ReasoningCampus · Online SHL preparation
Reverse Percentages: Our SHL Method and 6 Practice Questions
A price falls by 15%, and you are given the price after the discount. How do you get back to the original? Adding 15% to the new price feels reasonable, but it uses the wrong base. In this ReasoningCampus SHL numerical preparation lesson, we explain reverse percentages from the beginning and practise the difference between a change, a multiplier and a percentage point.
Numerical reasoning often asks you to reconstruct information rather than calculate forwards. A table may show this year’s revenue and its growth rate, leaving you to find last year’s value. The useful question is “What fraction of the original am I looking at?” Once that is clear, the arithmetic is usually short.
View full preparationRecognise the percentage question before choosing a formula
Our numerical preparation separates the following relationships because they look similar in a chart but require different bases. This is a ReasoningCampus teaching map for interpreting numerical information, not a claim that SHL publishes a fixed exhaustive list of percentage item types.
| Task | Relationship | What to check |
|---|---|---|
| Find a new value | New = original × (1 ± r) | Write r as a decimal: 15% = 0.15. |
| Find the original | Original = new ÷ (1 ± r) | Reverse the multiplier, not the percentage sign. |
| Find relative change | (New − original) ÷ original × 100% | The original is the denominator. |
| Find a whole from a part | Whole = part ÷ decimal share | The share refers to the whole. |
| Combine successive changes | Final = original × first multiplier × second multiplier | Each change acts on the value then in place. |
| Compare two percentage rates | Percentage-point difference = new rate − old rate | A point difference is not a relative percentage change. |
Try a quick sense-check: after a discount, the original must be larger than the final price. If your reverse calculation makes it smaller, revisit the equation before selecting an answer. After a rise, the direction is reversed: the original should be smaller.
Reverse percentages: the ReasoningCampus multiplier method
Write “original × multiplier = final” before inserting numbers. With two changes, write both multipliers in that same equation. We then ask you to check the recovered amount by applying the changes forwards. This return check catches a division in the wrong direction and shows why subtracting the same percentage does not undo a percentage increase.
When a value rises by 20%, the final amount is 120% of the original: original × 1.20. To recover the original, divide the final amount by 1.20. Subtracting 20% of the final value uses a different base.
Original = final ÷ multiplier.
Increase by p%: multiplier = 1 + p/100.
Decrease by p%: multiplier = 1 − p/100.
- Label the known amount. Is it the original, the final value, a change or a share?
- Translate the wording. “After a 15% discount” means 85% remains, so use 0.85.
- Choose the direction. Multiply to move forward. Divide to reverse a known change.
- Check by going forward again. Apply the stated change to your proposed original and confirm that it returns the given final amount.
A share of a total is a different question
If 120 is 40% of a total, the total is 120 ÷ 0.40 = 300. Do not divide by 1.40: no 40% increase has been stated. The words “is 40% of” and “has increased by 40%” describe different relationships.
| Statement | Equation | Unknown amount |
|---|---|---|
| 240 after a 20% increase | Original × 1.20 = 240 | 200 |
| 240 after a 20% decrease | Original × 0.80 = 240 | 300 |
| 240 is 20% of the total | Total × 0.20 = 240 | 1,200 |
Percentage multipliers: reverse, then verify forwards
If an item costs 80 after a 20% discount, record “original × 0.80 = 80”. The original is 100, and the forward check is 100 × 0.80 = 80. Adding 20% to 80 produces 96, which fails that check. Explain the changing base in words before attempting the two-discount exercise below.
Successive changes: multiply, then compare with the original
Example 1: why equal rises and falls do not cancel
A price starts at 100. A 20% reduction makes it 100 × 0.80 = 80. A subsequent 20% increase gives 80 × 1.20 = 96. The increase is calculated on 80, not on the original 100.
The combined multiplier is 0.80 × 1.20 = 0.96. The final value is 96% of the original, so the net change is a 4% decrease. To restore 80 to 100, the required increase is 20 ÷ 80 = 25%.
