MathsICSE Class 10 · Goods and Services Tax (GST)

Inverse GST Problems: Finding Price, Rate and Discount from the Amount Paid (ICSE Class 10)

Updated 15 October 2026 · Fact-checked

In an inverse GST problem you know the amount paid including tax and must work back. The taxable value is V = 100A/(100 + r), where A is the amount paid and r% the GST rate. Never subtract r% of A. From V you can then find the GST, the discount or the marked price.

The amount paid is 112 parts, so divide by 112 and multiply by 100; taking 12% of the amount paid is wrong.

Illustration drawn for QuestionBuddy.

Understand Inverse GST Problems

In a forward GST question you start with a price and add tax. In an inverse question you are told the result, usually the amount paid including GST, and must recover something earlier in the chain:

$Marked price M \to Taxable value V \to Amount paid A .$

The CISCE syllabus says 'problems based on corresponding inverse cases are also included', and they are easy to get wrong, because the tempting shortcut is wrong.

The key relation. Since $A = V (1 + \frac{r}{100}) = \frac{( 100 + r ) V}{100}$ ,

$V = \frac{100 \times A}{100 + r} and GST = A - V = \frac{r \times A}{100 + r} .$

So if ₹2,240 includes 12% GST, $V = \frac{100 \times 2240}{112} = 2000$ and the GST is ₹240. Taking 12% of ₹2,240 (₹268.80) is wrong, because the tax was 12% of ₹2,000, not of ₹2,240. Think of the amount paid as 112 parts, of which 100 parts are the price and 12 parts are the tax.

The four inverse questions.

1. Price before tax: $V = \frac{100 A}{100 + r}$ .
2. The GST rate: find the GST amount ( $A - V$ , or $2 \times$ CGST) and the taxable value, then $r = \frac{GST}{V} \times 100$ . If a discount is involved, use the taxable value after the discount.
3. The discount: find $V$ from $A$ , then discount $= M - V$ and $d = \frac{M - V}{M} \times 100$ .
4. The marked price: find $V$ from $A$ , then $M = \frac{100 V}{100 - d}$ for a percentage discount, or $M = V + k$ for a flat discount of ₹ $k$ .

Two unknown items in a bill. Sometimes a bill total and the total GST are given for items at two different rates. Let the taxable values be $x$ and $y$ : then $x + y = bill - GST$ and $\frac{r _{1} x}{100} + \frac{r _{2} y}{100} = GST$ , a pair of simple linear equations.

Always check forward. An inverse answer can be tested in one line: put your price back in, apply the discount and the tax, and you must land on the amount given.

Inverse GST Problems: formulas to remember

Taxable value from the amount paid: $V = \frac{100 \times A}{100 + r}$; $A$ = amount paid including GST, $r %$ = GST rate.
GST contained in the amount paid: $GST = A - V = \frac{r \times A}{100 + r}$; For 18% GST the tax is $\frac{18}{118}$ of the amount paid.
Rate of GST: $r = \frac{GST}{V} \times 100$; Use the full GST: double a CGST or SGST amount.
Discount percentage: $d = \frac{M - V}{M} \times 100$; $V$ found from the amount paid first.
Marked price: $M = \frac{100 \times V}{100 - d}$; For a flat discount of ₹ $k$ : $M = V + k$ .
Change in amount when the rate changes: $Δ A = \frac{( r _{1} - r _{2} )}{100} \times V$; The taxable value is the same; only the tax changes.

How to solve Inverse GST Problems questions

Work backwards along the chain amount paid → taxable value → marked price, one link at a time.

1Identify what the given amount includes: GST only, or GST after a discount.
2If a CGST or SGST rate or amount is given, double it to get the full GST.
3Find the taxable value: $V = \frac{100 A}{100 + r}$ , or $V = A - GST$ when the GST amount is known.
4For the rate: $r = \frac{GST}{V} \times 100$ .
5For the discount: $M - V$ , then as a percentage of $M$ . For the marked price: $M = \frac{100 V}{100 - d}$ .
6For two items at different rates, form two equations: one for the taxable values and one for the GST.
7Check forward: apply discount and tax to your answer and confirm you get the amount given.

