💰 Math Formulas
69 formulas in 18 sections. For each one you get the formula, what every symbol means, where it is used, an explanation in plain words and a worked example.
Sections: Percentage · Profit & Loss · Discount · Simple Interest · Compound Interest · Ratio & Proportion · Average · Speed, Distance, Time · Tax (GST) · Growth & Depreciation · Time & Work · Mixtures & Alligation · Partnership · Trains · Boats & Streams · Pipes & Cisterns · Ages · Clocks
Value = (Percentage ÷ 100) × Number
| % | the percentage (per hundred) |
|---|---|
| Number | the total amount |
Where it is used: To find a part of a total when the part is given as a percentage — used in discounts, marks, tax, tips and statistics.
In simple words: 'Per cent' means 'out of 100'. To find 20% of something, split it into 100 equal parts and take 20 of them.
Example: 20% of 50 = (20 ÷ 100) × 50 = 10
Percentage = Fraction × 100
| Fraction | a part written as a/b |
|---|
Where it is used: To express a fraction or a score out of a total as a percentage, so different quantities can be compared easily.
In simple words: Multiply the fraction by 100 to see how much it would be out of 100.
Example: 3/4 = 3/4 × 100 = 75%
Percentage = (Part ÷ Whole) × 100
| Part | the amount you have |
|---|---|
| Whole | the total amount |
Where it is used: To find what percentage one number is of another — such as marks obtained out of total marks, or attendance.
In simple words: Divide the part by the whole, then multiply by 100 to turn it into a percentage.
Example: 45 marks out of 60 = (45 ÷ 60) × 100 = 75%
Increase % = (Increase ÷ Original) × 100
| Increase | new value − old value |
|---|---|
| Original | the starting value |
Where it is used: To measure growth as a percentage — price rise, population growth, salary hike.
In simple words: It shows how much bigger something got, compared to what it was, out of 100.
Example: Price ₹200 → ₹250: increase 50, so 50/200 × 100 = 25%
Decrease % = (Decrease ÷ Original) × 100
| Decrease | old value − new value |
|---|---|
| Original | the starting value |
Where it is used: To measure a fall as a percentage — price drop, weight loss, reduction in pollution.
In simple words: It shows how much smaller something got, compared to what it was, out of 100.
Example: Price ₹200 → ₹150: decrease 50, so 50/200 × 100 = 25%
Percentage = Decimal × 100
| Decimal | the number in decimal form |
|---|
Where it is used: To turn a decimal or fraction into a percentage for easy comparison.
In simple words: Move the decimal point two places right (multiply by 100) to get the percentage.
Example: 0.25 = 0.25 × 100 = 25%
Net = 100 − [x + y − (xy ÷ 100)] % off
| x, y | the two discount percents |
|---|
Where it is used: To find the single equivalent discount when two discounts are applied one after another.
In simple words: Two discounts don't just add — the second acts on the reduced price, so this gives the real total off.
Example: 20% then 10%: 20+10−2 = 28% off
Error % = (|Error| ÷ Actual value) × 100
| Error | measured − actual |
|---|---|
| Actual | true value |
Where it is used: To express how far a measurement is from the true value, as a percentage.
In simple words: How wrong a measurement is compared to the correct value, out of 100.
Example: Measured 52 vs actual 50 → 2/50×100 = 4%
Net % = a − b − (ab ÷ 100)
| a | increase % |
|---|---|
| b | decrease % |
Where it is used: To find the overall change when a value rises by a% and then falls by b%.
In simple words: The rise and fall don't simply cancel — this gives the true overall change.
Example: +20% then −20% → 20−20−4 = −4%
B is less than A by [x ÷ (100+x)] × 100 %
| x | the percent A is more than B |
|---|
Where it is used: To convert 'A is x% more than B' into 'B is how much % less than A'.
In simple words: The percent more one way is not the same as the percent less the other way.
Example: A is 25% more than B → B is 20% less than A
Percentage = (Ratio value) × 100
| Ratio value | the fraction form of the ratio |
|---|
Where it is used: To convert a ratio into a percentage for comparison.
