- Estimation means finding an answer which is close to the correct answer by doing a rough calculation of the value, number, quantity on an extent of something.
- Estimation is a very easy method to find the answer quick.
- In daily life, estimation is handy tool.
- When we want to find an answer which belongs to our daily life's problem and we don't need an exact answer then we can use estimation method.
- In estimation method, it is important that the answer must make a sense and work with our problem.

Let's take an example to understand estimation.

- A bottle is filled with water as shown in a figure. If someone asks, to what level is the bottle filled , then we can say that half of the bottle is filled with water.
- In this example, we are saying that half bottle is filled, this answer is estimated answer and is working with our problem.

Note: In this example, we don't need to give an exact answer so we can use estimation method.

- Rounding a number means to change its value slightly. It is not as exact as the original value but it is approximately close to the original value. A number can be rounded to a certain place values like tens, hundreds, thousands etc.

**Rounding Rules:**

The Rules which are used to change the number to the nearest power of ten are known as Rounding Rule.

**Rounding Rules for tens place:**

- To round off a 2-digit number or a higher number to the nearest tens place, analyze the digit at the ones place.
- If the digit at ones place of a two-digit number is less than five, then round the number down to the nearest tens place.
- If the digit at ones place of a two-digit number is greater than five, then round the number up to the nearest tens place.

**Rounding off Whole numbers (Tens place):**

- We can understand the rounding off whole numbers (Tens place) by following examples:

(A) \(59\)

- When we analyze the number, we find that the nearest tens to \(59\) are \(50\) and \(60\).
- \(9\) which is at ones place is greater than \(5\) and also \(59\) is closer to \(60\).
- So, we will round up the number to \(60\).

(B) \(73\)

- The nearest tens to \(73\) are \(70\) and \(80\).
- \(3\) which is at ones place is less than \(5\) and also \(73\) is closer to \(70\).
- So, we will round down the number to \(70\).

(C) \(25\)

- The nearest tens to \(25\) are \(20\) and \(30\).
- \(25\) is equidistant from both the tens.
- So, we will round up the number to \(30\).

- To round off the decimal to the nearest tenth's place, analyze the digit after decimal point i.e. at the hundredths place.
- If the hundredths place value is \(5\) or greater than \(5\) then digit of tenths place increases by \(1\) and the digit at hundredths place and thereafter become zero.
- If the hundredths place value is less than \(5\) then digit of tenths place remains same and the digit at hundredths place and thereafter becomes zero.

**Rounding off decimal (hundredths place):**

- Let's take an example to understand it.

Example (1): \(49.38\)

- Let's use place value chart.

Tens | Ones | Tenths | Hundredths | |

4 | 9 | . | 3 | 8 |

- In place value chart analyze the hundredths place.
- Here, the digit at the hundredths place is \(8\) and \(8\) is greater than \(5\).
- Thus, the digit of tenths place increases by \(1\) and the digit at the hundredths place become zero.
- So, the nearest tenths to \(49.38\) after round-off is \(49.4\).

Example (2): \(33.42\)

- Let's use place value chart.

Tens | Ones | Tenths | Hundredths | |

3 | 3 | . | 4 | 2 |

- In place value chart analyze the hundredths place.
- Here, the digit at the hundredths place is \(2\) and \(2\) is less than \(5\).
- Thus, the digit at tenths place remains same and the digit at the hundredths place become zero.
- So, the nearest tenths to \(33.42\) after round-off is \(33.4\).

A \(56.4\)

B \(57.0\)

C \(56.3\)

D \(56.0\)

- To round off a decimal to the nearest hundredth place, analyze the digit at the thousandths place.
- If the digit at the thousandths place is \(5\) or greater than \(5\) , then the digit at the hundredth place increases by \(1\) and the digits at the thousandth place and thereafter become zero.
- If the digit at the thousandths place is less than \(5\) , then the digit at hundredth place remains same and the digits at the thousandth place and thereafter become zero.

**Rounding off Decimal Number (Thousandths place):**

Rounding up to hundredths place or thousandths place is similar to tenth place. Let's look at an example to understand it.

Example: \(23.759\)

Let's use the place value chart.

Tens | Ones | Tenths | Hundredths | Thousandths | |

2 | 3 | . | 7 | 5 | 9 |

- Analyze the digit at the thousandths place in the given decimal number.
- Here the digit at the thousandths place is greater than \(5\).
- Thus, the digit at the hundredths place increases by \(1\) and the digit at the thousandth place becomes zero.
- So, the nearest hundredths to \(23.759\) after round off is \(23.76\).

- So, the nearest hundredths to \(23.759\) after round off is \(23.76\) .

A \(28.38\)

B \(28.391\)

C \(28.3\)

D \(28.39\)

**Rounding**

- Rounding a number means to change its value slightly. It is not as exact as the original value but it is approximately close to the original value. A number can be rounded to a certain place values like tens, hundreds, thousands etc.

**Rounding Rules:**

The rules which are used to change the number to the nearest power of ten are known as Rounding Rules.

**Rounding Rules for hundreds place:**

- To round off a whole number to the nearest hundreds place, analyze the digit at the tens place.
- If the digit at the tens place is \(5\) or greater than \(5\), round the number up to the nearest hundreds place.
- If the digit at the tens place is less than \(5\), round the number down to the nearest hundreds place.

**Rounding off a whole number (Hundreds place)**

- Let's take an example to understand the rounding off a whole number (hundreds place).

