74 concept-first Class 8 Maths questions across 11 chapters — each with a clear model answer, the why behind it, and a memory trick to make it stick. NCERT-aligned and free. Build the base now, and your board exams, NEET and JEE feel a whole lot easier later.
Solving an equation means undoing the operations to leave the unknown by itself.
Answer
Subtract 5 from both sides: 3x = 20 - 5 = 15. Divide both sides by 3: x = 15/3 = 5. So x = 5. Check: 3(5) + 5 = 15 + 5 = 20, which is correct.
Isolate the variable using inverse operations
•3x = 20 - 5 = 15
•x = 15/3 = 5
•Check: 3(5) + 5 = 20
Why learn this
Every 'find the missing value' problem - money, coding, physics - is a hidden equation.
💡 Memory trick
Do the opposite, in reverse order: undo the +5 first, then the x3.
MathematicsSolving a linear equationeasy
Solve for x: 3x + 5 = 20.
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What it is
A linear equation in one variable has one solution found by isolating the variable.
Answer
Start with 3x + 5 = 20. Subtract 5 from both sides: 3x = 15. Divide both sides by 3: x = 5. Check: 3(5) + 5 = 15 + 5 = 20, which is correct.
ax + b = c => x = (c - b)/a
•Keep both sides balanced
•3x + 5 = 20 -> 3x = 15
•x = 15/3 = 5
•Verify by substitution
Why learn this
It is the basic tool for solving real-life problems with one unknown.
💡 Memory trick
Do the same operation on both sides to keep the balance; move terms by changing their sign.
MathematicsWord problem on linear equationsmedium
The sum of two consecutive natural numbers is 45. Find the numbers.
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What it is
Word problems are solved by translating the statement into an equation and solving it.
Answer
Let the first number be x, so the next consecutive number is x + 1. Their sum is x + (x + 1) = 45, which gives 2x + 1 = 45. So 2x = 44 and x = 22. The two consecutive numbers are 22 and 23. Check: 22 + 23 = 45.
x + (x + 1) = sum
•Let the numbers be x and x + 1
•2x + 1 = 45
•x = 22
•Numbers are 22 and 23
Why learn this
It turns everyday situations, like ages or costs, into solvable maths.
💡 Memory trick
Let the unknown be x, form the equation from the sentence, then solve.
Squares and Square Roots4 questions
MathematicsSquare root by prime factorisationmedium
Find the square root of 1225 by prime factorisation.
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What it is
A square root asks: which number multiplied by itself gives this number?
Answer
Factorise: 1225 = 5 x 5 x 7 x 7 = 5^2 x 7^2. Pair the equal factors and take one from each pair: sqrt(1225) = 5 x 7 = 35. So the square root of 1225 is 35.
sqrt(a^2 x b^2) = a x b
•1225 = 5^2 x 7^2
•Take one factor from each pair
•sqrt(1225) = 5 x 7 = 35
Why learn this
Square roots power areas, geometry and, later, all of trigonometry.
💡 Memory trick
Prime-factorise, pair them up, take one factor from each pair.
MathematicsSide from area of a squareeasy
A square has an area of 144 square units. Find its side. Slide the area to explore.
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What it is
The side of a square is the square root of its area.
⬜Side = √ area
Side12.00 units
Answer
Side = sqrt(area) = sqrt(144) = 12 units, because 12 x 12 = 144. Finding the side of a square from its area is exactly what taking a square root means.
side = sqrt(area)
•Side = sqrt(area)
•Reverses area = side^2
•Area 144 -> side 12
Why learn this
It is the everyday use of a square root.
💡 Memory trick
Since area = side^2, side = sqrt(area).
MathematicsProperties of square numberseasy
State two properties of perfect squares. Is 3728 a perfect square?
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What it is
A perfect square ends only in 0, 1, 4, 5, 6 or 9, and never in 2, 3, 7 or 8.
Answer
Two properties of perfect squares are: a perfect square never ends in the digits 2, 3, 7 or 8, and the number of zeros at the end of a perfect square is always even. The number 3728 ends in 8, so it cannot be a perfect square.
Perfect square never ends in 2, 3, 7 or 8
•Squares end in 0,1,4,5,6,9 only
•Never end in 2,3,7,8
•Even number of trailing zeros
•3728 ends in 8 -> not a perfect square
Why learn this
This quick test tells you a number is not a perfect square at a glance.