In simple words: Turn the ratio into a fraction, then multiply by 100.
Example: 3:5 → 3/8 × 100 = 37.5%
Percentage = (Marks obtained ÷ Total marks) × 100
| Marks obtained | your score |
|---|---|
| Total marks | maximum |
Where it is used: To find an exam percentage — the most common everyday use of percentage.
In simple words: Divide your marks by the total and multiply by 100.
Example: 425 out of 500 → 85%
Profit = Selling Price − Cost Price
| SP | selling price (price you sell at) |
|---|---|
| CP | cost price (price you bought at) |
Where it is used: Used in business and shopkeeping to find the gain made when something is sold for more than it cost.
In simple words: If you sell a thing for more than you paid, the extra money is your profit.
Example: Bought at ₹80, sold at ₹100 → Profit = 100 − 80 = ₹20
Loss = Cost Price − Selling Price
| CP | cost price (price you bought at) |
|---|---|
| SP | selling price (price you sell at) |
Where it is used: Used in trade to find how much money is lost when something is sold for less than it cost.
In simple words: If you sell a thing for less than you paid, the money you fall short by is your loss.
Example: Bought at ₹100, sold at ₹80 → Loss = 100 − 80 = ₹20
Profit % = (Profit ÷ Cost Price) × 100
| Profit | SP − CP |
|---|---|
| CP | cost price |
Where it is used: To compare how good a deal is, as a percentage of the cost — the standard way businesses state their gain.
In simple words: It tells you your gain compared to what you spent, out of 100. Higher percent = better deal.
Example: Profit ₹20 on CP ₹80 → 20/80 × 100 = 25%
Loss % = (Loss ÷ Cost Price) × 100
| Loss | CP − SP |
|---|---|
| CP | cost price |
Where it is used: To state a loss as a percentage of the cost price, so losses on different items can be compared.
In simple words: It shows how big your loss is compared to what you paid, out of 100.
Example: Loss ₹20 on CP ₹100 → 20/100 × 100 = 20%
SP = CP × (100 + Profit %) ÷ 100
| CP | cost price |
|---|---|
| Profit % | gain percent |
Where it is used: To work out the price to sell at so as to make a required percentage of profit.
In simple words: Add the profit percent to 100, then take that percent of the cost price to get the selling price.
Example: CP ₹200, profit 10% → SP = 200 × 110/100 = ₹220
CP = (SP × 100) ÷ (100 + Profit %)
| SP | selling price |
|---|---|
| Profit % | gain percent |
Where it is used: To work backwards to the cost price when the selling price and profit percent are known.
In simple words: Given the selling price and the profit percent, this recovers what it originally cost.
Example: SP ₹220 at 10% profit → CP = 22000/110 = ₹200
If sold at same price, one at +x% and one at −x%, net = −(x²/100)%
| x | the percent |
|---|
Where it is used: A classic result: selling two items at the same price with equal profit and loss percents gives an overall loss.
In simple words: Equal profit and loss percents don't cancel — you always end with a small loss.
Example: x=10 → net loss = 100/100 = 1%
SP = CP × (100+P%)/100; MP set so that after discount SP is reached
| CP | cost price |
|---|---|
| P% | desired profit |
Where it is used: To fix a marked price that still gives the wanted profit after a discount is offered.
In simple words: Shopkeepers mark the price high so that even after a discount they still make their profit.
Example: CP ₹100, want 20% profit after 20% discount → mark ₹150
Gain % = (Error ÷ (True − Error)) × 100
| True | true weight (e.g. 1000 g) |
|---|---|
| Error | shortfall in weight |
Where it is used: To find the extra profit a trader makes by giving less than the true weight.
In simple words: If a shopkeeper's '1 kg' is actually less, this is the hidden profit they gain.
Example: Sells 900 g as 1 kg → 100/900 × 100 ≈ 11.1%
Discount = Marked Price − Selling Price
| MP | marked price (label price) |
|---|---|
| SP | selling price (price actually paid) |
Where it is used: Used in shops and sales to find the reduction given on the labelled price.
