Core Ideas in Plain Terms

Numbers live on a line: zero in the middle, positives to the right, negatives to the left. Adding moves you right, subtracting moves you left. Subtracting a negative is like removing a debt — it moves you right, so $5 - (-3) = 5 + 3 = 8$. Two negatives multiplied make a positive. The rule at work is $a - b = a + (-b)$: subtracting is adding the opposite.

A fraction like $\tfrac34$, the decimal $0.75$, and the percent $75\%$ are the same number in different clothes. "Percent" means "per hundred," so $75\%$ is $\tfrac{75}{100}$. To add fractions you need a common bottom number; to find a percent of something you multiply, using $\text{part} = \dfrac{p}{100} \times \text{whole}$.

A variable like $x$ just stands for a number you don't know yet. An equation is a balance scale: whatever you do to one side you must do to the other. To solve, peel away operations in reverse with inverse operations — undo addition with subtraction, undo multiplication with division. The licence to do this is the properties of equality: if $a = b$, then $a + c = b + c$ and $ac = bc$.

The product rule for exponents, $x^a \cdot x^b = x^{a+b}$: three 2s multiplied together, times four more 2s, is seven 2s — $2^3 \cdot 2^4 = 2^7 = 128$. Dividing subtracts exponents the same way, and anything (nonzero) to the power $0$ is $1$.

A number is divisible by $3$ exactly when its digit sum is, and by $9$ when its digit sum is divisible by $9$. Try $4{,}527$: digits add to $4+5+2+7 = 18$, so it is divisible by $9$ (indeed $4{,}527 = 9 \times 503$). Divisible by $5$? Just look at the last digit.

Turn the words into an equation phrase by phrase ("three more than twice a number" $\to 2n + 3$). Then undo operations in reverse order with inverse operations — and estimate first so a slip cannot hide.