💡 Memory trick
If a number ends in 2, 3, 7 or 8, it is NEVER a perfect square.
MathematicsSquare root by prime factorisationmedium
Find the square root of 324 by prime factorisation.
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What it is
The square root of a perfect square is found by pairing its prime factors and taking one from each pair.
Answer
Prime factorise 324: 324 = 2 x 2 x 3 x 3 x 3 x 3 = (2 x 2) x (3 x 3) x (3 x 3). Taking one factor from each pair gives 2 x 3 x 3 = 18. So the square root of 324 is 18. Check: 18 x 18 = 324.
sqrt(number) = product of one prime from each pair
•324 = 2^2 x 3^4
•Pair equal primes
•Take one from each pair: 2 x 3 x 3
•sqrt(324) = 18
Why learn this
It is a reliable method that also reveals whether a number is a perfect square.
💡 Memory trick
Make pairs of equal primes; take ONE factor from each pair and multiply.
Exponents and Powers3 questions
MathematicsStandard formmedium
Express 0.0000073 in standard (scientific) form.
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What it is
Standard form writes very large or very small numbers as a x 10^n.
Answer
Move the decimal point to just after the first non-zero digit to get 7.3. The point moved 6 places to the right, so we multiply by 10 raised to the power -6. Therefore 0.0000073 = 7.3 x 10^-6.
Standard form: a x 10^n, where 1 <= a < 10
•Bring one non-zero digit before the decimal
•Count places moved = 6 (moving right -> negative power)
•= 7.3 x 10^-6
Why learn this
Scientists write an atom's mass or a galaxy's size this way - it kills long strings of zeros.
💡 Memory trick
Move the point, count the hops; hopping right gives a negative power.
MathematicsLaws of exponentsmedium
Simplify using laws of exponents: 2^3 x 2^4 and 5^6 / 5^2.
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What it is
The laws of exponents let us multiply, divide and take powers of numbers written with exponents.
Answer
Using the law a^m x a^n = a^(m+n): 2^3 x 2^4 = 2^(3+4) = 2^7 = 128. Using the law a^m / a^n = a^(m-n): 5^6 / 5^2 = 5^(6-2) = 5^4 = 625. Also recall that a^0 = 1 for any non-zero a.
a^m x a^n = a^(m+n) ; a^m / a^n = a^(m-n)
•a^m x a^n = a^(m+n)
•a^m / a^n = a^(m-n)
•(a^m)^n = a^(mn)
•a^0 = 1 ; a^(-m) = 1/a^m
Why learn this
They simplify very large and very small numbers and appear throughout algebra.
💡 Memory trick
Multiply -> add powers; Divide -> subtract powers; Power of a power -> multiply powers.
MathematicsStandard form of small numbersmedium
Express 0.00000045 in standard form.
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What it is
Very small numbers are written compactly in standard form using negative powers of 10.
Answer
To write 0.00000045 in standard form, move the decimal point so that there is one non-zero digit before it. The decimal moves 7 places to the right, giving 4.5. Since the number is small, the power of 10 is negative: 0.00000045 = 4.5 x 10^(-7).
a x 10^n, where 1 <= a < 10
•Standard form: a x 10^n with 1 <= a < 10
•Small numbers use negative powers of 10
•Decimal moved 7 places
•0.00000045 = 4.5 x 10^(-7)
Why learn this
Scientists write tiny sizes, like a cell's diameter, neatly this way.
💡 Memory trick
Move the decimal to get one non-zero digit before it; count moves = the power of 10.
Comparing Quantities23 questions
MathematicsProfit percenteasy
A shopkeeper buys a bag for 500 rupees and sells it for 575 rupees. Find the profit percent.
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What it is
Profit percent measures your gain compared with the cost price, not the selling price.
Answer
Profit = selling price - cost price = 575 - 500 = 75 rupees. Profit percent = (profit / cost price) x 100 = (75 / 500) x 100 = 15 percent. So the shopkeeper makes a 15 percent profit.
Profit% = (Profit / Cost Price) x 100
•Profit = SP - CP = 75
•Profit% = (Profit / CP) x 100
•= (75/500) x 100 = 15%
Why learn this
It's the number every shop, startup and investor lives and dies by.
💡 Memory trick
Always divide by CP - the Cost Price is your 'Compare Point'.
MathematicsSimple interesteasy
Find the simple interest on 2000 at 5% per year for 3 years. Slide the values to explore.