In simple words: The money taken off the printed price during a sale is the discount.
Example: MP ₹500, SP ₹400 → Discount = ₹100
Discount % = (Discount ÷ Marked Price) × 100
| Discount | MP − SP |
|---|---|
| MP | marked price |
Where it is used: To express a sale reduction as a percentage of the marked price — how sales are advertised ('20% off').
In simple words: It shows how much is taken off, out of every 100 rupees of the printed price.
Example: Discount ₹100 on MP ₹500 → 100/500 × 100 = 20%
SP = MP × (100 − Discount %) ÷ 100
| MP | marked price |
|---|---|
| Discount % | percent off |
Where it is used: To find the price you actually pay after a percentage discount is applied.
In simple words: Take the printed price, knock off the discount percent, and that's what you pay.
Example: MP ₹500, 20% off → SP = 500 × 80/100 = ₹400
MP = (SP × 100) ÷ (100 − Discount %)
| SP | selling price |
|---|---|
| Discount % | percent off |
Where it is used: To recover the marked (label) price when the paid price and discount are known.
In simple words: Works backwards from the paid price and discount to the printed price.
Example: SP ₹400, 20% off → MP = 40000/80 = ₹500
SI = (P × R × T) ÷ 100
| P | principal (money borrowed or saved) |
|---|---|
| R | rate of interest per year (%) |
| T | time in years |
Where it is used: To find the extra money (interest) paid or earned on a loan or deposit, when interest is the same each year.
In simple words: It's the rent you pay for borrowing money. Bigger loan, higher rate, or longer time all make it more.
Example: P=₹1000, R=5%, T=2 yrs → SI = 1000×5×2/100 = ₹100
A = P + SI
| A | total amount to be paid back |
|---|---|
| P | principal |
| SI | simple interest |
Where it is used: To find the total money returned at the end of a loan or the maturity value of a deposit.
In simple words: Add the interest to the money you started with to get the total you'll have or owe.
Example: P=₹1000, SI=₹100 → A = 1000 + 100 = ₹1100
R = (SI × 100) ÷ (P × T)
| SI | interest |
|---|---|
| P | principal |
| T | time (years) |
Where it is used: To find the yearly interest rate when the interest, principal and time are known.
In simple words: Rearranges the simple-interest formula to solve for the rate.
Example: SI=100, P=1000, T=2 → R = 5%
T = (SI × 100) ÷ (P × R)
| SI | interest |
|---|---|
| P | principal |
| R | rate (%) |
Where it is used: To find how long money was invested/borrowed from the interest, principal and rate.
In simple words: Rearranges simple interest to solve for the time.
Example: SI=100, P=1000, R=5 → T = 2 years
T = 100 ÷ R
| R | rate per year (%) |
|---|
Where it is used: To find how many years it takes money to double under simple interest.
In simple words: Divide 100 by the rate to see how long the money takes to double.
Example: R = 5% → T = 100/5 = 20 years
P = (SI × 100) ÷ (R × T)
| SI | interest |
|---|---|
| R | rate (%) |
| T | time (years) |
Where it is used: To find the original sum when the interest, rate and time are known.
In simple words: Rearranges the simple-interest formula to find the starting money.
Example: SI=100, R=5, T=2 → P = 10000/10 = ₹1000
A = P × (1 + R/100)^T
| A | final amount |
|---|---|
| P | principal |
| R | rate per year (%) |
| T | time in years |
Where it is used: For loans and savings where interest is added to the principal each year, so you earn interest on interest.
In simple words: Each year the interest joins your money, and next year you earn interest on that bigger amount too — it grows faster and faster.
Example: P=₹1000, R=10%, T=2 → A = 1000×(1.1)² = ₹1210
CI = A − P
| A | final amount |
|---|---|
| P | principal |
Where it is used: To find just the interest part when interest is compounded, by removing the original money.
In simple words: Take away the money you started with from the final amount to see how much extra you gained.
