**Worksheet on Average :**

Worksheet given in this section is much useful to the students who would like to practice problems on average.

**Problem 1 :**

Find the average of first 20 natural numbers.

**Problem 2 :**

Find the average of first 15 odd numbers.

**Problem 3 : **

Find the average of first 20 natural numbers which are the multiples of 7.

**Problem 4 :**

The average of four consecutive odd numbers is 10. Find the smallest of these numbers.

**Problem 5 : **

The average age of students of a class is 15.8 years. The average age of boys in the class is 16.4 years and that of the girls is 15.4 years. Find the ratio of the number of boys to the number of girls in the class.

**Problem 1 :**

Find the average of first 20 natural numbers.

**Solution : **

**Step 1 :**

Write the formula to find sum of first "n" natural numbers.

S_{n} = n(n + 1) / 2

**Step 2 :**

Use the formula given in step 1 to find the sum of first 20 natural numbers.

S_{20} = 20(20 + 1) / 2

S_{20} = 20 ⋅ 21 / 2

S_{20} = 210

**Step 3 :**

Find the average of first 20 natural numbers.

Average of first 20 natural numbers is

= Sum of first 20 natural numbers / 20

= 210 / 20

= 10.5

Hence, the average of 20 natural numbers is 10.5

**Problem 2 :**

Find the average of first 15 odd numbers.

**Solution : **

**Step 1 :**

Write the formula to find sum of first "n" odd numbers.

S_{n} = n^{2}

**Step 2 :**

Use the formula given in step 1 to find the sum of first 15 odd numbers.

S_{n} = n^{2}

S_{15} = 15^{2}

S_{15} = 225

**Step 3 :**

Find the average of first 15 odd numbers.

Average of first 15 odd numbers is

= Sum of first 15 odd numbers / 15

= 225 / 15

= 15

Hence, the average of 15 odd numbers is 15.

**Problem 3 : **

Find the average of first 20 natural numbers which are the multiples of 7.

**Solution : **

**Step 1 :**

The first natural number which is a multiple of 7 is 7.

The next numbers which are the multiples of 7 are 14, 21..... **Step 2 :**

Write the first twenty natural numbers which are the multiples of 7.

They are 7,14,21,28........ up to 20 terms. **Step 3 :**

Find the sum of all the above numbers.

That is,

= 7 + 14 + 21 + 28.........up to 20 term

Because all of the above numbers are multiples of 7, we can factor 7.

= 7(1 + 2 + 3 + 4 +.........+ 20)

= 7 ⋅ 210

= 1470

**Step 4 : **

Find the average.

Average = (Sum of all 20 numbers) / 20

Average = 1470 / 20

Average = 73.5

Hence average of first 20 natural numbers which are the multiples of 7 is 73.5

**Problem 4 :**

The average of four consecutive odd numbers is 10. Find the smallest of these numbers.

**Solution : **

**Step 1 :**

Let "x' be the first odd number.

Then the four consecutive even numbers are

x, x + 2, x + 4, x + 6

**Step 2 :**

**Given :** Average of the four consecutive odd numbers is 10.

So, we have

(x + x + 2 + x + 4 + x + 6) / 4 = 10

(4x + 12) / 4 = 10

Multiple by both sides by 4.

4x + 12 = 4 ⋅ 10

4x + 12 = 40

Subtract 12 from both sides.

4x = 28

Divide both sides by 4.

x = 7

Hence the smallest of four consecutive odd numbers is 7.

**Problem 5 : **

The average age of students of a class is 15.8 years. The average age of boys in the class is 16.4 years and that of the girls is 15.4 years. Find the ratio of the number of boys to the number of girls in the class.

**Solution : **

**Step 1 :**

Let "x" be the number of boys and "y" be the no. of girls.

We have to find the ratio x : y.

And also, total number of students in the class is (x + y).

**Step 2 :**

**Given : **The average age of students of in the class is 15.8 years.

That is,

(Sum of the ages of all students) / No. of students = 15.8

(Sum of the ages of all students) / (x + y) = 15.8

Sum of the ages of all students = 15.8(x + y)

Sum of the ages of all students = 15.8x + 15.8y ----(1)

**Step 3 :**

**Given : **The average age of boys of in the class is 16.4 years.

That is,

(Sum of the ages of all boys) / No. of boys = 16.4

(Sum of the ages of all boys) / x = 16.4

Sum of the ages of all boys = 16.4x ----(2)

**Step 4 : **

**Given : **The average age of girls of in the class is 15.4 years.

That is,

(Sum of the ages of all girls) / No. of boys = 15.4

(Sum of the ages of all girls) / y = 15.4

Sum of the ages of all girls = 15.4y ----(3)

**Step 4 : **

Sum of the ages of all students is equal to sum of the total ages of boys and girls.

That is,

(1) = (2) + (3)

15.8x + 15.8y = 16.4x + 15.4y

Simplify.

0.4y = 0.6x

0.4 / 0.6 = x / y

4 / 6 = x / y

2 / 3 = x / y

2 : 3 = x : y

Hence, the ratio of the number of boys to the number of girls in the class is 2 : 3.

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