu plucked 20 flowers.

3. Sam loves watering plants. Her mom asked her to water the pots which were marked in the order of the multiples of 8. However, she missed a few pots. Can you help her identify the pots that she missed in the following list: 8, 16, __, 32, __, 48, 56, 64, __?

Solution:
Given that Sam’s mom asked her to water the pots which were marked in the order of the multiples of 8.
Let us start counting the multiplication table of 8: 8 × 1 = 8, 8 × 2 = 16, 8 × 3 = 24, 8 × 4 = 32, 8 × 5 = 40, 8 × 6 = 48, 8 × 7 = 56, 8 × 8 = 64, 8 × 9 = 72.

The missed pots are 24, 40, and 72.

Rounding off the numbers means shortening the length of the number from long digits by replacing it with the nearest value. Round of to the nearest 1000 means minimizing the given decimal number to its nearest 1000 value. Check out the complete concept to learn the process to Round off the Numbers to Nearest 1000. We have also given Solved examples for your best practice.

Also, See:

• Round off to Nearest 10
• Round off to Nearest 100

How to Round off the Numbers to Nearest 1000?

Based on the below steps, we can easily round the numbers to the nearest 1000.
1. First, Find out the thousand’s digit in the number.
2. Next, choose the next smallest number (that is the hundredths digit of the number).
3. Now, check the hundred’s digit is either <5 (That means 0, 1, 2, 3, 4) or > = 5 (That is 5, 6, 7, 8, 9).
(i) If the digit is < 5, then the hundreds place is replaced with the digit ‘0’.
(ii) If the digit is > = 5, then the hundred’s digit is replaced with the digit ‘0’, and the thousand’s place digit is increased by 1 digit.

For example, Number 3350 Round to the Nearest 1000.
Step 1: Thousand’s digit of the number is 3.
Step 2: Hundreds digit of the number is 3.
Step 3: The hundred’s digit ‘3’ is < 5. So, we have to apply 3(i) conditions. That is, the hundred’s placed is replaced with the digit ‘0’.
3350 Rounding of the nearest 1000 is equal to 3000.

Rounding to Nearest 1000 Examples

1. Round of the number 2850 to nearest 1000.

Solution:
The given decimal number is 2850.
Step 1: Thousand’s digit of the number 2850 is ‘2’.
Step 2: Hundred’s digit of the number 2850 is ‘8’.
Step 3: The hundred’s digit of the number ‘8’ is > 5. So, the hundred’s digit is replaced by ‘0’ and the thousand’s digit is increased by ‘1’. That is
3000.

By rounding the number 2850 to its nearest 1000, it is equal to 3000.

2. Round of the number 5059 to nearest 1000.

Solution:
The given decimal number is 5059.
Step 1: Thousand’s digit of the number 5059 is ‘5’.
Step 2: Hundred’s digit of the number 5059 is ‘0’.
Step 3: The hundred’s digit of the number ‘0’ is < 5. So, the hundred’s digit is replaced by ‘0’. That is
5000.

By rounding the number 5059 to its nearest 1000, it is equal to 5000.

3. Round of the number 7985 to nearest 1000.

Solution:
The given decimal number is 7985.
Step 1: Thousand’s digit of the number 7985 is ‘7’.
Step 2: Hundred’s digit of the number 7985 is ‘9’.
Step 3: The hundred’s digit of the number ‘9’ is > 5. So, the hundred’s digit is replaced by ‘0’ and the thousand’s digit is increased by ‘1’. That is
8000.

By rounding the number 7985 to its nearest 1000, it is equal to 8000.

4. Round of the number 6500 to nearest 1000.

Solution:
The given decimal number is 6500.
Step 1: Thousand’s digit of the number 6500 is ‘6’.
Step 2: Hundred’s digit of the number 6500 is ‘5’.
Step 3: The hundred’s digit of the number ‘5’ is = 5. So, the hundred’s digit is replaced by ‘0’ and the thousand’s digit is increased by ‘1’. That is

By rounding the number 6500 to its nearest 1000, it is equal to 7000.