Example: A=₹1210, P=₹1000 → CI = ₹210
A = P × (1 + R/200)^(2T)
| P | principal |
|---|---|
| R | annual rate (%) |
| T | time in years |
Where it is used: For deposits/loans where interest is added twice a year instead of once.
In simple words: When interest is added every 6 months, halve the rate and double the number of periods.
Example: P=₹1000, R=10%, T=1 → A = 1000×(1.05)² = ₹1102.50
CI − SI = P × (R ÷ 100)²
| P | principal |
|---|---|
| R | rate (%) |
Where it is used: A quick way to find how much more compound interest gives than simple interest over 2 years.
In simple words: Over two years, the extra that compounding adds is just P times (R/100) squared.
Example: P=1000, R=10% → CI−SI = 1000×0.01 = ₹10
Years to double ≈ 72 ÷ R
| R | annual compound rate (%) |
|---|
Where it is used: A quick estimate of how long money takes to double under compound interest.
In simple words: Divide 72 by the interest rate for a fast estimate of the doubling time.
Example: R = 8% → about 72/8 = 9 years
A = P × (1 + R/400)^(4T)
| P | principal |
|---|---|
| R | annual rate (%) |
| T | years |
Where it is used: For interest added four times a year (every 3 months).
In simple words: Quarter the rate and quadruple the periods when interest is added every three months.
Example: P=1000, R=8%, T=1 → 1000×(1.02)⁴ ≈ ₹1082.4
Ratio = a : b = a ÷ b
| a, b | the two quantities being compared |
|---|
Where it is used: To compare two quantities of the same kind — mixing, sharing, map scales, recipes.
In simple words: It tells you how many of one thing there are for each of another, like 2 cups of rice for every 1 cup of dal.
Example: 20 boys and 30 girls → ratio 20:30 = 2:3
a : b = c : d ⇒ a × d = b × c
| a, b, c, d | four quantities in proportion |
|---|
Where it is used: To check if two ratios are equal, or to find a missing value when scaling recipes, maps or prices.
In simple words: When two ratios are equal, the cross-products match. This lets you find a missing amount when keeping the same ratio.
Example: 2:3 = 4:6 because 2×6 = 3×4 = 12
Share = (its ratio part ÷ total parts) × Amount
| Amount | the total to divide |
|---|
Where it is used: To split money or quantity between people in a given ratio.
In simple words: Add the ratio numbers, then give each person their share of the total.
Example: ₹600 in 1:2 → 200 and 400
y = k × x (y/x is constant)
| k | the constant |
|---|---|
| x, y | the two quantities |
Where it is used: When two quantities rise and fall together — cost vs quantity, distance vs time at fixed speed.
In simple words: If one doubles, the other doubles; their ratio stays the same.
Example: 5 pens ₹50 → 8 pens ₹80
x × y = constant
| x, y | the two quantities |
|---|
Where it is used: When one quantity rises as the other falls — speed vs time, workers vs days.
In simple words: If one doubles, the other halves; their product stays the same.
Example: 4 workers 6 days → 8 workers 3 days
Average = Sum of values ÷ Number of values
| Sum | total of all the numbers |
|---|---|
| Number | how many numbers there are |
Where it is used: To find a single typical value for a set of numbers — average marks, runs, temperature or income.
In simple words: Add everything up and share it out equally. It's the 'middle' amount if everyone had the same.
Example: Marks 10, 20, 30 → (10+20+30)/3 = 20
Average speed = Total distance ÷ Total time
| Total distance | sum of all distances |
|---|---|
| Total time | sum of all times |
Where it is used: To find the true average speed of a journey with several stages or stops.
In simple words: Add up all the distance and all the time, then divide — don't just average the speeds.
Example: 120 km in 3 h total → 40 km/h
New avg = (Old sum ± change) ÷ New count
| Old sum | total before |
|---|---|
| New count | number of items after |
Where it is used: To update an average when a value is added, removed or replaced.
In simple words: Adjust the total, then divide by the new number of items.