Inverse GST Problems in real life

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Price before tax from a GST-inclusive receipt

Suppose your supermarket receipt shows a total of ₹1,344 for some kitchen items, 'inclusive of GST at 12%'.

How much of the ₹1,344 was the price of the goods and how much was tax?

Try it first, then show the solutionSolution

Taxable value $= \frac{100 \times 1344}{112} = 1200$ .
GST $= 1344 - 1200 = 144$ (check: 12% of 1200 = 144).

Answer: ₹1,200 for the goods and ₹144 GST.

Memory hook: Out of every ₹112 paid at 12% GST, ₹100 is price and ₹12 is tax.

Rate per unit before tax

Imagine your family books a hotel room for 3 nights and is charged ₹13,440 in all, including GST at 12%. The family's budget was ₹4,000 per night before tax.

Find the room rate per night before GST. Was it within the budget?

Try it first, then show the solutionSolution

Taxable value for 3 nights $= \frac{100 \times 13440}{112} = 12000$ .
Per night $= 12000 \div 3 = 4000$ .

Answer: ₹4,000 per night before tax, exactly the budget.

Memory hook: Strip the tax first, then share it out per night.

Tricks to remember Inverse GST Problems

Short memory aids. Say each one aloud once, then use it on a question the same day.

100 parts price, r parts tax
At $r %$ GST, the amount paid is $100 + r$ equal parts: $100$ are price and $r$ are tax. Divide the amount into $100 + r$ parts.
Use it when: Finding $V$ or the GST from an inclusive amount.
Divide, don't subtract
To remove 18% GST, divide by 1.18. Subtracting 18% of the total removes too much.
Use it when: Every inverse question; this is the classic trap.
Back along the arrows
M → V → A forward; A → V → M backward. Never jump from A to M in one step without writing V.
Use it when: Marked price and discount questions.
Check forward in one line
Your $M$ × (1 − d/100) × (1 + r/100) must equal the given $A$ .
Use it when: Catching errors before you move on.

Inverse GST Problems: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: GST inside an inclusive price · 1 mark · MCQBoard level

An article costs ₹2,240 including GST at 12%. The GST paid is:
(a) ₹268.80
(b) ₹240
(c) ₹224
(d) ₹120

Try it first, then show the solutionSolution

Taxable value $= \frac{100 \times 2240}{112} = 2000$ .
GST $= 2240 - 2000 = 240$ .

Answer: (b) ₹240

Example 2: Price before tax and the CGST · 3 marks · Short answerBoard level

A customer pays ₹6,372 for a mobile phone, including GST at 18%. The sale is intra-state. Find (a) the price of the phone before GST, (b) the CGST.

Try it first, then show the solutionSolution

(a) $V = \frac{100 \times 6372}{118} = 5400$ .
GST $= 6372 - 5400 = 972$ .
(b) CGST $= \frac{972}{2} = 486$ (check: 9% of 5400 = 486).

Answer: (a) ₹5,400 (b) ₹486

Example 3: Finding the rate after a discount · 3 marks · Short answerHarder board level

An article marked at ₹3,500 is sold at a discount of 20%. The customer pays ₹2,940 including GST. Find the rate of GST.

Try it first, then show the solutionSolution

Taxable value $= 3500 \times 0.8 = 2800$ .
GST $= 2940 - 2800 = 140$ .
Rate $= \frac{140}{2800} \times 100 = 5%$ .

Answer: 5%

Example 4: Finding the marked price · 4 marks · Short answerHarder board level

After a discount of 15% on the marked price, a customer pays ₹10,030 for a refrigerator, including GST at 18%. Find the marked price and the discount.

Try it first, then show the solutionSolution

Taxable value $= \frac{100 \times 10030}{118} = 8500$ .
$M \times 0.85 = 8500 \Rightarrow M = 10000$ .
Discount $= 10000 - 8500 = 1500$ .
Check: $10000 \times 0.85 \times 1.18 = 10030$ .

Answer: Marked price ₹10,000; discount ₹1,500.