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What it is
Simple interest is the extra money paid for borrowing, or earned for saving, a sum.
💰Simple interest = P × R × T ÷ 100
Interest300 ₹
Answer
Simple interest = P x R x T / 100 = 2000 x 5 x 3 / 100 = 300. So the interest is 300 in the same currency as the principal. Increasing the principal, the rate or the time each increases the interest.
SI = P x R x T / 100
•SI = P x R x T / 100
•P = principal, R = rate % per year, T = time in years
•Interest grows with P, R and T
Why learn this
It's everyday money maths - loans, savings and deposits.
💡 Memory trick
SI = P x R x T / 100. More principal, rate or time -> more interest.
MathematicsPercentageeasy
A student scores 45 out of 180. What percentage is that? Slide the values to explore.
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What it is
A percentage expresses a part out of a whole as a number per hundred.
📊Percentage = (part ÷ whole) × 100
Percentage25.0 %
Answer
Percentage = (part / whole) x 100 = (45 / 180) x 100 = 25%. A percentage rescales any fraction to 'out of 100', which makes different quantities easy to compare.
percentage = (part / whole) x 100
•% = (part / whole) x 100
•It means 'per hundred'
•45 out of 180 = 25%
Why learn this
It's how marks, discounts, interest and statistics are compared fairly.
💡 Memory trick
% = (part / whole) x 100.
MathematicsDiscounteasy
A 2000 item has a 25% discount. How much do you save? Slide the values to explore.
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What it is
A discount is a reduction on the marked price, found as a percentage of that price.
🏷️Discount = price × rate ÷ 100
You save500 ₹
Answer
Discount = marked price x rate / 100 = 2000 x 25 / 100 = 500. So you save 500 and pay 2000 - 500 = 1500. A higher rate or a higher marked price gives a bigger saving.
Find the compound interest on 1000 at 10% per year for 2 years. Slide the values to explore.
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What it is
Compound interest is calculated on the principal plus the interest already earned, so it grows faster than simple interest.
📈CI: A = P(1 + R÷100)^T
Compound interest210 ₹
Answer
Amount A = P(1 + R/100)^T = 1000 x (1 + 10/100)^2 = 1000 x 1.21 = 1210. Compound interest = A - P = 1210 - 1000 = 210. This is more than the simple interest (200) because the second year's interest is earned on 1100, not just 1000.
A = P(1 + R/100)^T ; CI = A - P
•A = P(1 + R/100)^T
•CI = A - P
•Interest is earned on interest, so it beats simple interest
Why learn this
It's how savings, loans and investments really grow over time.
💡 Memory trick
A = P(1 + R/100)^T, then CI = A - P.
MathematicsProfit and loss percentageeasy
An item costing 2000 is sold for 2500. Find the profit percentage. Slide the values to explore.
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What it is
Profit percentage is the profit expressed as a percentage of the cost price.
💹Profit % = (SP − CP) ÷ CP × 100
Profit25.0 %
Answer
Profit = SP - CP = 2500 - 2000 = 500. Profit % = (profit / CP) x 100 = (500 / 2000) x 100 = 25%. The percentage is always taken on the cost price; if the selling price is below the cost price the answer is negative, showing a loss.
profit % = (SP - CP) / CP x 100
•Profit % = (SP - CP) / CP x 100
•Always taken on the cost price (CP)
•A negative value means a loss
Why learn this
It's how shops and businesses measure how good a deal is.
💡 Memory trick
Profit % = (SP - CP) / CP x 100. Negative means a loss.
MathematicsPercentage changeeasy
A value rises from 80 to 100. Find the percentage change. Slide the values to explore.
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What it is
Percentage change is the increase or decrease compared with the original value, expressed as a percentage.
🔁Change % = (new − old) ÷ old × 100
Percentage change25.0 %
Answer
Percentage change = (new - old) / old x 100 = (100 - 80) / 80 x 100 = 20 / 80 x 100 = 25%. It is always taken on the original (old) value; if the new value is smaller, the change is negative, showing a decrease.
percentage change = (new - old) / old x 100
•Change % = (new - old) / old x 100
•Always on the old value
•Negative means a decrease
Why learn this
It's how price rises, discounts and growth are reported.
💡 Memory trick
Change % = (new - old) / old x 100. Take it on the OLD value.
