Approximate Values & Estimation: Your Guide to Smart Guessing!

Hey there! Ever been asked for the time and you say "it's about half past ten"? Or when you're shopping, you think "this will cost around $50"? That's exactly what this chapter is about! We're going to learn how to work with numbers that are "close enough" to the real value. This is a super useful skill in real life, from planning a party to checking if your homework answer makes sense. Let's get started!


What are Approximate Values? The Art of Rounding

An approximate value is a number that is close to the exact value, but is simpler and easier to use. The main way we find an approximate value is by rounding. You've probably done some rounding before, but we're going to level up your skills!

Rounding to a Certain Number of Decimal Places (d.p.)

This is all about how many numbers you want to keep after the decimal point.

Here’s the Secret Rule: The Next-Door Neighbour Trick

To round to a certain decimal place, we look at the digit immediately to its right (its next-door neighbour).

  • If the neighbour is 5 or more (5, 6, 7, 8, or 9), you round up the digit you're keeping. ("5 or more, raise the score!")
  • If the neighbour is 4 or less (4, 3, 2, 1, or 0), you leave the digit as it is. ("4 or less, let it rest!")
Step-by-Step Example:

Let's round the number 8.736

1. To 1 decimal place (1 d.p.):

  • The first decimal place is the 7. We write: 8.7 | 36
  • Look at its neighbour: it's a 3.
  • Since 3 is "4 or less", the 7 gets to rest.
  • Answer: 8.7

2. To 2 decimal places (2 d.p.):

  • The second decimal place is the 3. We write: 8.73 | 6
  • Look at its neighbour: it's a 6.
  • Since 6 is "5 or more", the 3 has to raise its score... to 4!
  • Answer: 8.74
Watch out for this common mistake!

After you round, make sure to drop all the digits that came after the rounding place. For example, when we rounded 8.736 to 8.74, the 6 at the end disappeared. Don't keep it!


A New Challenge: Significant Figures! (s.f.)

Don't worry if this seems tricky at first! Significant figures (or s.f.) are just the "important" digits in a number. They are the ones that carry meaningful information about its size.

How to Spot a Significant Figure:
  1. All non-zero digits (1-9) are ALWAYS significant.
    Example: In 38.2, there are 3 s.f. (the 3, 8, and 2).
  2. Zeros between non-zero digits are ALWAYS significant.
    Example: In 507, there are 3 s.f. (the 5, 0, and 7).
  3. Zeros at the beginning of a number are NEVER significant.
    Example: In 0.0025, there are only 2 s.f. (the 2 and 5). The zeros are just placeholders.
  4. Zeros at the end of a decimal number are significant.
    Example: In 4.500, there are 4 s.f. (the 4, 5, 0, and 0). They tell us the number is precisely four and a half.
How to Round to a Certain Number of Significant Figures:

The steps are almost the same as rounding to decimal places!

Step 1: Starting from the left, find the first non-zero digit. This is your first significant figure.

Step 2: Count from there to find the significant figure you need to round to.

Step 3: Use the "Next-Door Neighbour Trick" to decide whether to round up or let it rest.

Step 4: Change any remaining digits to the right into zeros if they are before the decimal point. Drop them if they are after the decimal point.

Step-by-Step Example:

Let's round the number 52,817

1. To 1 significant figure (1 s.f.):

  • The first s.f. is 5. We write: 5 | 2,817
  • Its neighbour is 2. (Let it rest!)
  • The 5 stays as 5. The other digits (2, 8, 1, 7) become placeholders (zeros).
  • Answer: 50,000

2. To 2 significant figures (2 s.f.):

  • The second s.f. is 2. We write: 52 | 817
  • Its neighbour is 8. (Raise the score!)
  • The 2 rounds up to 3. The digits after it become zeros.
  • Answer: 53,000

Now let's try a decimal: 0.04065

1. To 2 significant figures (2 s.f.):

  • The first s.f. is 4 (we skip the leading zeros). The second is 0. We write: 0.040 | 65
  • Its neighbour is 6. (Raise the score!)
  • The 0 rounds up to 1.
  • Answer: 0.041
Quick Review Box

Decimal Places (d.p.): Count spots AFTER the decimal point.
Significant Figures (s.f.): Start counting from the FIRST non-zero digit.