(1) \(365\)

- After analyzing the tens place of the number, we see that the nearest hundreds to \(365\) are \(300\) and \(400\). Tens place digit, \(6\) is greater than \(5\) and the number \(365\) is closer to \(400\).

So, we shall round the number up to \(400\).

Note:

The values from \(301\) to \(349\) are round off to \(300\) and the values from \(350\) to \(399\) are round off to \(400\).

**Decimal Rounding on Number line**

Decimal number can be rounded off by using a number line to the Nearest Quarter.

- We can easily understand the decimal number rounding with the help of number line.
- Let's look at an example.

Example:

(A) \(.57\)

- Decimal number \(0.57\) comes between the whole numbers \(0\) and \(1\). So firstly, draw a number line and divide it into four equal parts between \(0\) and \(1\).

- Now, plot the decimal number \(0.57\) on it. \(0.57\) lies in between \(0.50\) and \(0.75\).
- Now, we can see that \(0.57\) is closest to quarter \(0.50\).
- So, we will round it off to \(0.50\).

(B) \(6.27\)

We can round it to nearest whole number.

- The decimal number \(6.27\) comes between the whole number \(6\) and \(7\). So draw a number line and divide it into quarters between \(6\) to \(7\).

- Now plot the decimal number \(6.27\) on it. \(6.27\) lies in between \(6.25\) and \(6.50\).

- Now we can see that \(6.27\) is closest to \(6\).
- So, we shall round off it to whole number \(6\).

Another way to round off decimal is by using rounding rules.

Rounding Rules for a decimal number [Tenths place]:

- To round off the decimal number to the whole number, firstly make a place value chart for given decimal number and analyze the first digit after the decimal point, i.e. at the tenth's place.

Place Value Chart

Tens | Ones | Tenths | Hundredths | Thousandths |

- If the digit at the tenth's place is \(5\) or greater than \(5\) then the digit of ones place increases by \(1\) and the digits to the right of decimal point, i.e. at tenths place and thereafter become zero.
- If the digit at the tenth's place is less than \(5\) then the digit of ones place remains same and the digits to the right of decimal point, i.e. at tenths place and thereafter become zero.

**Rounding off decimal number up to whole number**

Let's take an example for understanding of rounding off decimal number.

**Example 1**: Round off \(23.6\) to the nearest whole number.

- In place value chart analyze the first digit after decimal point i.e. at the tenths place.

Place value chart for given decimal number is

Tens | ones | tenths | hundredths | |

2 | 3 | . | 6 |

- Here, the digit at the tenths place is \(6\) and \(6\) is greater than \(5\).
- Thus, the digit of ones place is increased by \(1\).
- Therefore, the digit to the right of decimal point i.e. at tenths place becomes zero.
- So, the nearest whole number of \(23.6\) is \(24\).

**Example 2**: Round off \(47.3\) to the nearest whole number.

- Analyze the first digit after decimal point i.e. at the tenth place in given decimal number.

Place value chart for given decimal number is

Tens | ones | tenths | hundredths | |

4 | 7 | . | 3 |

- Here the digit at the tenths place is \(3\) and \(3\) is less than \(5\).
- Thus, the digit of ones place remains same.
- Therefore, the digit to the right of decimal point i.e. at tenth's place becomes zero.

So, the nearest whole number of \(47.3\) is \(47\).

A \(75\)

B \(76\)

C \(74\)

D \(73\)

**Rounding off fraction to the nearest half**

- To round off a fraction to the nearest whole number or nearest half, analyze the numerator and the denominator.
- We have three main values to round to \(0\), \(\dfrac{1}{2}\) , \(1\), when we round a fraction to the nearest half.
- We round the fraction to whichever half, it is closest.
- If the numerator is almost as large as the denominator round the number up to the next whole number.
- If the numerator is about half of the denominator round the fraction to \(\dfrac{1}{2}\).
- If the numerator is equidistant from both the nearest half then round the fraction up.

Example: \(\dfrac{7}{8}\)

The numerator \(7\) is almost as large as the denominator \(8\), so we shall round off it up to \(1\).

Example: \(\dfrac{3}{5}\)

The numerator \(3\) is about half of the denominator \(5\), so round the fraction to \(\dfrac{1}{2}\).

- A mixed fraction or a mixed number lies between two nearest whole numbers.
- A mixed fraction includes both whole numbers and a fraction.
- To round off the mixed fraction to the nearest whole number, analyze the mixed fraction.
- We need to figure out whether a fraction is larger than \(\dfrac{1}{2}\) or less than \(\dfrac{1}{2}\).

\(\to\) If the fraction part of mixed fraction is greater than or equal to \(\dfrac{1}{2}\), then the fraction is round-up.

\(\to\) If the fraction part of mixed fraction is less than \(\dfrac{1}{2}\) then the fraction is round-down.

Example: (i) \(1\dfrac{2}{3}\)

- In this mixed fraction, the fraction part \(\dfrac{2}{3}\) is greater than \(\dfrac{1}{2}\) i.e. \(\dfrac{2}{3}>\dfrac{1}{2}\)
- So, mixed fraction \(1\dfrac{2}{3}\) is round up to nearest whole number i.e. \(2.\)

(ii) \(15\dfrac{1}{4}\)

- In this mixed fraction, the fraction part \(\dfrac{1}{4}\) is less than \(\dfrac{1}{2}\) i.e. \(\dfrac{1}{4}<\dfrac{1}{2}\)
- So, mixed fraction \(15\dfrac{1}{4}\) is round down to nearest whole number i.e.\(15.\)