Example: Avg of 4 nums=10 (sum 40); add 20 → 60/5 = 12
Average = (n + 1) ÷ 2
| n | how many numbers |
|---|
Where it is used: A quick average for 1, 2, 3, … up to n.
In simple words: The average of the first n counting numbers is just the middle value.
Example: Average of 1..9 = 10/2 = 5
Speed = Distance ÷ Time
| Distance | how far (m or km) |
|---|---|
| Time | how long (s or hours) |
Where it is used: To find how fast something moves — vehicles, trains, runners — and to plan travel times.
In simple words: It tells you how much distance is covered in one unit of time. Cover more ground in less time and your speed is higher.
Example: 120 km in 2 hours → Speed = 120/2 = 60 km/h
Distance = Speed × Time
| Speed | how fast |
|---|---|
| Time | how long |
Where it is used: To find how far something travels when its speed and travel time are known.
In simple words: Multiply how fast you go by how long you go for, to find how far you get.
Example: 60 km/h for 3 hours → Distance = 60×3 = 180 km
Time = Distance ÷ Speed
| Distance | how far |
|---|---|
| Speed | how fast |
Where it is used: To find how long a journey takes when the distance and speed are known.
In simple words: Divide the distance by the speed to find how long the trip will take.
Example: 180 km at 60 km/h → Time = 180/60 = 3 hours
Average speed = 2xy ÷ (x + y)
| x | speed one way |
|---|---|
| y | speed the return way |
Where it is used: To find the average speed for a round trip covering the same distance at two different speeds.
In simple words: You can't just average the two speeds — this formula gives the correct average for equal distances.
Example: x=60, y=40 → 2·60·40/100 = 48 km/h
m/s = km/h × (5 ÷ 18)
| 5/18 | conversion factor |
|---|
Where it is used: To convert a speed from kilometres per hour to metres per second (and ×18/5 for the reverse).
In simple words: Multiply km/h by 5/18 to get metres per second.
Example: 72 km/h = 72 × 5/18 = 20 m/s
Relative speed = a − b
| a | faster speed |
|---|---|
| b | slower speed |
Where it is used: To find how fast one moving object gains on another going the same way.
In simple words: When both move the same way, subtract their speeds to see how fast one catches up.
Example: 60 and 40 km/h same way → gap closes at 20 km/h
Relative speed = a + b
| a, b | the two speeds |
|---|
Where it is used: To find how fast two objects approach each other head-on.
In simple words: When moving toward each other, add their speeds to see how fast the gap shrinks.
Example: 60 and 40 km/h toward each other → 100 km/h
km/h = m/s × (18 ÷ 5)
| 18/5 | conversion factor |
|---|
Where it is used: To convert a speed from metres per second to kilometres per hour.
In simple words: Multiply m/s by 18/5 to get km/h.
Example: 20 m/s = 20 × 18/5 = 72 km/h
Final Price = Price × (100 + GST %) ÷ 100
| Price | price before tax |
|---|---|
| GST % | tax rate |
Where it is used: To find the final bill amount after Goods and Services Tax is added.
In simple words: Add the tax percent on top of the price to get the total you pay at the counter.
Example: Price ₹1000, GST 18% → 1000 × 118/100 = ₹1180
Value = P × (1 ± R/100)^T
| P | starting value |
|---|---|
| R | rate per period (%) |
| T | number of periods |
| ± | + for growth, − for depreciation |
Where it is used: To find future population (growth, +) or the falling value of machines/vehicles (depreciation, −).
In simple words: Same idea as compound interest: use plus for things that grow, minus for things that lose value.
Example: Car ₹5,00,000, depreciates 10%/yr, 2 yrs → 5,00,000×(0.9)² = ₹4,05,000
Past value = Present ÷ (1 + R/100)ⁿ
| R | growth rate (%) |
|---|---|
| n | years back |
Where it is used: To find a past population or value from the present one and the growth rate.
In simple words: Divide instead of multiply to step backwards in time.