Key Takeaway for Approximate Values

Rounding makes numbers simpler. Whether you're using decimal places or significant figures, the rule is the same: look at the next digit to the right to decide whether to round up or stay the same.


Superpower Skill: Estimation!

Estimation is about finding a quick, rough answer to a calculation. It's like a mental shortcut to check if your final answer is in the right ballpark. Think about it: if you calculate `48 x 102` and get 500, estimation ( `50 x 100 = 5000` ) tells you something went wrong!

Strategy 1: Rounding Off

This is the most common strategy. You just round each number in the problem to an easier number (usually to 1 significant figure) and then do the calculation.

Example: Estimate the value of $$59.2 \times 3.14$$

  • Round 59.2 to the nearest ten (or 1 s.f.), which is 60.
  • Round 3.14 to the nearest whole number (or 1 s.f.), which is 3.
  • Now, the calculation is easy: $$60 \times 3 = 180$$.
  • The exact answer is 185.768. Our estimate of 180 is pretty close!

Strategy 2: Rounding Up

Sometimes, you need to make sure your estimate is definitely on the high side. This means you round all the numbers up to the next convenient value.

Real-World Example: You are buying snacks for a party. A bag of chips is $23.50, a bottle of soda is $12.80, and a pack of cookies is $18.90. You want to make sure you have enough money.

  • Round $23.50 up to $24.
  • Round $12.80 up to $13.
  • Round $18.90 up to $20.
  • Your estimated total is $$24 + 13 + 20 = $57$$.
  • By rounding everything up, you can be confident that you have enough cash!

Strategy 3: Rounding Down

This is the opposite of rounding up. Here, you want to be sure your estimate is on the low side. You round all the numbers down to a more convenient value.

Real-World Example: You have a 250 cm roll of wrapping paper. You need to wrap presents that each need 48 cm of paper. You want to estimate the minimum number of presents you can definitely wrap.

  • You could round 48 cm up to 50 cm to make the division easier and safer.
  • Your estimated calculation is $$250 \div 50 = 5$$.
  • You can be confident that you can wrap at least 5 presents.
Did you know?

Engineers and scientists use estimation all the time! When building a bridge, they first make estimations to see if a design is possible before they spend months on exact calculations. A quick estimation can save a lot of time and money!

Key Takeaway for Estimation Strategies

Estimation is about making calculations easier. Use rounding off for a general guess. Use rounding up when you need to be sure you have enough (like with money or materials). Use rounding down (or rounding the divisor up) when you want to find a minimum number you can achieve.


Challenge Zone: Becoming an Estimation Expert!

The truly powerful part of estimation is choosing the best strategy for the situation and judging if the result is reasonable. This is what you do in real life without even thinking about it!

Let's look at a problem:

A school is planning a trip for 127 students. The bus company says each bus can hold a maximum of 30 students. Estimate how many buses the school needs to book.

Thinking it through:

What's the goal? To make sure EVERY student gets a seat. We can't leave anyone behind!

Exact Calculation: $$127 \div 30 = 4.233...$$ buses.

Applying an Estimation Strategy:

  • If we round 127 down to 120, we get $$120 \div 30 = 4$$ buses. But what about the other 7 students? This isn't a good strategy here.
  • If we round 4.233... down to 4 buses, 7 students will be left at school. That's not a reasonable result!
  • In this situation, even if the answer was 4.1, you would need to round up to the next whole number. You can't book 0.1 of a bus!

Conclusion: The school needs to book 5 buses. This is a situation where you must round up to make sure everyone is included. The context of the problem is the most important clue!

You've done an amazing job! Keep practising and soon you'll be estimating like a pro, making your maths journey (and your shopping trips) much easier. You've got this!