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Daily Maths

Daily Maths

Reverse percentages: finding the original

Percentages- Boxes Paper 1 + 2 10 min read

On this page
  1. In this note
  2. Formula bank
  3. Why you can't just take the percentage off
  4. Spotting what the given amount is
  5. Undoing two changes
  6. Ways they test it
  7. Mistakes to avoid
  8. Summary
  9. Practice
  10. Can I do this?
  11. Quiz

In this note

Syllabus 1.13: reverse percentages
  • 1.13Spot a reverse percentage question: you know the amount AFTER the change and want the ORIGINAL.
  • 1.13Find the original by dividing by the multiplier, or with boxes (no calculator).
  • 1.13Find a price before tax, a cost price from a profit or loss, or the whole from a part.

Before you start: percentage increase and decrease (multipliers). 0580: Extended. 4024: everyone.

Formula bank

Reverse percentageLearn it\(\text{original}=\dfrac{\text{new amount}}{\text{multiplier}}\)the multiplier is the one that went FROM the original TO the new amount.Use it when you know the amount after the change.
After an increase of r%Learn itnew = \((100+r)\%\) of the originale.g. a price including 15% tax is 115% of the price before tax.Use it when tax, pay rise, profit.
After a decrease of r%Learn itnew = \((100-r)\%\) of the originale.g. a sale price after 15% off is 85% of the original.Use it when sale, loss, depreciation.
Given the part or the changeLearn it\(\text{whole}=\dfrac{\text{part}}{r/100}\)"$18 is 12% of the price" → 18 ÷ 0.12.Use it when the number given IS the r%.

Why you can't just take the percentage off

A price goes up 20% and is now $60. Taking 20% of $60 off gives $48, but \(48\times1.2=57.60\), not $60. The 20% was of the OLD price, so $60 is 120% of the original, not 100%.

General methodHow to answer any reverse percentage question

  1. Decide what percentage of the original the given amount is: increase → 100 + r; decrease → 100 − r; a part or the change itself → r.
  2. Write it as a multiplier (115% → 1.15, 85% → 0.85).
  3. Divide: original = given amount ÷ multiplier.
  4. Check by going forwards: original × multiplier must give the amount in the question.

Example In a sale prices are reduced by 15%. A jacket now costs $68. Find its original price.

$68 is 85% of the original, so divide by 0.85.

  1. \(100\%-15\%=85\%\) → multiplier 0.85
  2. \(68\div0.85=80\)
  3. check: \(80\times0.85=68\) ✓

Answer $80

No calculator? Use boxes. 85% and 100% are both made of 5% boxes: 85% = 17 boxes = $68, so one box = $4 and 100% = 20 boxes = $80.

85% is 17 boxes worth $68, so one box is $4 and 20 boxes are $80Original: 100% = 20 boxes of 5%85% = $6815% offOne box = 5% = $68 ÷ 17 = $4$4
The box method for $68 = 85%.

Try it yourself

1
After a 10% increase a price is $77. What was it before?
Hint
$77 is 110%.
Show answer

Answer $70

2
A coat costs $42 after a 30% reduction. Original price?
Hint
$42 is 70%.
Show answer

Answer $60

3
$38 is 95% of a number. Find the number.
Show answer

Answer 40

Spotting what the given amount is

The question gives…and wants…The given amount is…
a sale price after r% offthe price before the sale\((100-r)\%\)
a price including r% taxthe price before tax\((100+r)\%\)
a salary after an r% risethe salary before\((100+r)\%\)
a selling price at r% profitthe cost price\((100+r)\%\)
a selling price at r% lossthe cost price\((100-r)\%\)
the change ("the discount was $18, 12%")the original\(r\%\)
a part ("36 students are 45%")the whole\(r\%\)

Example The discount on a bike is $18, which is 12% of the original price. Find the original price.

$18 IS the 12%, so divide by 0.12.

  1. \(18\div0.12=150\)

Answer $150

Undoing two changes

General methodHow to undo two or more changes

  1. Write the multiplier for each change.
  2. Divide by ALL of them: original = final ÷ \((m_1\times m_2)\). The same change n times: ÷ \(m^{n}\).

Example A price went up 10% and then down 20%. It is now $440. What was it at the start?

Divide by \(1.1\times0.8\).

  1. \(1.1\times0.8=0.88\)
  2. \(440\div0.88=500\)

Answer $500

Try it yourself

1
A machine loses 10% of its value each year. After 3 years it is worth $5832. Value when new?
Hint
÷ \(0.9^{3}\).
Show answer

Answer $8000

Ways they test it

Type 1 Price before tax / service

"A TV costs $92 including 15% tax. Find the price before tax." (2 marks)

Spot it "including", "after tax".

  1. Given = 115%.
  2. ÷ 1.15.

For example $92 including 15% tax

  1. \(92\div1.15=80\)

Answer $80

Type 2 Original price before a sale

"After a 25% reduction shoes cost $45. Find the original price."

Spot it "sale price", "reduced to".