5. Round of the number 1287 to nearest 1000.

Solution:
The given decimal number is 1287.
Step 1: Thousand’s digit of the number 1287 is ‘1’.
Step 2: Hundred’s digit of the number 1287 is ‘2’.
Step 3: The hundred’s digit of the number ‘2’ is < 5. So, the hundred’s digit is replaced by ‘0’. That is
1000.

By rounding the number 1287 to its nearest 1000, it is equal to 1000.

6. Round off the below numbers to the nearest 1000.
(i) 50,105.
(ii) 25, 657
(iii) 3562
(iv) 9254
(v) 4895
(vi) 78962

Solution:
(i) The given decimal number is 50,105.
Step 1: Thousand’s digit of the number 50,105 is ‘0’.
Step 2: Hundred’s digit of the number 50,105 is ‘1’.
Step 3: The hundred’s digit of the number ‘1’ is < 5. So, the hundred’s digit is replaced by ‘0’. That is
50,000.
By rounding off the number50,105 to its nearest 1000, it is equal to 50,000.
(ii) The given decimal number is 25,657.
Step 1: Thousand’s digit of the number 25,657 is ‘5’.
Step 2: Hundred’s digit of the number 25,657 is ‘6’.
Step 3: The hundred’s digit of the number ‘6’ is > 5. So, the hundred’s digit is replaced by ‘0’, and the thousand’s digit of the number is increased by ‘1’. That is
26,000.
By rounding the number 25,657 to its nearest 1000, it is equal to 26,000.
(iii) The given decimal number is 3562.
Step 1: Thousand’s digit of the number 3562 is ‘3’.
Step 2: Hundred’s digit of the number 3562 is ‘5’.
Step 3: The hundred’s digit of the number ‘5’ is = 5. So, the hundred’s digit is replaced by ‘0’, and the thousand’s digit of the number is increased by ‘1’. That is
4000.
By rounding the number 3562 to its nearest 1000, it is equal to 4000.
(iv) The given decimal number is 9254.
Step 1: Thousand’s digit of the number 9254 is ‘9’.
Step 2: Hundred’s digit of the number 9254 is ‘2’.
Step 3: The hundred’s digit of the number ‘2’ is < 5. So, the hundred’s digit is replaced by ‘0’. That is
9000.
By rounding of the number 9254 to its nearest 1000, it is equal to 9000.
(v) The given decimal number is 4895.
Step 1: Thousand’s digit of the number 4895 is ‘4’.
Step 2: Hundred’s digit of the number 4895 is ‘8’.
Step 3: The hundred’s digit of the number ‘8’ is > 5. So, the hundred’s digit is replaced by ‘0’, and the thousand’s digit of the number is increased by ‘1’. That is
5000.
By rounding the number 4895 to its nearest 1000, it is equal to 5000.
(vi) The given decimal number is 78,962.
Step 1: Thousand’s digit of the number 78,962 is ‘8’.
Step 2: Hundred’s digit of the number 78,962 is ‘9’.
Step 3: The hundred’s digit of the number ‘9’ is > 5. So, the hundred’s digit is replaced by ‘0’, and the thousand’s digit of the number is increased by ‘1’. That is
79,000.
By rounding the number 78,962 to its nearest 1000, it is equal to 79,000.

18 Times Table is one of the difficult tables below 20. To make you learn 18-time table easily, we have given 18 Times Table Multiplication Chart. 18 times table values are double the values of 9 times table. This is a very important table for children to quickly solve the solutions and for mental ability. There are various ways to learn the 18 Multiplication Table. We have provided the different Math Tables along with the explanation below. Check out all the ways and make your learning simple.

How to Read Table of 18?

One time eighteen is 18

Two times eighteen are 36

Three times eighteen are 54

Four times eighteen are 72

Five times eighteen are 90

Six times eighteen are 108

Seven times eighteen are 126

Eight times eighteen are 144

Nine times eighteen are 162

Ten times eighteen are 180

Eleven times eighteen are 198

Twelve times eighteen are 216

Multiplication Table of 18 up to 20

Check out the multiplication table of 18 and remember the output to make your math-solving problems easy.

