Formula Manipulation and Basic Concepts

Changing the subject of a formula is like solving a puzzle—you just need to isolate the letter you want on one side. The subject is the letter that appears alone on the left-hand side of the equation. To change it, use inverse operations (opposite operations) to rearrange.

For example, if you have 5p - 7 = c and want to make p the subject, add 7 to both sides first $5p = c + 7$, then divide by 5 $p = (c + 7)/5$. It's that simple!

Ratios help you compare quantities and can be simplified just like fractions. If you have 15:25, divide both numbers by their highest common factor (5) to get 3:5. Remember to convert units when needed—3km becomes 3000m when comparing with metres.

Quick Tip: Triangle types are easy to remember—equilateral (all equal), isosceles (two equal), scalene (all different). Angles can be acute (less than 90°), right (exactly 90°), or obtuse (more than 90°).

Inequalities and Number Operations

Inequality symbols are your friends once you remember the basics: > means "greater than" and < means "less than". Think of the symbol as a hungry mouth that always wants to eat the bigger number! The symbols ≥ and ≤ include "or equal to" in the comparison.

For example, x ≥ 3 means x can be 3 or any number larger than 3. These symbols help you describe ranges of values rather than just exact numbers.

Standard form makes huge or tiny numbers manageable by writing them as a number between 1 and 10, multiplied by a power of 10. So 723,000 becomes 7.23 × 10⁵, and 0.0000063 becomes 6.3 × 10⁻⁶. The power tells you how many places to move the decimal point.

Expanding brackets means multiplying everything inside by what's outside. With 3$x + 2$, multiply both x and 2 by 3 to get 3x + 6. For double brackets like $x + 2$$x + 6$, multiply each term in the first bracket by each term in the second.

Quick Tip: Prime factors are just the prime numbers that multiply together to make your original number—use factor trees to break numbers down step by step.

Mastering Standard Form

Standard form (also called standard index form) is brilliant for handling numbers that would otherwise be ridiculously long to write out. Scientists use it constantly because they deal with everything from the size of atoms to the distance between galaxies!

The trick is splitting any number into two parts: a number between 1 and 10, then multiply by the right power of 10. For large numbers like 50,000, you get 5 × 10⁴ $since 10⁴ = 10,000$. The positive power tells you it's a big number.

Small numbers use negative powers. When you see something like 4.78 × 10⁻³, the negative power means you're dealing with a decimal. 10⁻³ = 0.001, so 4.78 × 10⁻³ = 0.00478.

Converting back to ordinary numbers is straightforward—just do the multiplication. Move the decimal point right for positive powers, left for negative powers. The power tells you exactly how many places to move.

Quick Tip: Remember that 10⁰ = 1, and each time the power increases by 1, you add another zero to your number (10¹ = 10, 10² = 100, 10³ = 1000).