Example: Now 12100 at 10%/yr → 2 yrs ago = 12100/1.21 = 10000
1/T = 1/a + 1/b
| a | days A alone takes |
|---|---|
| b | days B alone takes |
| T | days together |
Where it is used: To find how long two people/pipes take together when each one's solo time is known.
In simple words: Add how much each does in a day; the total per day tells how fast they finish together.
Example: A=6 days, B=12 days → 1/T = 1/6+1/12 = 1/4, so T = 4 days
M₁ × D₁ = M₂ × D₂
| M | number of workers |
|---|---|
| D | number of days |
Where it is used: To find how many workers or days are needed for the same job.
In simple words: More workers means fewer days; the total worker-days stays the same.
Example: 10 men × 6 days = 15 men × 4 days
Work = Rate × Time
| Rate | work per unit time |
|---|---|
| Time | time worked |
Where it is used: To find how much of a job is completed in a given time.
In simple words: Multiply how fast you work by how long you work to find how much gets done.
Example: Rate 1/6 per day, 3 days → 3/6 = half the job
Cheaper : Dearer = (D − Mean) : (Mean − C)
| C | cost of cheaper part |
|---|---|
| D | cost of dearer part |
| Mean | cost of the mixture |
Where it is used: To find the ratio in which two ingredients at different prices must be mixed for a target price.
In simple words: Balances how much of the cheap and the costly item to mix to hit an average price.
Example: ₹10 and ₹15 to make ₹12 → (15−12):(12−10) = 3:2
Final = Initial × (1 − x/V)ⁿ
| V | total volume |
|---|---|
| x | amount replaced each time |
| n | number of times |
Where it is used: To find how much pure liquid remains after repeated removal and replacement with water.
In simple words: Each time you remove some mixture and top up with water, the pure part shrinks by a fixed fraction.
Example: 40 L, remove 4 L twice → 40×(0.9)² = 32.4 L pure
Profit ratio = (Capital × Time) of each partner
| Capital | money invested |
|---|---|
| Time | months invested |
Where it is used: To split business profit fairly among partners by their investment and duration.
In simple words: Each partner's share depends on how much they put in and for how long.
Example: A: ₹5000×12, B: ₹6000×10 → 60000:60000 = 1:1
Time = Length of train ÷ Speed
| Length | length of the train |
|---|---|
| Speed | train's speed |
Where it is used: To find how long a train takes to cross a pole or a standing person.
In simple words: The train only has to move its own length past a pole, so divide length by speed.
Example: 200 m at 20 m/s → 200/20 = 10 s
Time = (Train length + Platform length) ÷ Speed
| Speed | train's speed |
|---|
Where it is used: To find how long a train takes to fully cross a platform or bridge.
In simple words: The train must cover its own length plus the platform's length.
Example: (200+300) m at 25 m/s → 500/25 = 20 s
Down = b + s, Up = b − s
| b | speed of boat in still water |
|---|---|
| s | speed of stream |
Where it is used: To find a boat's effective speed with or against a river current.
In simple words: Going with the flow adds the current's speed; going against subtracts it.
Example: Boat 10, stream 2 → down 12, up 8 km/h
1/T = 1/a + 1/b
| a, b | hours each pipe alone takes |
|---|---|
| T | hours together |
Where it is used: To find how long two pipes take to fill a tank together (same idea as time & work).
In simple words: Add how much each pipe fills per hour to find the combined filling time.
Example: a=6, b=12 → 1/T = 1/4, T = 4 hours
Future age = present + n, Past age = present − n
| n | the number of years |
|---|
Where it is used: The basis of age word-problems — everyone ages by the same amount over time.
In simple words: Add the years for the future, subtract them for the past — for every person equally.
Example: Now 20 → in 5 years 25, 5 years ago 15
Angle = |30H − 5.5M| degrees
| H | the hour |
|---|---|
| M | the minutes |
Where it is used: To find the angle between the hour and minute hands at a given time.
In simple words: A formula that gives the angle between the two clock hands from the time shown.
Example: At 3:00 → |90 − 0| = 90°