Rounding and approximations Revision Notes for OCR GCSE Maths

Rounding and approximations

1. Rounding

Definition: Rounding is the process of simplifying a number while keeping its value close to what it originally was. When you round a number, you reduce the number of digits but keep the overall magnitude of the number intact.

Steps for Rounding:

Identify the Key Digit
Check the Unwanted Digit
Be Cautious of the Number 9

Identify the Key Digit:

The key digit is the last digit you want to keep in your rounded number.

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Example: If rounding $1.0602$ to two decimal places, the key digit is the second digit after the decimal, which is $6$ .

Check the Unwanted Digit:

Look at the digit immediately to the right of the key digit (the unwanted digit).
If the unwanted digit is $5$ or above, add one to the key digit.
If the unwanted digit is less than $5$ , leave the key digit as it is.

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Example: In $1.0602$ , if you are rounding to two decimal places, the unwanted digit is $0$ . Since $0$ is less than $5$ , the key digit $6$ remains unchanged, so $1.0602$ rounds to $1.06$ .

Be Cautious of the Number 9:

If the key digit is $9$ and the unwanted digit is $5$ or above, rounding will change the $9$ to $0$ and increase the digit to the left by one.

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Example: Rounding $3.1092$ to three decimal places involves a key digit $9$ and an unwanted digit $2$ . Since $2$ is less than $5$ , the key digit $9$ stays $9$ , so $3.1092$ rounds to $3.109$ .

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Worked Example: Question: Round $3.1092×5.95$ to two decimal places.

Step-by-Step Solution:

Multiply the Numbers: $3.1092×5.95=18.49014$

Identify the Key Digit:

If rounding $18.49014$ to two decimal places, the key digit is the second digit after the decimal, which is $9$ .

Check the Unwanted Digit:

The unwanted digit is $0$ , which is less than $5$ .

Round the Number:

Since the unwanted digit is less than $5$ , the key digit stays the same. $18.49014$ rounds to $18.49$

(a) Decimal Places

Definition:

Decimal Places (dp): This refers to how many digits you keep after the decimal point. For example, rounding $5.963$ to $2$ decimal places means keeping two digits after the decimal, like $5.96$ .
Key Rule: If the question asks for a certain number of decimal places, you must give exactly that number of decimal places—no more, no less.

Worked Examples:

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Example 1: Problem: Round $5.639$ to $1$ decimal place.

Step-by-Step Solution:

Identify the Key Digit:

The question asks for $1$ decimal place, so the key digit is the first digit after the decimal point, which is $6$ .
Write it down as:

5.\underline{6}

Check the Unwanted Digit:

Look at the digit to the right of the key digit, which is $3$ .
Since $3$ is less than $5$ , the key digit $6$ stays the same.

Final Answer:

The number rounded to $1$ decimal place is: $5.6$

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Example 2: Problem: Round $12.0482$ to $2$ decimal places.

Step-by-Step Solution:

Identify the Key Digit:

The question asks for $2$ decimal places, so the key digit is the second digit after the decimal point, which is $4$ .
Write it down as:

12.0\underline{4}82

Check the Unwanted Digit:

Look at the digit to the right of the key digit, which is $8$ .
Since $8$ is $5$ or above, you add $1$ to the key digit $4$ .

Final Answer:

The number rounded to $2$ decimal places is:

\underline{12.05}

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Example 3: Rounding to $3$ Decimal Places Problem: Round $25.72037$ to $3$ decimal places.

Step-by-Step Solution:

Identify the Key Digit:

The question asks for $3$ decimal places, so the key digit is the third digit after the decimal point, which is $0$ .
Write it down as:

25.72\underline{0}37

Check the Unwanted Digit:

Look at the digit to the right of the key digit, which is $3$ .
Since $3$ is less than $5$ , the key digit $0$ stays the same.

Final Answer:

The number rounded to $3$ decimal places is:

25.720

Important Note: Even though the last digit is $0$ , it must be included to reflect the rounding to $3$ decimal places. This ensures accuracy and meets the requirement of the question.

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Example 4: Rounding to $2$ Decimal Places Problem: Round $3.7952$ to $2$ decimal places.

Step-by-Step Solution:

Identify the Key Digit:

The question asks for $2$ decimal places, so the key digit is the second digit after the decimal point, which is $9$ .
Write it down as:

3.7\underline{9}52

Check the Unwanted Digit:

Look at the digit to the right of the key digit, which is $5$ .
Since $5$ is $5$ or above, you add $1$ to the key digit $9$ .
This would turn $9$ into $10$ , but since you can't have a digit of $10$ , you round up the next digit to the left, turning $79$ into $80$ .

Final Answer:

The number rounded to $2$ decimal places is:

3.80

(b) Rounding to the Nearest Whole Number, $10, 100, 1000$ , etc.

Next, we'll look at how to round numbers to the nearest whole number, $10, 100$ , or even larger values. This type of rounding is useful for making quick estimates or simplifying large numbers.

Key Points to Remember:

Consistency in Size: The size of your rounded number should be similar to the original number. If you're rounding to the nearest $10$ , your answer should reflect that.
Use of Zeros: Sometimes, zeros are necessary to maintain the correct place value when rounding. These concepts are simple yet crucial for ensuring that your rounded numbers are accurate and appropriate for the context of the problem.

Worked Examples:

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Example 1: Problem: Round $3.825$ to the nearest whole number.

Step-by-Step Solution:

Identify the Key Digit:

The key digit is the digit representing the whole number, which is $3$ .
Write it down as:

\underline{3}.825

Check the Unwanted Digit:

The unwanted digit is the first digit after the decimal point, which is $8$ .
Since $8$ is more than $5$ , you add $1$ to the key digit.