Example 5: Rate and discount from the SGST · 4 marks · Short answerHOTS / competency

An article is marked at ₹2,000. In an intra-state sale, the SGST on it is ₹96 and the customer pays ₹1,792 in all. Find (a) the rate of GST, (b) the percentage discount given.

Try it first, then show the solutionSolution

GST $= 2 \times 96 = 192$ .
Taxable value $= 1792 - 192 = 1600$ .
(a) Rate $= \frac{192}{1600} \times 100 = 12%$ .
(b) Discount $= 2000 - 1600 = 400$ , that is $\frac{400}{2000} \times 100 = 20%$ .

Answer: (a) 12% (b) 20%

Example 6: Case study: two items, one total · 4 marks · Case-basedHOTS / competency

A bill for two items comes to ₹3,540, including GST. One item is taxed at 12% and the other at 18%, and the total GST on the bill is ₹480.
(a) Find the total taxable value.
(b) Find the taxable value of each item.
(c) Find the CGST on the item taxed at 18% (intra-state sale).

Try it first, then show the solutionSolution

(a) Total taxable value $= 3540 - 480 = 3060$ .
(b) Let the items be ₹ $x$ (12%) and ₹ $y$ (18%): $x + y = 3060$ and $0.12 x + 0.18 y = 480$ .
$0.12 (3060 - y) + 0.18 y = 480 \Rightarrow 367.20 + 0.06 y = 480 \Rightarrow y = 1880$ , so $x = 1180$ .
Check: $0.12 \times 1180 + 0.18 \times 1880 = 141.60 + 338.40 = 480$ .
(c) CGST $= 9%$ of $1880 = 169.20$ .

Answer: (a) ₹3,060 (b) ₹1,180 at 12% and ₹1,880 at 18% (c) ₹169.20

Where students lose marks in Inverse GST Problems

The slips examiners see most often, and what to write instead.

Mistake: Subtracting r% of the amount paid
The GST was r% of the taxable value, which is smaller than the amount paid.
Wrong working
₹2,240 includes 12% GST, so GST $= 12%$ of $2240 = 268.80$ .
Right working
$V = \frac{100 \times 2240}{112} = 2000$ ; GST $= 240$ .
Do this: Always divide by $(1 + \frac{r}{100})$ to get $V$ .
Usually costs: Every mark in the question.
Mistake: Using the marked price to find the rate when a discount was given
GST was charged on the discounted price.
Wrong working
GST $= 140$ , so rate $= \frac{140}{3500} \times 100 = 4%$ .
Right working
$V = 3500 \times 0.8 = 2800$ ; rate $= \frac{140}{2800} \times 100 = 5%$ .
Do this: Apply the discount first, then compare with the amount paid.
Usually costs: The method and answer marks.
Mistake: Dividing by the CGST factor instead of the GST factor
The buyer paid CGST and SGST, so the whole GST is in the amount.
Wrong working
CGST 9%: $V = \frac{6372}{1.09} = 5845.87$ .
Right working
GST 18%: $V = \frac{6372}{1.18} = 5400$ .
Do this: Double a CGST rate before working back.
Usually costs: The answer and all later parts.
Mistake: Not checking the answer forward
A slip in an inverse calculation is easy to spot by going forward.
Wrong working
Marked price ₹11,800 (divided by 0.85 before removing GST, then mixed up).
Right working
$10000 \times 0.85 \times 1.18 = 10030$ ✓.
Do this: Spend 20 seconds on a forward check of every inverse answer.
Usually costs: Marks that a check would have saved.

Exam tips

Write the formula $V = \frac{100 A}{100 + r}$ before substituting; it earns the method mark.
Give money answers to the paise when they are not whole (₹169.20), unless told to round.
When both a discount and GST are involved, write the three prices M, V and A as a list and fill them in.
In two-item inverse questions, the two equations are 'taxable values add up' and 'GSTs add up'.
Finish with a one-line forward check; examiners accept it as verification and it catches slips.