Example 2: recover the original after two discounts
A product costs 144 after discounts of 20% and then 10%. The retained proportions are 0.80 and 0.90. Together they give 0.72, so the original price is 144 ÷ 0.72 = 200. Check: 200 → 160 → 144.
Adding the discounts to make 30% would imply a multiplier of 0.70 and a different result. The second discount applies to the already reduced amount.
Example 3: percentage points versus percentage change
A rate moves from 18% to 24%. The absolute difference is 6 percentage points. Relative to the original rate, the change is (24 − 18) ÷ 18 × 100 = 33⅓%. Both describe the same movement but answer different questions.
If a statement says “the rate rose by 6%”, do not automatically interpret that as six percentage points. A 6% relative rise from 18% gives 18% × 1.06 = 19.08%.
When order matters
Two percentage multipliers commute if both changes apply to the entire running amount without intermediate rounding or other rules. For example, ×1.10 then ×0.90 gives the same final value as the reverse order. A fixed fee, threshold, cap or rounding step can change that. Adding 10 before a 20% discount is not the same as adding 10 afterwards.
Final check: preserve full precision until the last step unless the question explicitly tells you to round earlier.
Try a forward or reverse percentage calculation
This tool applies two percentage changes to the whole running amount, without fees, caps or intermediate rounding. Use negative numbers for decreases. The tool accepts amounts from 0 to 1,000,000,000 and changes from −99.99% to +1,000%, with up to two decimal places. It excludes a 100% decrease because that cannot be reversed. Displayed values are rounded to at most 12 significant digits; the calculation uses the unrounded values.
Reverse percentages: six explained practice questions
Try all six questions before opening the worked answers below. Submit your choices to see your score and individual feedback. You can change your choices before submission, or go straight to the answers and explanations for self-study.
This is an original ReasoningCampus learning check. One point is awarded per correct answer. The total is not an SHL score, a percentile, a pass prediction or a personality profile.
Answers and complete worked explanations
Open any question below to study the correct answer, the alternatives, the method and the common trap. These explanations are available without completing the quiz.
Review the reasoning in any missed or uncertain answer. The score describes this small set only.
Question 1: A discounted amount
Correct answer: B. 180
After a 15% discount, the customer pays 85% of the original price. If that original is x, the relationship is 0.85x = 153. Divide by 0.85 to obtain x = 180, answer B. Check it forwards: 15% of 180 is 27 and 180 − 27 = 153.
A adds 15% of 153, giving 175.95. That percentage is taken from the smaller final price, so it cannot restore the original discount. C takes another 15% off 153, giving 130.05, which moves in the wrong direction. D adds 15 currency units rather than reversing a 15% reduction. A quick preliminary check helps: the original must exceed 153, but that alone does not distinguish the three larger options. The multiplier equation does.
Question 2: Equal percentage changes
Correct answer: C. 99
Follow the index through the two changes. The rise takes it from 100 to 100 × 1.10 = 110. The subsequent fall is 10% of 110, which is 11, so the final index is 110 − 11 = 99. Equivalently, 100 × 1.10 × 0.90 = 99. C is correct.
A assumes equal percentage rates cancel, overlooking their different bases. B applies only the decrease to the original 100. D stops at the intermediate value after the increase. For any increase and then decrease by the same proportion r, the combined multiplier is (1 + r)(1 − r) = 1 − r². Here r = 0.10, so the net multiplier is 0.99: a 1% loss, not no change.
Question 3: Recovering a value after growth
Correct answer: A. 128
The final 160 represents 125% of the starting quantity. Set up 1.25x = 160, then divide: x = 160 ÷ 1.25 = 128. A is correct. To check without repeating the division, one quarter of 128 is 32 and 128 + 32 returns to 160.
B subtracts 25% of the final 160, giving 120. That removes 40, whereas the original increase was 32. C adds a further 25% to 160, producing 200. D subtracts 25 units rather than handling a percentage. These alternatives show why choosing the right base matters more than doing the arithmetic quickly. First label the 160 as the final value; then decide which operation reverses the multiplier that created it.