MathematicsLoss percentageeasy
An item costing 2000 is sold for 1600. Find the loss percentage. Slide the values to explore.
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What it is
Loss percentage is the loss expressed as a percentage of the cost price.
📉Loss % = (CP − SP) ÷ CP × 100
Loss20.0 %
Answer
Loss = CP - SP = 2000 - 1600 = 400. Loss % = (loss / CP) x 100 = (400 / 2000) x 100 = 20%. Like profit, it is always taken on the cost price; if the selling price is above the cost price, this becomes a profit instead.
loss % = (CP - SP) / CP x 100
•Loss % = (CP - SP) / CP x 100
•Always on the cost price
•SP above CP means a profit
Why learn this
It measures how bad a deal is when something sells below cost.
💡 Memory trick
Loss % = (CP - SP) / CP x 100.
MathematicsAmount with simple interesteasy
Find the amount on 2000 at 5% per year for 3 years. Slide the values to explore.
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What it is
The amount to be repaid is the principal plus the simple interest earned on it.
💰Amount = P + (P × R × T ÷ 100)
Amount2300 ₹
Answer
Simple interest = P x R x T / 100 = 2000 x 5 x 3 / 100 = 300. Amount = principal + interest = 2000 + 300 = 2300. The amount is always the principal plus whatever interest has built up.
amount = P + (P x R x T / 100)
•Amount = principal + simple interest
•SI = P x R x T / 100
•Amount = P + (P x R x T / 100)
Why learn this
It's the total you actually get back or pay after interest.
💡 Memory trick
Amount = P + (P x R x T / 100).
MathematicsSelling price from profit percentmedium
At what price should a 2000 item be sold to make a 25% profit? Slide to explore.
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What it is
The selling price for a required profit is the cost price increased by that profit percentage.
💹SP = CP × (1 + profit% ÷ 100)
Selling price2500 ₹
Answer
SP = CP x (1 + profit% / 100) = 2000 x (1 + 25/100) = 2000 x 1.25 = 2500. The profit of 500 is 25% of the cost price, so the selling price is 2500.
SP = CP x (1 + profit% / 100)
•SP = CP x (1 + profit% / 100)
•Profit is taken on the cost price
•25% on 2000 -> SP 2500
Why learn this
It's how a shopkeeper sets a price to earn a target profit.
💡 Memory trick
SP = CP x (1 + profit% / 100).
MathematicsPercentage of a numbereasy
What is 20% of 150? Slide the values to explore.
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What it is
A percentage of a number is that fraction out of a hundred, multiplied by the number.
📊Result = (percent ÷ 100) × number
Result30.0
Answer
Result = (percent / 100) x number = (20 / 100) x 150 = 0.2 x 150 = 30. Finding a percentage of a number just means multiplying the number by that fraction out of a hundred.
result = (percent / 100) x number
•Result = (percent / 100) x number
•20% of 150 = 30
•Per cent means per hundred
Why learn this
It's used for discounts, tips, marks and shares of a quantity.
💡 Memory trick
Result = (percent / 100) x number.
MathematicsDepreciationmedium
A 50000 machine depreciates 10% a year. Find its value after 2 years. Slide to explore.
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What it is
Depreciation reduces an asset's value by a fixed percentage each year, like compound interest in reverse.
📉Value = P × (1 − r÷100)ⁿ
Value now40500 ₹
Answer
Value = P x (1 - r/100)^n = 50000 x (1 - 10/100)^2 = 50000 x 0.9^2 = 50000 x 0.81 = 40500. Each year the value becomes 90% of the previous year, so it falls faster at first.
value = P x (1 - r/100)^n
•Value = P x (1 - r/100)^n
•Compound decrease, not a flat subtraction
•50000 at 10% -> 40500 after 2 years
Why learn this
It's how the value of vehicles and machines falls over time.
💡 Memory trick
Value = P x (1 - r/100)^n.
MathematicsCost price from selling pricemedium
An item sold for 2500 at a 25% profit. Find the cost price. Slide to explore.
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What it is
The cost price can be recovered from the selling price and the profit percentage by dividing.
💰CP = SP ÷ (1 + profit% ÷ 100)
Cost price2000 ₹
Answer
CP = SP / (1 + profit% / 100) = 2500 / (1 + 25/100) = 2500 / 1.25 = 2000. Because the selling price is 125% of the cost price, we divide by 1.25 to get back to 100%.