Final Answer:

The number rounded to the nearest whole number is: $4$

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Example 2: Problem: Round $32,825.2$ to the nearest hundred.

Step-by-Step Solution:

Identify the Key Digit:

The key digit is in the hundreds place, which is $8$ .
Write it down as:

32,\underline{8}25.2

Check the Unwanted Digit:

The unwanted digit is in the tens place, which is $2$ .
Since $2$ is less than $5$ , the key digit $8$ stays the same.

Final Answer:

The number rounded to the nearest hundred is: $32,800$

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Example 3: Problem: Round $4,365,901$ to the nearest thousand.

Step-by-Step Solution:

Identify the Key Digit:

The key digit is in the thousands place, which is $5$ .
Write it down as:

4,36\underline{5},901

Check the Unwanted Digit:

The unwanted digit is in the hundreds place, which is $9$ .
Since $9$ is more than $5$ , you add $1$ to the key digit, making it $6$ .

Final Answer:

The number rounded to the nearest thousand is: $4,366,000$

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Example 4: Problem: Round $3,999$ to the nearest ten.

Step-by-Step Solution:

Identify the Key Digit:

The key digit is in the tens place, which is $9$ .
Write it down as:

3,99\underline{9}

Check the Unwanted Digit:

The unwanted digit is in the ones place, which is also $9$ .
Since rounding this $9$ would push the next digit to $10$ , the number rounds up to the next hundred.

Final Answer:

The number rounded to the nearest ten is: $4000$

(c) Significant Figures

In GCSE Maths, significant figures (often abbreviated as sf or sig fig) are crucial for rounding numbers, especially in higher-level calculations. Significant figures help you maintain precision while simplifying a number.

Understanding Significant Figures:

The first significant figure is the first non-zero digit in a number.
The second significant figure is the digit immediately to the right of the first significant figure, and so on.
When rounding, the size of your rounded number should be similar to the original, meaning you may need to add zeros to maintain the correct place value.

Worked Examples:

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Example 1: Problem: Round $28.53$ to $1$ significant figure.

Step-by-Step Solution:

Identify the Key Digit:

The first significant figure is the first non-zero digit, which is $2$ .
Write it down as:

\underline{2}8.53

Check the Unwanted Digit:

The next digit is $8$ , which is more than $5$ , so you add $1$ to the key digit.

Final Answer:

The number rounded to $1$ significant figure is: $30$

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Example 2: Problem: Round $5,322$ to $2$ significant figures.

Step-by-Step Solution:

Identify the Key Digits:

The first significant figure is $5$ and the second is $3$ .
Write it down as:

5\underline{3}22

Check the Unwanted Digit:

The next digit is $2$ , which is less than $5$ , so the key digits stay the same.

Final Answer:

The number rounded to $2$ significant figures is: $5,300$

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Example 3: Problem: Round $0.027$ to $1$ significant figure.

Step-by-Step Solution:

Identify the Key Digit:

The first significant figure is the first non-zero digit, which is $2$ .
Write it down as:

0.0\underline{2}7

Check the Unwanted Digit:

The next digit is $7$ , which is more than $5$ , so you add $1$ to the key digit.

Final Answer:

The number rounded to $1$ significant figure is: $0.03$

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Example 4: Problem: Round $305,216$ to $3$ significant figures.

Step-by-Step Solution:

Identify the Key Digits:

The first three significant figures are $3$ , $0$ , and $5$ .
Write it down as:

\underline{305},216

Check the Unwanted Digit:

The next digit is $2$ , which is less than $5$ , so the key digits stay the same.

Final Answer:

The number rounded to $3$ significant figures is: $305,000$

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Example 5: Problem: Round $4.0044$ to $2$ significant figures.

Step-by-Step Solution:

Identify the Key Digits:

The first two significant figures are $4$ and $0$ .
Write it down as:

\underline{4}.\underline{0}04

Check the Unwanted Digit:

The next digit is $0$ , which is less than $5$ , so the key digits stay the same.

Final Answer:

The number rounded to $2$ significant figures is: $4.0$

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Example 6: Problem: Round $0.089722$ to $2$ significant figures.

Step-by-Step Solution:

Identify the Key Digits:

The first two significant figures are $8$ and $9$ .
Write it down as:

0.0\underline{89}722

Check the Unwanted Digit:

The next digit is $7$ , which is more than $5$ , so you add $1$ to the key digit $9$ .

Final Answer:

The number rounded to $2$ significant figures is: $0.090$

2. Approximations

Approximations are useful for simplifying tricky sums, especially when an exact answer isn't necessary or when working without a calculator.

Approximating a Complex Calculation

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Example Problem: You need to find the value of:

\frac{6.0602^2}{3.1092 \times 5.95}

This expression looks complicated, but with some rounding and BODMAS skills, we can simplify it to make it easier to solve.

Step 1: Round the Numbers

To simplify, round each number to the nearest whole number or to one significant figure:

Round $6.0602$ to $6$
Round $3.1092$ to $3$
Round $5.95$ to $6$ Now, the expression becomes:

\frac{6^2}{3 \times 6}

Step 2: Apply BODMAS

Using the BODMAS rule, solve the expression step-by-step:

Calculate the square:

6²=36

Multiply in the denominator:

3×6=18

Divide:

\frac{36}{18} = 2

So, the approximate value of the expression is $2$ .

Step 3: Comparing with the Exact Answer

If you use a calculator, the exact answer is:

1.98521838...

So, rounding gives us an approximation very close to the true value.

Final Approximation:

\frac{6.0602^2}{3.1092 \times 5.95} \approx 2

Here, the symbol ≈ means "approximately equal to."

Rounding and approximations (OCR GCSE Maths): Revision Notes