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neurologyYou need to know

• A percentage means ‘per hundred’, so `p\% = \frac{p}{100}` of a quantity.
• To convert a percentage to a decimal, divide by 100 (for example, `7.5\% = 0.075`).
• To convert a decimal to a percentage, multiply by 100 (for example, `0.36 = 36\%`).
• To convert a fraction to a percentage, write it as a decimal or an equivalent fraction out of 100 (for example, `\frac{3}{8}=0.375=37.5\%`).
• Percentages greater than 100% represent increases beyond the original amount (for example, 150% is 1.5 times the original).
• A percentage of an amount can be found by multiplying the amount by the decimal form of the percentage (for example, 12% of £80 is `0.12 \times 80`).
• Decimal percentages (for example, 2.5%) are handled the same way: convert to a decimal multiplier (2.5% = 0.025) before multiplying.
• ‘One quantity as a percentage of another’ is calculated using `\frac{\text{part}}{\text{whole}}\times 100\%`.
• Increasing by `p\%` is equivalent to multiplying by the multiplier `1+\frac{p}{100}`.
• Decreasing by `p\%` is equivalent to multiplying by the multiplier `1-\frac{p}{100}`.
• A multiplier greater than 1 produces an increase and a multiplier between 0 and 1 produces a decrease.
• Percentage change compares a change to the original value using `\frac{\text{new} - \text{original}}{\text{original}}\times 100\%`.
• Simple interest adds the same amount each time period, calculated by `I = P\times r \times t`, where `P` is principal, `r` is the interest rate per time period (as a decimal), and `t` is the number of time periods.
• The total after simple interest is `A = P + I` (equivalently `A = P(1+rt)`).
• Compound interest increases by the same percentage each time period, so the total is `A = P(1+r)^n`.
• Compound depreciation decreases by the same percentage each time period, so the total is `A = P(1-r)^n`.
• A reverse percentage problem works backwards by dividing by the multiplier (for example, if a price after a 20% increase is £60, the original is `60 \div 1.2`).
• For a percentage decrease, the reverse multiplier is `1-\frac{p}{100}` (for example, after 15% off, the multiplier is 0.85, so original `= \text{sale price} \div 0.85`).

• Convert between fractions, decimals, and percentages by using `\times 100`, `\div 100`, and fraction-to-decimal division, keeping place value accurate.
• Calculate any percentage (including decimal percentages) of an amount by converting the percentage to a decimal and multiplying, giving answers to an appropriate level of accuracy for the context (for example, money to 2 decimal places).
• Express one quantity as a percentage of another by forming the fraction `\frac{\text{part}}{\text{whole}}`, converting to a percentage, and stating the result with the correct units/context.
• Increase or decrease an amount by a given percentage by selecting the correct multiplier `1\pm \frac{p}{100}` and multiplying once, rather than doing separate ‘find p% then add/subtract’ steps.
• Use multipliers to solve multi-step percentage problems (for example, successive discounts) by multiplying the multipliers together before applying them to the original value.
• Calculate percentage change by computing the difference, dividing by the original amount, multiplying by 100, and stating whether it is an increase or decrease.
• Calculate simple interest totals over time by using `I = Prt` and `A = P + I`, ensuring the rate matches the time units (for example, per year).
• Calculate compound interest or depreciation totals by using `A = P(1\pm r)^n`, identifying `r` as a decimal rate per period and `n` as the number of periods.
• Solve reverse percentage problems by writing the forward multiplier, then dividing the final amount by this multiplier to find the original amount.