Practice questions on Inverse GST Problems

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · MCQBoard level
An amount of ₹1,050 includes GST at 5%. The price before GST is:
(a) ₹997.50
(b) ₹1,000
(c) ₹1,045
(d) ₹950
Hint
Divide by 1.05; do not subtract 5% of 1,050.
Answer
(b) ₹1,000
Q2 · 3 marks · Short answerBoard level
A customer pays ₹4,720 for an article, including GST at 18%, in an intra-state sale. Find the SGST.
Hint
Find the taxable value first.
Answer
$V = \frac{100 \times 4720}{118} = 4000$ ; GST = ₹720; SGST = ₹360.
Q3 · 3 marks · Short answerHarder board level
An article is marked at ₹1,750. A customer pays ₹1,568 for it, including GST at 12%. Find the percentage discount.
Hint
Taxable value first, then compare it with the marked price.
Answer
$V = \frac{100 \times 1568}{112} = 1400$ ; discount $= 350 = \frac{350}{1750} \times 100 = 20%$ .
Q4 · 1 mark · MCQHarder board level
After a discount of 10%, an article costs ₹2,124 including GST at 18%. Its marked price is:
(a) ₹1,800
(b) ₹2,000
(c) ₹2,124
(d) ₹2,360
Hint
₹2,124 ÷ 1.18, then ÷ 0.9.
Answer
(b) ₹2,000
Q5 · 1 mark · Assertion-reasonHarder board level
Assertion (A): If ₹3,360 includes GST at 12%, the GST is ₹360.
Reason (R): The GST contained in an amount A paid at r% GST is $\frac{r}{100 + r} \times A$ .
(a) (A) is true and (R) is false.
(b) (A) is false and (R) is true.
(c) Both (A) and (R) are true, and (R) is the correct explanation for (A).
(d) Both (A) and (R) are true, but (R) is not the correct explanation for (A).
Hint
Compute $\frac{12}{112} \times 3360$ .
Answer
(c). $\frac{12}{112} \times 3360 = 360$ , using exactly the relation in (R).
Q6 · 4 marks · Short answerHOTS / competency
When the GST rate on an article is cut from 28% to 18%, the amount a customer pays falls by ₹600. Find (a) the taxable value, (b) the amount paid before and after the cut.
Hint
The fall is 10% of the taxable value.
Answer
(a) $0.10 V = 600 \Rightarrow V = ₹6, 000$ . (b) Before: $6000 \times 1.28 = ₹7, 680$ ; after: $6000 \times 1.18 = ₹7, 080$ .
Q7 · 4 marks · Short answerHarder board level
A shopkeeper wants a customer to pay exactly ₹5,900, including GST at 18%, after he gives a discount of 20% on the marked price. What should the marked price be?
Hint
₹5,900 → taxable value → marked price.
Answer
$V = \frac{100 \times 5900}{118} = 5000$ ; $M = \frac{5000}{0.8} = ₹6, 250$ . Check: $6250 \times 0.8 \times 1.18 = 5900$ .

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.₹590 includes GST at 18%. Find the taxable value.Show answer
₹500.
2.₹2,100 includes GST at 5%. How much is the GST?Show answer
₹100.
3.Taxable value ₹1,500, amount paid ₹1,680. Rate of GST?Show answer
12%.
4.What fraction of an amount paid at 28% GST is tax?Show answer
$\frac{28}{128} = \frac{7}{32}$ .
5.Taxable value ₹900 after a 10% discount. Marked price?Show answer
₹1,000.
6.Why is subtracting 18% of the total wrong when removing 18% GST?Show answer
The 18% was charged on the smaller taxable value, so 18% of the total is more than the tax actually paid.

Inverse GST Problems: frequently asked questions

How do I find the price before GST from the total?

Multiply the total by 100 and divide by (100 + rate). For example, ₹1,180 at 18% GST gives 1180 × 100 ÷ 118 = ₹1,000.

How do I find the rate of GST if only the amounts are given?

Find the GST amount (amount paid minus taxable value, or twice the CGST), then divide it by the taxable value and multiply by 100.

What are inverse cases in the ICSE GST syllabus?

They are questions where you work backwards: from the amount paid to the price before tax, the rate, the discount or the marked price. The CISCE 2027 syllabus says they are included.

Can I just use 18/118 of the amount to get the GST?

Yes. At 18% GST, the tax is 18/118 of the amount paid and the price is 100/118 of it. Write the step clearly so the method mark is earned.