💰 Change, Income & Savings

• To change a ratio $a:b$ by adding the same number $x$ to both terms, build the new ratio $\dfrac{a+x}{b+x}$, set it equal to the target and cross-multiply for $x$. (To subtract, use $a-x$ and $b-x$; a negative answer just means you add instead — report the size.)

• The “let the common multiple be $x$” technique: whenever a quantity is “in the ratio $5:4$”, rewrite the parts as $5x$ and $4x$ so one unknown controls everything, then form one equation from the extra fact given.

• Income, expenditure and savings all hang on one rule: $\text{Savings}=\text{Income}-\text{Expenditure}$.

• If income is in the ratio $5:4$ to expenditure, set income $=5x$ and expenditure $=4x$; then savings $=5x-4x=x$. A given savings amount fixes $x$, and every other value follows.

• For two people, set incomes $=3x,\ 4x$ and use expenditure $=\text{income}-\text{savings}$ to write the expenditure ratio as an equation, then cross-multiply for $x$.

• For percentage shifts, work in units: new savings $=$ new income $-$ new expenditure, then percent change $=\dfrac{\text{new}-\text{old}}{\text{old}}\times 100$.