CP = SP / (1 + profit% / 100)
•CP = SP / (1 + profit% / 100)
•SP is (100 + profit%) of CP
•SP 2500 at 25% -> CP 2000
Why learn this
It's how you check what an item originally cost after a marked-up sale.
💡 Memory trick
CP = SP / (1 + profit% / 100).
MathematicsSales tax / GSTeasy
Find the tax on a 1000 item at 12%. Slide the values to explore.
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What it is
Sales tax is a percentage of the price added on top of it.
🧾Tax = price × rate ÷ 100
Tax amount120 ₹
Answer
Tax = price x rate / 100 = 1000 x 12 / 100 = 120. The final amount paid is the price plus this tax, so 1000 + 120 = 1120.
tax = price x rate / 100
•Tax = price x rate / 100
•Added on top of the price
•Final amount = price + tax
Why learn this
It's the GST added to almost everything you buy.
💡 Memory trick
Tax = price x rate / 100.
MathematicsRate from simple interestmedium
An interest of 300 comes from a principal of 2000 over 3 years. Find the rate. Slide to explore.
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What it is
The rate of interest is found by rearranging the simple interest formula.
📊Rate = SI × 100 ÷ (P × T)
Rate5.0 % / yr
Answer
Rate = SI x 100 / (P x T) = 300 x 100 / (2000 x 3) = 30000 / 6000 = 5% per year. This is SI = P x R x T / 100 rearranged to make R the subject.
rate = SI x 100 / (P x T)
•Rate = SI x 100 / (P x T)
•Rearranged from SI = PRT/100
•SI 300, P 2000, T 3 -> R 5%
Why learn this
It tells you what rate a deposit or loan is really charging.
💡 Memory trick
Rate = SI x 100 / (P x T).
MathematicsTime from simple interestmedium
An interest of 300 comes from a principal of 2000 at 5% per year. Find the time. Slide to explore.
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What it is
The time is found by rearranging the simple interest formula.
⏱️Time = SI × 100 ÷ (P × R)
Time3.0 yr
Answer
Time = SI x 100 / (P x R) = 300 x 100 / (2000 x 5) = 30000 / 10000 = 3 years. This is SI = P x R x T / 100 rearranged to make T the subject.
time = SI x 100 / (P x R)
•Time = SI x 100 / (P x R)
•Rearranged from SI = PRT/100
•SI 300, P 2000, R 5% -> T 3 years
Why learn this
It tells you how long money must stay invested to earn a given interest.
💡 Memory trick
Time = SI x 100 / (P x R).
MathematicsPrincipal from simple interestmedium
An interest of 300 is earned at 5% per year for 3 years. Find the principal. Slide to explore.
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What it is
The principal is found by rearranging the simple interest formula.
💰Principal = SI × 100 ÷ (R × T)
Principal2000 ₹
Answer
Principal = SI x 100 / (R x T) = 300 x 100 / (5 x 3) = 30000 / 15 = 2000. This is SI = P x R x T / 100 rearranged to make P the subject.
principal = SI x 100 / (R x T)
•Principal = SI x 100 / (R x T)
•Rearranged from SI = PRT/100
•SI 300, R 5%, T 3 -> P 2000
Why learn this
It works out how much was invested to earn a known interest.
💡 Memory trick
Principal = SI x 100 / (R x T).
MathematicsMarked price from selling pricemedium
An item sold for 1500 after a 25% discount. Find the marked price. Slide to explore.
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What it is
The marked price is recovered from the selling price and the discount percentage.
🏷️MP = SP ÷ (1 − discount% ÷ 100)
Marked price2000 ₹
Answer
MP = SP / (1 - discount% / 100) = 1500 / (1 - 25/100) = 1500 / 0.75 = 2000. Because the selling price is 75% of the marked price, we divide by 0.75 to get back to the full price.
MP = SP / (1 - discount% / 100)
•MP = SP / (1 - discount% / 100)
•SP is (100 - discount%) of MP
•SP 1500 at 25% off -> MP 2000
Why learn this
It tells you the original tag price before a discount.
💡 Memory trick
MP = SP / (1 - discount% / 100).
MathematicsDividing in a ratioeasy
Divide 500 in the ratio 2 : 3. Find the first share. Slide the values to explore.
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What it is
To divide a total in a given ratio, each share is the total times that part over the sum of the parts.
