Find the original amount from a percentage lesson | General Mathematics (Year 11, VCE Units 1 & 2, Percentages and ratios, Unitary method)

Reverse a percentage relationship

The known amount is a multiple of the unknown whole or original amount. Identify that multiplier before dividing

\begin{gathered}\text{Original}\\=\frac{\text{known amount}}{\text{multiplier}}\end{gathered}

A known part Use the given percentage directly when it describes the known part of the whole $\displaystyle m=\frac{p}{100}$
An increase Include the original whole and the added share $\displaystyle m=1+\frac{p}{100}$
A decrease Use the share that remains after the decrease $\displaystyle m=1-\frac{p}{100}$
Check forwards Applying the multiplier to the recovered original must return the known amount $\displaystyle 0.76\times\$120=\$91.20$
Unitary alternative For a known percentage part, first find one percent, then find one hundred percent $\displaystyle \begin{gathered}1\%\text{ amount}\\=\$78\div30=\$2.60\\100\%\text{ amount}\\=\$2.60\times100=\$260\end{gathered}$

Find the original workshop fee and the amount of the increase

\begin{gathered}\text{Increase}=16\%\\\text{New fee}=\$104.40\end{gathered}

$\displaystyle 100\%+16\%=116\%$The new fee includes the original fee and its increase
$\displaystyle 116\%=1.16$Convert the new fee's share to a multiplier
$\displaystyle \begin{gathered}\text{Original fee}\\=\$104.40\div1.16\end{gathered}$Division reverses the multiplication that produced the new fee
$\displaystyle =\$90$This is the original fee
$\displaystyle \text{Increase}=\$104.40-\$90$Subtract the original fee from the new fee to find the added amount
$\displaystyle =\$14.40$This is the amount of the increase

Using the changed amount as the base

✕$\displaystyle \begin{gathered}\text{Original}\\=\$112-0.40\times\$112\end{gathered}$✓$\displaystyle \text{Original}=\$112\div1.40$

The increase was a percentage of the original, so subtracting the same percentage of the changed amount removes too much

Dividing by the discount removed

✕$\displaystyle \text{Original}=\$91.20\div0.24$✓$\displaystyle \text{Original}=\$91.20\div0.76$

The sale price is the share left after the discount, not the amount removed

Stopping at the whole when asked for the remainder

✕$\displaystyle \text{Still needed}=\$126\div0.35$✓$\displaystyle \begin{gathered}\text{Still needed}\\=\$126\div0.35-\$126\end{gathered}$

Recovering the full target is an intermediate step. Subtract the amount already received to answer the remainder question

introRecover a whole from a known percentage part, then find a remainder

standardReverse a single increase or decrease by choosing its multiplier

challengeInterpret comparative statements and find an original amount and its change

Can I use the unitary method?+

Yes. Find the amount represented by one percent, then scale to the percentage you need

How can I check a recovered original?+

Apply the stated percentage relationship to it. You should obtain the known amount

Why does taking off the same percentage fail to undo an increase?+

The two percentages use different base amounts

Can I use a calculator?+

A calculator is allowed unless a question is marked no calculator

Which year level and chapter is this?+

Year 11, VCE Units 1 & 2, General Mathematics, chapter 1: Percentages and ratios