### Key Concepts

- Estimation
- Rounding Numbers
- Front-End Estimation

- Factors
- Composite and Prime Numbers
- L.C.M. and H.C.F.

- Multiples
- Multiples of Numbers
- Common Multiples
- Least Common Multiples

- Multiplication Using Models
- Array Model
- Area Model

**Introduction: **

In this chapter, we will be learning about the following:

- Rounding numbers to the nearest 10,000.
- Estimating sums using front-end estimation.
- Estimating differences using front-end estimation.
- Multiplying two numbers to find the product.
- Composite and Prime Numbers
- L.C.M. and H.C.F.
- Multiples of Numbers
- Common Multiples
- Least Common Multiples
- Multiplication Using Models

**Estimation in Maths**

**Definition of estimation:**

Estimation of numbers is the process of approximating or rounding off the numbers in which the value is used to avoid complicated calculations. There is a difference between the terms ‘exact’ and ‘estimation.’

**Example:** Estimate the sum of 34 and 56 by rounding off to the nearest hundred.

**Sol.:**

**Rounding numbers to Estimate Sum, Differences, Products, and Quotient.**

**Rounding:** A number can be rounded to the nearest ten or hundred by looking at the digit to the right of the tens or hundreds place to give an approximate value.

If it is less than 5, round down. If it is 5 or more, round up.

**Examples:**

1) Estimate 286 + 495.

** Sol.: **

** **The estimated sum rounded nearest to 100 is 800.

2) Estimate the difference of 786 – 214.

** Sol.:** 786 to the nearest hundred = 800

214 to the nearest hundred = 200

Required = 800 – 200 = 600

3) Estimate the product of 958 × 387.

** Sol.:** 958 rounds off as, 1000

387 rounds off as, 400

Required product = 1000 × 400 = 400,0000

4) Estimate the quotient of 2838 ÷ 125.

** Sol.:** 2838 rounds off as, 2800

125 rounds off as, 100

Required quotient = 2800 ÷ 100 = 28

**Front-End Estimation.**

Front-end estimation uses the leading digits in numbers to make an estimate.

**Example:** Find the sum of 9411 and 3849 by using front-end estimation.

**Sol.:**

**Example:** What is the front-end estimation of 8792?

**Sol.:** The front-end estimation of 8792 is 8000.

**Estimate to check whether the answer is reasonable.**

**Reasonable:** It is an estimate that is close to the actual answer.

**Example:** Find the sum of 254 + 143 to the nearest hundred.

**Sol:** 254 + 143 = 397

Estimate to check the answer is reasonable.

The answer is 400 reasonable.

[Since both numbers are rounded up, the estimate is greater than the actual sum.]

**Factors**

**Factor:** Multiplying two whole numbers gives a product. The numbers that we multiply are the factors of the product.

**Example:** Find the factors of 12.

**Sol.:**

**Example:** Find the factors of 14.

**Sol.:** 14 can be divided exactly by 7.

So, 7 is a factor 14.

Hence, the factors of 14 are 1, 7, 2, 14.

14 = 14 × 1

= 7 × 2

**Note: **A factor divides a number entirely without leaving any remainder.

**Composite and Prime Numbers.**

**Composite numbers:** Composite numbers have more than two factors. This means that apart from getting divided by 1 and itself, the composite numbers divide entirely by other numbers as well.

**Example:**

**Example:** 4, 6, 8, 9, 10, 12… are composite numbers.

**Prime numbers:** A prime number has exactly two factors, 1 and itself.

**Example:**

**Sol.:** 2, 3, 5, 7, 11, 13, 17…. are prime numbers.

**L.C.M. and H.C.F.**

**L.C.M.:** The least common multiple (L.C.M.) is the smallest number divisible by two or more given numbers.

**Example:** Find the L.C.M. of 18, 24, 96.

**Sol.:**

**Example:** Find out the L.C.M. of 32, 36.

**H.C.F.:** The highest common factor (H.C.F.) is the largest factor shared by two or more numbers.

**Example:** Find the H.C.F. of 144, 180, 192.

**Sol.:**

**Example:** Find the H.C.F. of 90 and 30.

H.C.F. = 2 × 3 × 5 = 30

**Multiples**

#### Finding multiples of a number.

Multiples are numbers that we get when we multiply a number by 1, 2, 3, 4 and so on.

**Example:** Find the multiples of 15.

**Sol.:**

**Example:** Find the multiples of 6.

**Sol.:** 6, 12, 18, 24, 30…60…

#### Finding Common Multiples of Two Whole Numbers.

**Common multiples:** The multiples that are common to two or more numbers are called the common multiples of those numbers.

**Example:** What are the common multiples of 3 and 4?

**Sol.:**

The common multiples of 3 and 4 are 12, 24, 36.

**Example:** What are the common multiples of 2 and 4?

**Sol.:** The multiples of 2 are 2, 4, 6, 8, 10, 12, 14, 16, 18…

The multiples of 4 are 4, 8, 12, 16, 20…

Common multiples are 4, 8, 12, 20…

### Finding the L.C.M. of Two Whole Numbers.

**L.C.M:** The smallest number of all the common multiples is called the L.C.M.

**Example:** What is the L.C.M. of 3 and 4?

**Sol.:**

**Multiplying Using Models**

**Multiplying Numbers Using the Array model.**

**Array model:** The array model of multiplication uses the number of rows and number of columns in an array to illustrate the product of two numbers.

**Example:** Draw an array model of 2 × 4.

**Sol.:**

**Example:**

Multiplying using Array Model.

**Sol.: **There are 5 columns and 2 rows.

Product = 5 × 2

= 10

**Multiplying Numbers Using Area Model.**

**Area model:** An area model is a rectangular diagram or model used for multiplication. In the diagram, the factors define the length and width of the rectangle.

**Example:** Find 73 × 64 by using the area model.

**Sol.:**

**Example:** Find 27 × 35 using the area model?

**Sol.:** 27 × 35 = (20 + 7) × (30 + 5)

600 + 100 + 210 + 35 = 945

Therefore, 27 × 35 = 945

### Exercise:

- Estimate the value of 294 to the nearest hundreds.
- Estimate the sum of 256 + 342 to the nearest hundreds.
- Estimate the difference of 525 – 430 to the nearest hundreds.
- Estimate the product of 164 × 630 to the nearest hundreds.
- Estimate the quotient of 5463 ÷ 630 to the nearest hundreds.
- Using front-end estimation, find the sum of 2456 and 5432.
- Write the factors of 15.
- Find the factors of 54 and then decide whether it is prime or composite?
- Find the factors of 17 and then decide whether it is prime or composite?
- Find out the L.C.M. of 44 and 63.
- Find out the H.C.F. of 50 and 64.
- Write the common multiples of 6 and 8.
- What are the least common multiples of 5 and 10?
- Multiply using the array model.
- Find the product using the area model for 25 × 18.

### What have we learned:

In this chapter, we learned:

- When two factors are multiplied, the product is a multiple of both numbers.
- About knowing the factors and multiples of numbers can help in estimating products and quantities.
- About using front-end estimation to check the reasonableness of products, sums and differences.
- About multiplication using models.