  1. Given = 75%.
  2. ÷ 0.75 (or boxes of 25%).

Type 3 Cost price from profit or loss

"A lamp is sold for $162 at a 35% profit. Find the cost price."

Spot it "profit" or "loss" and a selling price.

  1. Profit: given = 135%; loss: given = 65%.
  2. Divide by the multiplier.

For example $162 at 35% profit

  1. \(162\div1.35=120\)

Answer $120

Type 4 The whole from a part

"36 girls are 45% of the year group. How many students?"

Spot it "is r% of".

  1. Given = r%.
  2. ÷ r/100.

For example 36 is 45%

  1. \(36\div0.45=80\)

Answer 80 students

Type 5 Find the tax / charge itself

"A bill of $287.50 includes a 15% service charge. How much is the charge?" (3 marks)

Spot it asks for the extra, not the original.

  1. Find the original first.
  2. Charge = given − original.

For example $287.50 including 15%

  1. \(287.50\div1.15=250\)
  2. \(287.50-250=37.50\)

Answer $37.50

Mistakes to avoid

  • ✗ $60 after +20%: 60 − 20% of 60 = $48✓ \(60\div1.2=50\)The 20% was of the original, not of $60.
  • ✗ Multiplying by the opposite multiplier (× 0.8 to undo × 1.2)✓ divide by 1.2× 0.8 is not the reverse of × 1.2.
  • ✗ Service charge = 15% of $287.50✓ $287.50 − $250 = $37.50The charge is 15% of the bill BEFORE the charge.
  • ✗ Two changes: ÷ 1.3 for +10% then +20%✓ ÷ \((1.1\times1.2)\)Undo each multiplier, never add percentages.

Summary

  • Reverse percentage: you know the amount AFTER the change.
  • Find what % of the original it is (100 + r, 100 − r, or r), then divide by that multiplier.
  • No calculator: boxes (e.g. 5% boxes for 85%).
  • Check by going forwards.

If you see …

  • "original", "before", "cost price"a reverse percentage: divide.
  • "including tax"÷ (1 + tax rate).
  • "the discount was $x, which is r%"÷ r/100.

Practice

Practice: easy, medium, hard

Easy

1
After a 20% pay rise a salary is $1080 a month. What was it before?
Show working
  1. \(1080\div1.2=900\)

Answer $900

2
In a 10% off sale a shirt costs $23.40. Original price?
Show working
  1. \(23.40\div0.9=26\)

Answer $26

3
The discount on a book is $9, which is 15% of the original price. Original price?
Show working
  1. \(9\div0.15=60\)

Answer $60

Medium

1
After a 10% increase the rent of a flat is $352 a week. Rent before the increase?
Guide me

What is it asking? the rent before

What do we know? $352 = 110%

Plan ÷ 1.1

Show working
  1. \(352\div1.1=320\)

Answer $320

2
A car is sold for $9200. This is 80% of the price paid. Find the price paid.
Guide me

What is it asking? the price paid

What do we know? $9200 = 80%

Plan ÷ 0.8

Show working
  1. \(9200\div0.8=11\,500\)

Answer $11 500

3
An item sells for $255 at a 15% loss. Find the cost price.
Hint
A loss: $255 is 85% of the cost.
Guide me

What is it asking? the cost price

What do we know? loss of 15%

Plan ÷ 0.85

Show working
  1. \(255\div0.85=300\)

Answer $300

Hard

1
A restaurant bill of $287.50 includes a 15% service charge. Calculate the service charge.
Hint
The charge is 15% of the bill BEFORE the charge.
Guide me

What is it asking? the service charge

What do we know? $287.50 = 115%

Plan find 100% first, then subtract

Show working
  1. \(287.50\div1.15=250\)
  2. \(287.50-250=37.50\)

Answer $37.50

PartAnswer / workingMarksGuidance
\(287.50\div1.15\) [= 250]M1
\(287.50-250\)M1
37.50A1
Total: 3 marks
2
A phone sells for $4650 at a 24% profit. Find (a) the cost price (b) the profit.
Guide me

What is it asking? cost and profit

What do we know? $4650 = 124% of the cost

Plan ÷ 1.24, then subtract

Show working
  1. \(4650\div1.24=3750\)
  2. \(4650-3750=900\)

Answer (a) $3750 (b) $900

PartAnswer / workingMarksGuidance
(a)\(4650\div1.24\)M1
3750A1
(b)900B1
Total: 3 marks
3
A price went up 20% then up 4%. It is now $624. What was it at first?
Guide me

What is it asking? the first price

What do we know? two increases

Plan ÷ \((1.2\times1.04)\)

Show working
  1. \(1.2\times1.04=1.248\)
  2. \(624\div1.248=500\)

Answer $500

PartAnswer / workingMarksGuidance
\(624\div(1.2\times1.04)\)M1
500A1
Total: 2 marks

Can I do this?

Anything unticked: revise that section first. All ticked: take the quiz.

Quiz

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