Fractions, decimals, percentages and ratioIB MYP Maths Standard: Revision notes

A percentage is a fraction out of $100$. To convert: fraction to decimal, divide the top by the bottom ($\frac{3}{8}=3\div8=0.375$); decimal to percentage, multiply by $100$ ($0.375=37.5\%$); percentage to fraction, put it over $100$ and simplify ($35\%=\frac{35}{100}=\frac{7}{20}$). Some fractions give recurring decimals: $\frac13=0.\dot{3}=33.\dot{3}\%$. Learn the common set: $\frac12=0.5=50\%$, $\frac14=0.25=25\%$, $\frac15=0.2=20\%$.

A ratio compares quantities of the same kind, written $a:b$, and can be simplified by dividing both parts by their common factor: $1080:360=3:1$. The order matters: walkers to non-walkers $=7:13$ is not $13:7$. If $\frac{7}{20}$ walk, the others are $\frac{13}{20}$, so the ratio is $7:13$. To share in a ratio, add the parts, find one part, then multiply: share $56$ in the ratio $3:5$: $3+5=8$ parts, one part is $7$, shares are $21$ and $35$. A rate compares different units, such as km per hour, AED per kg or dollars per euro.

To find a percentage of an amount, write it as a decimal and multiply: $12\%$ of $850=0.12\times850=102$. Use a multiplier: an increase of $12\%$ has multiplier $1.12$ and a decrease of $12\%$ has multiplier $0.88$ (that is $100\%-12\%=88\%$). So the sale price is 850\times0.88=\748$. To find a percentage change, use $\text{percentage change}=\frac{\text{change}}{\text{original}}\times100.$ Always divide by the original amount.

For a change repeated over several periods, multiply the multipliers. A rise of $6\%$ then a fall of $4\%$ has overall multiplier $1.06\times0.96=1.0176$, a net rise of $1.76\%$ (not $2\%$). Compound interest of $r\%$ for $n$ years uses $\text{amount}\times\left(1+\frac{r}{100}\right)^n$: \8000$at$2.5%$for$2$years is$8000\times1.025^2=$8405$. A rise of$12%$followed by a fall of$12%$ does not return to the start, because each percentage is of a different amount.

A reverse percentage problem gives the amount after a change and asks for the original. Find the percentage the new amount represents, then divide. After a $20\%$ discount the price is $80\%$ of the original. If $80\%=1440$, then $1\%=18$ and $100\%=1800$. Equivalently, original $=1440\div0.8=1800$. Check by applying the change forwards: $1800\times0.8=1440$.

Real situations (sales, savings, profit, inflation) need you to choose the right method, show clear steps and judge whether the answer makes sense. Profit as a percentage of cost $=\frac{\text{profit}}{\text{cost}}\times100$. If savings grow at $2.5\%$ a year but prices rise at $3\%$ a year, the savings lose buying power. Compare using the same multiplier method: 8000\times1.03^5=\9274.19$is needed to keep up, but$8000\times1.025^5=$9051.27$ is available. Always state a conclusion in words.