18	×	1	=	18
18	×	2	=	36
18	×	3	=	54
18	×	4	=	72
18	×	5	=	90
18	×	6	=	108
18	×	7	=	126
18	×	8	=	144
18	×	9	=	162
18	×	10	=	180
18	×	11	=	198
18	×	12	=	216
18	×	13	=	234
18	×	14	=	252
18	×	15	=	270
18	×	16	=	288
18	×	17	=	306
18	×	18	=	324
18	×	19	=	342
18	×	20	=	360

Tricks to Remember 18 Times Table

(i) If you know the 9 times table, then you can easily remember the 18 times table. Yes, add the resultant values of the 9 times table to the 9 times table. That is,
9 X 1 = 9 + 9 = 18 = 18 X 1 = 18.
9 X 2 = 18 + 18 = 36 = 18 X 2 = 36.
9 X 3 = 27 + 27 = 54 = 18 X 3 = 54.
9 X 4 = 36 + 36 = 72 = 18 X 4 = 72.
9 X 5 = 45 + 45 = 90 = 18 X 5 = 90.

(ii) If you know the 17 times table, then it is very easy to remember 18 times table. Yes,
17 X 1 = 17 + 1 = 18 = 18 X 1 = 18.
17 X 2 = 34 + 2 = 36 = 18 X 2 = 36.
17 X 3 = 51 + 3 = 54 = 18 X 3 = 54.
17 X 4 = 68 + 4 = 72 = 18 X 4 = 72.
17 X 5 = 85 + 5 = 90 = 18 X 5 = 90.
……17 X 10 = 170 + 10 = 180 = 18 X 10 = 180.

(iii) One more tip to remember 18 times table is
19 X 1 = 19 – 1 = 18 = 18 X 1 = 18.
19 X 2 = 38 – 2 = 36 = 18 X 2 = 36.
19 X 3 = 57 – 3 = 54 = 18 X 3 = 54.
19 X 4 = 76 – 4 = 72 = 18 X 4 = 72.
19 X 5 = 95 -5 = 90 = 18 X 5 = 90.
…..19 X 10 = 190 – 10 = 180 = 18 X 10 = 180.

Get More Tables:

0 Times Multiplication Chart	1 Times Multiplication Chart	2 Times Multiplication Chart
3 Times Multiplication Chart	4 Times Multiplication Chart	5 Times Multiplication Chart
6 Times Multiplication Chart	7 Times Multiplication Chart	8 Times Multiplication Chart
9 Times Multiplication Chart	10 Times Multiplication Chart	11 Times Multiplication Chart
12 Times Multiplication Chart	13 Times Multiplication Chart	14 Times Multiplication Chart
15 Times Multiplication Chart	16 Times Multiplication Chart	17 Times Multiplication Chart
19 Times Multiplication Chart	20 Times Multiplication Chart	21 Times Multiplication Chart
22 Times Multiplication Chart	23 Times Multiplication Chart	24 Times Multiplication Chart
25 Times Multiplication Chart

Solved Example on Eighteen Times Table

1. By using the 18 Times Table find the (i) 18 times 4 (ii) 18 times 6 minus 4 (iii) 18 times 2 plus 6 (iv) 18 times 3 multiple of 2?

Solution:
(i) 18 Times 4.
By using the 18 times table,
18 Times 4 in mathematical is equal to 18X 4 = 72.
So, 18 Times 4 is equal to 72.
(ii) 18 times 6 minus 4.
By using the 18 Times table,
18 Times 6 minus 4 can be written as 18 X 6 – 4 in mathematical.
18 X 6 – 4 = 108 – 4 = 104.
So, 18 times 6 minus 4 is equal to 104.
(iii) 18 times 2 plus 6
By using the 18 times table,
We can write the 18 times 2 plus 6 as 18 X 2 + 6.
18 X 2 + 6 = 36 + 6 = 42.
Therefore, 18 Times 2 plus 6 is equal to 42.
(iv) 18 times 3 multiple of 2
By using the 18 times table,
We can write the 18 times 3 multiple of 2 as 18 X 3 X 2.
18 X 3 X 2 = 54 X 2 = 108.
Therefore, 18 Times 3 multiple of 2 is equal to 108.

