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Find the original amount from a percentage

Start a set of 8
Step 3 of 9

Recover the whole

First identify what percentage the known amount represents. Divide by that percentage as a decimal. You can also use the unitary method from the previous section

Known percentage part

Here T is a fundraising target in dollars, and $78 is 30% of that target

\[\begin{aligned}0.30T&=78\\T&=78\div0.30\\&=260\end{aligned}\]

After a decrease

Subtract the decrease from the original one hundred percent. The remaining share is the multiplier

\[\begin{aligned}100\%-24\%&=76\%\\\text{Multiplier}&=0.76\end{aligned}\]

After an increase

Add the increase to the original one hundred percent

\[\begin{aligned}100\%+18\%&=118\%\\\text{Multiplier}&=1.18\end{aligned}\]

A named comparison

In a comparison using 'more than' or 'less than', the amount named after 'than' is the reference. Use the corresponding increase or decrease multiplier

Find a remainder last

Recover the whole before subtracting the known part. Keep both amounts in the same unit

\[\text{Remainder}=\text{whole}-\text{known part}\]
Year 9, Chapter 1, 3 difficulty levels, Calculator allowedBuild a worksheet

Reverse a percentage relationship

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

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

Common mistakes

Using the changed amount as the base
✕\(\displaystyle \text{Original}=\$112-0.40\times\$112\)✓\(\displaystyle \text{Original}=\$112\div1.40\)
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
✕\(\displaystyle \text{Still needed}=\$126\div0.35\)✓\(\displaystyle \text{Still needed}=\$126\div0.35-\$126\)
Recovering the full target is an intermediate step. Subtract the amount already received to answer the remainder question

Difficulty levels

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

Questions about this topic

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 9 Mathematics, chapter 1: Number and financial mathematics

In this chapter

Evaluate an integer expressionFactors, multiples, HCF, LCM and primesRound to decimal places or significant figuresEstimate an expression using rounded inputsConvert between fractions, mixed numbers and decimalsClassify real numbers and compare exact valuesAdd, subtract, multiply and divide fractions and mixed numbersWrite a ratio from countsWrite ratios in simplest formComplete equivalent ratiosUse a constant rateDivide a quantity in a given ratioDecide which option is the better valueConvert between fractions, decimals and percentagesFind a percentage of a quantityExpress one quantity as a percentage of another1F Percentages and money: Find the original amount from a percentageIncrease or decrease a quantity by a percentageFind percentage changePercentage error against a reference value1G Percentage change and percentage error: Find the original amount from a percentageProfit, mark-up and discountIncome and taxSimple interestCompound interest (optional extension)