Combining arithmetic operations

Welcome to Number Operations: Combining Arithmetic Operations!

Ever tried to follow a recipe and realized that the order of steps matters? If you put the cake in the oven before mixing the flour and eggs, you’ll end up with a mess! Mathematics is exactly the same. When we have a calculation with many different signs (like \(+\), \(\times\), or \(\sqrt{}\)), we need a set of rules to tell us which part to do first. In this guide, we’ll master the "rules of the road" for math so you can solve any calculation with confidence.

Did you know? Without a standard order of operations, a simple sum like \(2 + 3 \times 5\) could have two different answers! Is it 25 or 17? (Spoiler: It’s 17!) These rules ensure that everyone, everywhere in the world, gets the same result.

The Secret Code: BIDMAS

To remember the priority of operations, we use a handy mnemonic called BIDMAS. It acts like a ladder—you start at the top and work your way down.

B - Brackets: Always solve anything inside brackets first. \( ( ... ) \)
I - Indices: This includes Powers (like \(5^2\)), Roots (like \(\sqrt{16}\)), and Reciprocals (like \(\frac{1}{x}\)).
D / M - Division and Multiplication: These are equally important. Do them in the order they appear from left to right.
A / S - Addition and Subtraction: These are also equal. Do them in the order they appear from left to right.

Memory Aid: Some people use BODMAS (where O stands for "Orders"), but for your OCR J560 syllabus, BIDMAS is the most common way to remember it!

Key Takeaway: If a calculation looks scary, just look for the B first, then the I, and so on. Breaking it down makes it much easier.

Understanding the "I" (Indices, Roots, and Reciprocals)

This is where many students get tripped up, but don't worry! In the J560 curriculum, the "I" stands for Indices, but it’s actually a "bucket" for three things that share the same priority level:

1. Powers: Small numbers floating above a base, like \(3^2\) (which means \(3 \times 3 = 9\)).
2. Roots: The opposite of a power, like \(\sqrt{25}\) (which is \(5\)).
3. Reciprocals: One divided by a number. For example, the reciprocal of \(4\) is \(\frac{1}{4}\).

Example: \( 10 + 2^3 \)
Even though addition comes first in the sentence, the Power (Indices) has a higher priority.
Step 1: Calculate \( 2^3 = 8 \)
Step 2: \( 10 + 8 = 18 \)

Key Takeaway: Powers and roots are more "powerful" than multiplying or adding, so they almost always happen near the start of your calculation.

The "Left to Right" Rule

Multiplication and Division are "best friends"—they have the same level of priority. The same goes for Addition and Subtraction. When you see both in a sum, you simply work from left to right, just like reading a book.

Example: \( 10 \div 2 \times 5 \)
Since Division and Multiplication are equal, we move left to right.
Step 1: \( 10 \div 2 = 5 \)
Step 2: \( 5 \times 5 = 25 \)

Common Mistake to Avoid: Don't assume Multiplication always comes before Division just because "M" comes after "D" in the word BIDMAS. They are a team! Do whichever one comes first in the expression.

Step-by-Step: Putting it All Together

Let's try a complex-looking problem: \( 5 + (12 - 2) \div \sqrt{25} \)

Step 1: Brackets (B)
Look inside the brackets: \( 12 - 2 = 10 \).
Our sum now looks like: \( 5 + 10 \div \sqrt{25} \)

Step 2: Indices/Roots (I)
Find the root: \( \sqrt{25} = 5 \).
Our sum now looks like: \( 5 + 10 \div 5 \)

Step 3: Division (D)
Division is higher priority than addition: \( 10 \div 5 = 2 \).
Our sum now looks like: \( 5 + 2 \)

Step 4: Addition (A)
Final step: \( 5 + 2 = 7 \).
The final answer is 7!

Key Takeaway: If you feel stuck, write "B I D M A S" on the side of your paper and tick off each letter as you check the calculation for those symbols.

Quick Review Box

• Brackets are the ultimate priority.
• Powers, Roots, and Reciprocals come next.
• Multiplication and Division are equal (Left to Right).
• Addition and Subtraction are the final steps (Left to Right).
• Common Error: Adding before Multiplying. Remember: \( 2 + 3 \times 4 \) is \( 2 + 12 = 14 \), not \( 5 \times 4 = 20 \)!

Using Inverse Operations to Check

The syllabus (Section 1.04a) mentions using inverse operations. This is a great trick to check your work! Inverse means "opposite."

• The inverse of Addition is Subtraction.
• The inverse of Multiplication is Division.
• The inverse of a Power is a Root.

If you calculate \( 50 \times 6 = 300 \), you can check it by doing \( 300 \div 6 = 50 \). It’s like walking backward to make sure you didn't drop your keys!

Final Encouragement: Combining operations can feel like untangling a knot. Just take it one string at a time using BIDMAS, and you'll find the answer every time!