 

A cumulative frequency is the sum of frequency values of class or basic value. The frequency values are equal to the number of times the score or basic value or class is repeated. For Example, Class : 1 ,2, 1, 1, 1, 3,3, 3, 5, 5, 5, 6, 6, 7, 7, 7, 8, 8. The cumulative frequency of a value of a variable is the collection of data of a number of values less than or equal to the value of the variable. The cumulative frequency of a class interval that is overlapping or nonoverlapping is the sum of the frequencies of earlier class intervals and the concerned class interval.

Class	Frequency	Cumulative Frequency
1	4	4
2	1	5 (that is 4 +1)
3	3	8 (that is 5 + 3)
5	3	11 (that is (8 + 3)
6	2	13 (that is 11 + 2)
7	3	16 (that is 13 + 3)
8	2	18 (that is 16 + 2)

Also, Read: Medians and Altitudes of a Triangle

Cumulative Frequency Examples

1. The Following Table gives the frequency distribution of marks obtained by the 30 students. Find the Cumulative frequency based on the below values?

Solution: Based on the student’s marks and the frequency of the marks, we can easily find out the cumulative frequency. Cumulative frequency is the sum of the frequency of marks of the students. That is,

So, the cumulative frequency is 5, 17 ( 5 + 12), 27 (17 + 10), and 30 (27 + 3).

2. The below table gives the mass of 30 objects with the frequency. Find out the cumulative frequency for the objects?

Solution: As per the given information We have mass objects and the frequency of the mass of objects. The cumulative frequency is the sum of the frequency of mass of objects. That is

Finally, the cumulative frequency of the mass of objects is 10, 16, 36, and 51.

3. The below-given details are the ages of the employees in a particular company and the frequency of the ages of employees. Find the cumulative frequency for the given data?

Solution: As per the given details,
Ages of the employees in a company and the frequency of ages of the employees are noted. The cumulative frequency of the ages of employees is

4. A cloth store contains different colors of clothes. The color details, the cumulative frequency of some colors, and the frequency of the colors are given below, find the final cumulative frequency?

Solution: The given details are colors of cloths are white, brown, black, red, and pink.
The frequency of the colors is 10, 18, 20, 2, and 6.
The cumulative frequency of colors is the sum of the frequency of the colors. That is,
White – 10
Brown – 10 + 18 = 28
Black – 28+ 20 = 48
Red – 48 + 2 = 50
Pink – 50 + 6 = 56.
So, the final cumulative frequency of the colors is equal to 56.

5. For the collection of numbers 10, 12, 35, 10, 10, 12, 12, 35, 35, 35, 35, 10, 13, 11, 11, 13, 11, 13, and 10? What is the cumulative frequency of 13?

Solution: As per the given information
The given numbers are10, 12, 35, 10, 10, 12, 12, 35, 35, 35, 35, 10, 13, 11, 11, 13, 11, 13, and 10.
The frequency of the numbers is
Number – frequency
10 – 5
11 – 3
12 – 3
13 – 3
Cumulative frequency is equal to the sum of the frequency and the cumulative frequency of the 13 is equal to the sum of the frequency of less than or equal to 13. That is
5 + 3+ 3 + 3 = 14.
Therefore, the cumulative frequency of 13 is equal to 14.

6. The marks of 100 students are given below with the frequency. Find the cumulative frequency and answer the following questions.
(i) How many students obtain less than 41 % marks?
(ii) How many students obtain at least 51% marks?

Solution: The cumulative frequency is

(i) How many students obtain less than 40 marks?
The number of students obtaining less than 41% of marks is 31 – 40% cumulative frequency = 65.
(ii) How many students obtain at least 51% marks?
The number of students obtaining at least 51% of marks =total number of students – the number of students obtaining less than or equal to 41 – 50%.
= 100 – 75 = 25.

So, the number of students obtaining at least 51% of marks is equal to 25.