Question 4: A change in a rate
Correct answer: D. 25%
There are two valid descriptions of this change, but only one answers the question. The difference is 10% − 8% = 2 percentage points. The relative increase compares that difference with the original rate: 2 ÷ 8 × 100 = 25%. D is therefore correct.
A confuses percentage points with percentage change. B uses the new rate as the denominator, calculating 2 ÷ 10 = 20%, which answers a different question. C calculates 10 ÷ 8 × 100 = 125%: the new rate is 125% of the old, meaning it is 25% higher. You can check the answer by increasing 8% by a quarter of itself: 8% + 2% = 10%. Keep the labels “points”, “of” and “increase” distinct.
Question 5: A part of a total
Correct answer: B. 300
Here 120 is a part of the organisation, not a value after growth. Let the total be T. The statement means 0.40T = 120, so T = 120 ÷ 0.40 = 300. B is correct. Another route is to find 10% first: if 40% is 120, then 10% is 30 and 100% is 300.
A increases 120 by 40%, obtaining 168, but no increase is described. C finds 40% of the part rather than recovering the whole. D fails a forward check: 40% of 200 is 80, not 120. Before calculating, ask whether the known number is the whole, a part or a changed value. The same percentage can require multiplication or division depending on that relationship.
Question 6: Two discounts
Correct answer: C. 288
Apply each discount to the amount that exists at that stage. After the first reduction, 400 × 0.80 = 320 remains. The second discount removes 10% of 320, or 32, leaving 288. The one-line calculation is 400 × 0.80 × 0.90 = 288, answer C.
A subtracts 30% from the original price, treating both discounts as if they shared the same base. B reports the intermediate 320 and omits the second discount. D applies only the 10% discount to 400. The combined multiplier is 0.72, so the overall reduction is 28%, not 30%. Keep intermediate amounts in your working, but underline the requested final amount before selecting an option.
ReasoningCampus SHL numerical preparation: percentages in context
ReasoningCampus provides SHL numerical preparation with theory, exercises and worked solutions. This lesson demonstrates how we connect the calculation to the wording. Use the wider preparation materials to practise recognising the same relationship when it appears in a business table or a multi-step problem.
Choose your next practice step
- You used the final amount as the original base. Rewrite the equation with an unknown original amount and solve it before checking the choices.
- You added successive percentage changes. Multiply the factors and apply them forwards to verify the recovered starting value.
- You confused a rate change with percentage points. Return to the worked rate example and label the two quantities separately in your practice notes.
Explore the ReasoningCampus SHL preparation materials and choose the areas named in your assessment invitation.
Reverse percentages: practical FAQs
Why can’t I subtract the same percentage to reverse an increase?
Because the increase was measured against the original amount. To reverse +20%, divide the final value by 1.20; subtracting 20% of the final value uses a larger base.
Can I add two percentage increases together?
Not for successive changes to a running amount. A 10% rise followed by another 10% rise gives ×1.10 ×1.10 = ×1.21, or 21% growth.
How do I know whether the question asks for percentage points?
Look for a difference between two rates. If it asks for the relative percentage increase, divide that difference by the original rate. Keep the unit in your final answer.
What if the original value is zero?
The usual relative-change formula divides by the original and is undefined at zero. Report the absolute change or follow a different explicitly supplied measure.
Does this tool include VAT rules, fees or currency conversion?
No. It demonstrates pure percentage changes. If an exercise adds fees, rounding instructions, taxes or exchange rates, model those operations separately and in the stated order.
What should I practise after getting these right?
Use unfamiliar charts and word problems where the original base must first be identified. ReasoningCampus numerical preparation develops this interpretation step alongside the calculation.
Official sources and how this lesson was prepared
The SHL references below provide the assessment context. The teaching method, fictional examples, diagrams, tools and explained practice answers are ReasoningCampus learning material; they are not official SHL questions or scoring rules.
AI assistance was used in research, drafting and implementation. Calculations, logic and the page tools were checked during preparation. These lessons are independent educational material; the methods are not SHL scoring rules. Read about ReasoningCampus or send a correction or question.
