## Approximation and Estimation Part 1

#### What is approximation?

Approximation means rounding off. We round off numbers to simplify them so they are easy to use and easy to remember.

*Examples:*- You bought a pair of shoes for $99. If someone asks you the price of the shoes you are more likely to say that the price is $100.
- This is called
**rounding up**where the new number is bigger than the given number. - $99 ≈ $100 (You say it as:
*$99 is approximately equal to $100*) - A dress costs $81. You will more likely to say that the cost of the dress was $80.
- This is called
**rounding down**where the new number is smaller than the given number. - $81 ≈ $80 (You say it as:
*$81 is approximately equal to $80*)

####
1. a) Round off 77 250 to the nearest 1000

b) Round off 77 500 to the nearest 1000

c) Round off 77 876 to the nearest 1000

**Rounding off to the nearest thousand**

When rounding off to the nearest 1000, you only care about the part of the number up to the thousands place. So,

- Check the digit in the hundreds place.
- If it is
*less than 5*, keep the digit in the thousands place unchanged (i.e. round down) - If it is
*equal to or more than 5*, add 1 to the digit in the thousands place (i.e. round up) - Change the digits in the hundreds place, tens place and ones place to 0s.

- 77 250 ≈ 77 000 (when rounded off to the nearest 1000, it is rounded down because it is closer to 77 000 than to 78 000)
- 77 500 ≈ 78 000 (when rounded off to the nearest 1000, it is rounded up because it is closer to 78 000 than to 77 000)
- 77 876 ≈ 78 000 (when rounded off to the nearest 1000, it is rounded up because it is closer to 78 000 than to 77 000)

*Notes:*- All numbers to the left of the midpoint on the number line will be rounded down.
- All numbers to the right of the midpoint including the midpoint itself will be rounded up.

####
2. a) Round off 1425 to the nearest 100

b) Round off 1450 to the nearest 100

c) Round off 1483 to the nearest 100

**Rounding off to the nearest hundred**

When rounding off to the nearest 100, you only care about the part of the number up to the hundreds place. So,

- Check the digit in the tens place.
- If it is
*less than 5*, keep the digit in the hundreds place unchanged (i.e. round down) - If it is
*equal to or more than 5*, add 1 to the digit in the hundreds place (i.e. round up) - Change the digits in the tens place and ones place to 0s.

- 1425 ≈ 1400 (when rounded off to the nearest 100, it is rounded down because it is closer to 1400 than to 1500)
- 1450 ≈ 1500 (when rounded off to the nearest 100, it is rounded up because it is closer to 1500 than to 1400)
- 1483 ≈ 1500 (when rounded off to the nearest 100, it is rounded up because it is closer to 1500 than to 1400)

####
3. a) Round off 1402 to the nearest 10

b) Round off 1405 to the nearest 10

c) Round off 1408 to the nearest 10

**Rounding off to the nearest ten**

When rounding off to the nearest 10, you only care about the part of the number up to the tens place. So,

- Check the digit in the ones place.
- If it is
*less than 5*, keep the digit in the tens place unchanged (i.e. round down) - If it is
*equal to or more than 5*, add 1 to the digit in the tens place (i.e. round up) - Change the digits in the ones place to 0.

- 1402 ≈ 1400 (when rounded off to the nearest 10, it is rounded down because it is closer to 1400 than to 1410)
- 1405 ≈ 1410 (when rounded off to the nearest 10, it is rounded up because it is closer to 1410 than to 1400)
- 1408 ≈ 1410 (when rounded off to the nearest 10, it is rounded up because it is closer to 1410 than to 1400)

#### 4. If Jack has an income of $6000 when rounded off to the nearest thousand. What could his actual income be?

- When a number given to you has been round off to the nearest thousand, then there are 1000 possible options that the actual number could be.
- Half of those numbers (500 of them) would be less than the given number, and
- Half of those numbers (500 of them) would be equal to or greater than the given number.
- When rounding off to the nearest 1000, all numbers from $5500 to $5999 will be rounded up to give $6000.
- When rounding off to the nearest 1000, all numbers from $6001 to $6499 will be rounded down to give $6000.

So, Jack's actual income could be anywhere

**between $5500 and $6499**.

#### 5. If Jack has an income of $5920 and Harry has an income of $4120, estimate the sum of money they earn together.

Jack's income = $5920 ≈ $6000

Harry's income = $4120 ≈ $4000

Total = Jack's income + Harry's income

≈ $6000 + $4000

= $10 000

Jack and Harry earn approximately

Harry's income = $4120 ≈ $4000

Total = Jack's income + Harry's income

≈ $6000 + $4000

= $10 000

Jack and Harry earn approximately

**$10 000**together.