E1.12 - Reverse percentages (Extended)

E1.12 - Reverse percentages (Extended)

A sale price is $144 after a 20% reduction. Taking another 20% off cannot reveal the old price: the old price must be larger. E1.12 uses division by percentage multipliers to recover original amounts, including cost prices and values before successive changes.

Recovering the original

Which amount represents 100%? In a reverse-percentage question, the given final amount often represents more or less than 100% of an unknown starting amount. Rebuild the forward calculation before reversing it.

Reversing a percentage multiplier

final=original×qoriginal=finalq\begin{aligned} \text{final}&=\text{original}\times q\\ \text{original}&=\frac{\text{final}}q \end{aligned}

Here qq is the multiplier for the stated change and must be non-zero. A 100% reduction loses all information about the original amount, so it cannot be reversed from the final amount alone.

Not in the supplied formula list: learn the relationship.

Dividing reverses multiplication. After a 20% reduction, the final amount is 80% of the original, so dividing by 0.800.80 recovers 100%. Adding 20% of the smaller final amount uses the wrong base.

Worked example 1: reversing an increase

Question. A ticket costs $144 after a 20% price increase. Calculate its original price.

Plan. The final price is 120% of the original, so divide by 1.201.20.

Original×1.20=144Original=144÷1.20=120\begin{aligned} \text{Original}\times1.20&=144\\ \text{Original}&=144\div1.20=120 \end{aligned}

Answer. $120.00.

Check. 20% of 120is120 is 24. Adding gives $144, the stated final price.

Keep the final amount and percentage the same, but change an increase to a reduction. Now the original must be above the final price.

Worked example 2: reversing a decrease

Question. A ticket costs $144 after a 20% reduction. Calculate its original price.

Plan. The final price is 80% of the original, so divide by 0.800.80.

Original×0.80=144Original=144÷0.80=180\begin{aligned} \text{Original}\times0.80&=144\\ \text{Original}&=144\div0.80=180 \end{aligned}

Answer. $180.00.

Check. 20% of 180is180 is 36. Subtracting gives 144.Incontrast,144. In contrast, 144\times1.20=172.80wouldreduceby20would reduce by 20% to138.24, not $144.

Checking forward tests both the recovered amount and the chosen multiplier. A plausible-looking original can still fail to produce the stated final amount.

Watch out

A 20% decrease is undone by dividing by 0.800.80, which is multiplication by 1.251.25. The required reverse increase is therefore 25%, because it acts on the smaller amount.

Write the forward multiplier before finding the next original price.

Profit and selling price

A percentage profit is normally based on cost price. If the profit is 15% of cost, the selling price is 115% of cost. When the selling price is known, this is a reverse-percentage problem.

Selling-price multiplier

For a profit of r%r\% of cost price,

selling price=cost price×(1+r100)\text{selling price}=\text{cost price}\times\left(1+\frac r{100}\right)

Recover the cost price by dividing the selling price by this multiplier. The profit amount is then selling price minus cost price.

Keep the words “of cost” in mind. Profit as a percentage of selling price would be a different comparison and must be explicitly stated if intended.

Worked example 3: recovering the trader's cost

Question. A trader sells a cabinet for $230, making a profit of 15% of its cost price. Calculate the cost price and profit amount.

Plan. Treat $230 as 115% of cost, then subtract the recovered cost from the sale.

Cost price=230÷1.15=200Profit=230−200=30\begin{aligned} \text{Cost price}&=230\div1.15=200\\ \text{Profit}&=230-200=30 \end{aligned}

Answer. Cost price 200;profit200; profit 30.

Check. 30÷200×100%=15%30\div200\times100\%=15\%, as stated.

The selling price stays the same, but the profit percentage rises. A larger share of the sale is profit, so the original cost must be lower.

Worked example 4: changing the profit percentage

Question. A trader sells another cabinet for $230, making a profit of 25% of its cost price. Calculate the cost price and profit amount.

Plan. Divide by 1.251.25, because the sale represents 125% of cost.

Cost price=230÷1.25=184Profit=230−184=46\begin{aligned} \text{Cost price}&=230\div1.25=184\\ \text{Profit}&=230-184=46 \end{aligned}

Answer. Cost price 184;profit184; profit 46.

Check. A quarter of 184 is 46, so the profit is exactly 25% of cost. Adding 184+46184+46 gives 230.

With the same selling price, the larger percentage profit corresponds to the smaller cost price. This agrees with the direction of both recovered answers.

Watch out

230×0.75=172.50230\times0.75=172.50 does not reverse a 25% profit. It subtracts 25% of the selling price, even though the stated profit is based on cost. Recover the cost by division first.

Identify the percentage represented by the full selling price.

Successive changes

To recover an amount before two changes, undo both multipliers. Work backwards through the story, or divide by the product of the multipliers. Both routes reverse the same forward calculation.

Problem-solving

Write a forward chain from the unknown original to the known final amount. Then reverse its arrows with division. This keeps the changing bases visible.

The first example has two rises. Each must be reversed even though only one final amount is given.

Worked example 5: reversing two increases

Question. A membership fee increases by 10%, then increases by a further 20%. The final fee is $264. Calculate the original fee.

Plan. Undo the 20% rise first, then undo the 10% rise.

Before the second rise=264÷1.20=220Original fee=220÷1.10=200\begin{aligned} \text{Before the second rise}&=264\div1.20=220\\ \text{Original fee}&=220\div1.10=200 \end{aligned}

Equivalently, 264÷(1.10×1.20)=264÷1.32=200264\div(1.10\times1.20)=264\div1.32=200.

Answer. $200.00.

Check. 200×1.10=220200\times1.10=220, then 220×1.20=264220\times1.20=264.

Change the second rise to a reduction. Dividing by its multiplier now makes the amount larger during the first backward step.

Worked example 6: reversing a rise and a reduction

Question. A membership fee increases by 10%, then is reduced by 20%. The final fee is $264. Calculate the original fee.

Plan. Undo the reduction by dividing by 0.800.80, then undo the increase by dividing by 1.101.10.

Before the reduction=264÷0.80=330Original fee=330÷1.10=300\begin{aligned} \text{Before the reduction}&=264\div0.80=330\\ \text{Original fee}&=330\div1.10=300 \end{aligned}

Answer. $300.00.

Check. The forward calculation is 300×1.10×0.80=264300\times1.10\times0.80=264. Its overall multiplier is 0.880.88, so the final amount should be below the original.

Dividing by the combined multiplier is shorter, while reversing the stages individually makes each intermediate amount visible. Choose either route and keep every stated change.

Watch out

Do not add the two percentage changes and reverse that total. Two rises of 10% and 20% give multiplier 1.321.32, rather than 1.301.30. A rise and reduction use their own multipliers, rather than the difference between their percentages.

For the opening sale price, 144÷0.80=180144\div0.80=180. Checking the forward calculation confirms the recovered original rather than relying on its size alone.

Quick Recap

  • Identify which amount represents 100%.
  • Divide the final amount by the forward multiplier to recover the original.
  • Percentage profit normally uses cost price as its base.
  • Reverse every multiplier in a sequence and check by going forwards.

A 144salepriceaftera20144 sale price after a 20% reduction comes from an original price of 180.