Approximation and estimation

Welcome to Approximation and Estimation!

Ever been in a shop and tried to work out if you have enough money for three snacks and a drink? Or wondered how long it will take to walk to a friend's house? You weren't doing exact math; you were estimating! In this chapter, we learn how to make numbers easier to work with while keeping our answers "good enough" for the real world. We will cover rounding, estimation, and finding the "buffer zone" of a measurement, known as bounds.

1. Rounding Numbers

Rounding makes numbers simpler to talk about and use. It’s like saying a concert had "about 20,000 people" instead of "19,842 people."

The Golden Rule of Rounding

Look at the next digit to the right of the one you are rounding to:

• If the digit is 5 or more, round UP (add 1 to your target digit).
• If the digit is 4 or less, round DOWN (keep the target digit the same).

Memory Aid: "5 to 9, climb the vine! 0 to 4, stay on the floor!"

Decimal Places (dp)

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

Example: Round \( 3.461 \) to 1 decimal place (1 dp).
1. The first digit after the decimal is 4.
2. The next digit is 6.
3. Since 6 is "5 or more," we round the 4 up to a 5.
4. Answer: \( 3.5 \)

Significant Figures (sf)

Significant figures are the digits that carry meaning. They are the "important" numbers starting from the very first non-zero digit.

Rules for Zeros:
1. Zeros at the start of a number (like \( 0.005 \)) are not significant.
2. Zeros in the middle (like \( 505 \)) are significant.
3. Zeros at the end after a decimal point (like \( 5.20 \)) are significant because they show precision.

Example: Round \( 0.00452 \) to 1 significant figure (1 sf).
1. The first significant figure is 4.
2. The next digit is 5.
3. Since it is 5, round the 4 up.
4. Answer: \( 0.005 \)

Quick Review: Always count from the first non-zero digit for significant figures!

Key Takeaway: Rounding makes numbers manageable. Whether it's to the nearest 10, 100, or a specific decimal place, always look one digit to the right to decide whether to round up or stay the same.

2. Estimation

Estimation is used to find an approximate answer quickly. It is also a fantastic way to check if a calculator answer looks "sensible."

The Strategy: Round to 1 Significant Figure

For most GCSE estimation problems, the trick is to round every number in the calculation to 1 significant figure first, then do the math.

The Symbol: We use \( \approx \) which means "is approximately equal to."

Example (Real World): Estimate the cost of \( 2.8 \text{ kg} \) of potatoes at \( 68\text{p} \) per kg.
1. Round \( 2.8 \) to 1 sf: \( 3 \)
2. Round \( 68\text{p} \) to 1 sf: \( 70\text{p} \)
3. Multiply: \( 3 \times 70\text{p} = 210\text{p} \) or \( £2.10 \)
4. Answer: \( \approx £2.10 \)

Complex Estimation

Don't worry if a calculation looks scary with square roots or decimals! Just use 1 sf for everything.

Example: Estimate \( \sqrt{\frac{2.9}{0.051 \times 0.62}} \)
1. Round \( 2.9 \rightarrow 3 \)
2. Round \( 0.051 \rightarrow 0.05 \)
3. Round \( 0.62 \rightarrow 0.6 \)
4. Calculation: \( \sqrt{\frac{3}{0.05 \times 0.6}} = \sqrt{\frac{3}{0.03}} = \sqrt{100} = 10 \)
5. Answer: \( \approx 10 \)

Common Mistake: Don't try to be too exact! If you don't round to 1 sf at the start, the multiplication or division becomes much harder than it needs to be.

Key Takeaway: To estimate, round every number to 1 significant figure before you do any calculating. Use the \( \approx \) symbol to show your answer isn't exact.

3. Upper and Lower Bounds

When a number is rounded, the "true" value could be slightly higher or lower. Bounds tell us the maximum and minimum possible values of that original number.

Finding the Bounds

A simple trick: Take the unit you rounded to (e.g., the nearest 10), halve it (5), and then add/subtract that from your number.

Example: A length is \( 20 \text{ cm} \) rounded to the nearest 10 cm.
1. Half of 10 is 5.
2. Lower Bound (LB): \( 20 - 5 = 15 \text{ cm} \)
3. Upper Bound (UB): \( 20 + 5 = 25 \text{ cm} \)

Inequality Notation (Error Intervals)

We write bounds using symbols: \( \le \) (less than or equal to) and \( < \) (less than).
For our \( 20 \text{ cm} \) example: \( 15 \le \text{length} < 25 \)
Note: We use \( < \) for the Upper Bound because if it were exactly 25, it would round up to 30!

Truncating vs. Rounding

Truncating is like "chopping off" the end of a number without rounding up.
Example: If \( x = 2.1 \) truncated to 1 dp, the value must be at least 2.1 but less than 2.2.
Error Interval: \( 2.1 \le x < 2.2 \)

Discrete vs. Continuous Quantities

Continuous: Things you measure (weight, height, time). Use the \( < \) symbol for the Upper Bound.
Discrete: Things you count (cars, people).
Example: You have 200 cars to the nearest hundred.
The number of cars \( n \) satisfies: \( 150 \le n < 250 \). Since you can't have half a car, we could also say \( 150 \le n \le 249 \).

Calculating with Bounds (Higher Tier Tip)

To find the Maximum Area of a rectangle, multiply the Upper Bound of the length by the Upper Bound of the width.
To find the Minimum Area, multiply the Lower Bounds together.

Key Takeaway: Bounds create a "safety zone" around a rounded number. Use \( \le \) for the bottom of the zone and \( < \) for the top.

Summary Quick Review

Rounding: Use "5 or more, round up." Significant figures start at the first non-zero digit.
Estimation: Round everything to 1 significant figure before calculating. Use \( \approx \).
Bounds: Add and subtract half of the rounding unit to find the maximum and minimum possibilities. Express this as an inequality like \( \text{LB} \le x < \text{UB} \).

Don't worry if bounds seem tricky at first! Just remember the "half-way" rule, and you'll be on the right track!