🧩First share = total × a ÷ (a + b)
First share200.0
Answer
First share = total x a / (a + b) = 500 x 2 / (2 + 3) = 500 x 2 / 5 = 200. The second share is 500 x 3 / 5 = 300, and the two shares add back to 500.
share = total x part / sum of parts
•First share = total x a / (a + b)
•Shares add back to the total
•500 in 2 : 3 -> 200 and 300
Why learn this
It's how money, sweets or profits are split fairly.
💡 Memory trick
First share = total x a / (a + b).
MathematicsPercentageeasy
A student scored 45 marks out of 60. Find the percentage of marks.
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What it is
A percentage is a fraction with denominator 100; it compares a part to a whole out of 100.
Answer
Percentage = (marks obtained / total marks) x 100 = (45/60) x 100 = 75%. So the student scored 75%.
Percentage = (part / whole) x 100
•Percent = per hundred
•Percentage = (part/whole) x 100
•(45/60) x 100 = 75%
•a% of b = (a/100) x b
Why learn this
Marks, discounts and interest are all expressed as percentages.
💡 Memory trick
Percent means 'per hundred'. To find a% of b, compute (a/100) x b.
MathematicsProfit and loss percentagemedium
An article bought for 500 rupees is sold for 600 rupees. Find the profit percent.
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What it is
Profit or loss is always calculated as a percentage of the cost price.
Answer
Profit = selling price - cost price = 600 - 500 = 100 rupees. Profit percent = (profit / cost price) x 100 = (100/500) x 100 = 20%. So there is a 20% profit.
Profit% = (SP - CP)/CP x 100
•Profit = SP - CP = 100
•Profit% = (profit/CP) x 100
•(100/500) x 100 = 20%
•Always on cost price
Why learn this
Shopkeepers and buyers use it to know real gain or loss.
💡 Memory trick
Profit% and Loss% are BOTH found on Cost Price (CP), not selling price.
MathematicsDiscountmedium
The marked price of a shirt is 800 rupees and a discount of 15% is given. Find the selling price.
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What it is
A discount is a reduction given on the marked price of an article.
Answer
Discount = 15% of marked price = (15/100) x 800 = 120 rupees. Selling price = marked price - discount = 800 - 120 = 680 rupees. So the shirt is sold for 680 rupees.
SP = MP - discount ; discount = discount% of MP
•Discount is on the marked price
•Discount = (15/100) x 800 = 120
•SP = MP - discount
•SP = 800 - 120 = 680
Why learn this
It is how sale prices are calculated in shops.
💡 Memory trick
Discount is on the Marked Price (MP). Selling Price = MP - discount.
MathematicsCompound interesthard
Find the compound interest on 10000 rupees for 2 years at 10% per annum, compounded annually.
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What it is
Compound interest is interest calculated on the amount, which grows each period as interest is added to the principal.
Answer
Using A = P(1 + R/100)^n, we get A = 10000 x (1 + 10/100)^2 = 10000 x (1.1)^2 = 10000 x 1.21 = 12100 rupees. Compound interest = A - P = 12100 - 10000 = 2100 rupees.
A = P(1 + R/100)^n ; CI = A - P
•A = P(1 + R/100)^n
•A = 10000 x (1.1)^2 = 12100
•CI = A - P = 2100
•Interest is added to principal each year
Why learn this
Banks and loans use compound interest, so it grows faster than simple interest.
💡 Memory trick
Amount A = P(1 + R/100)^n. Compound interest = A - P.
Algebraic Expressions and Identities3 questions
MathematicsStandard identitymedium
Use a standard identity to expand (x + 5)^2.
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What it is
Algebraic identities are shortcuts that expand or factor expressions instantly.
Answer
Use the identity (a + b)^2 = a^2 + 2ab + b^2 with a = x and b = 5. So (x + 5)^2 = x^2 + 2(x)(5) + 5^2 = x^2 + 10x + 25.
(a + b)^2 = a^2 + 2ab + b^2
•(a + b)^2 = a^2 + 2ab + b^2
•Here a = x, b = 5
•= x^2 + 10x + 25
Why learn this
They save time in almost every algebra, physics and competitive-exam problem.
💡 Memory trick
Square the first, twice the product, square the last: a^2 + 2ab + b^2.
MathematicsIdentity (a + b)^2medium
Use an identity to expand (x + 5)^2 and to evaluate 103^2.
Reveal answerHide answer▾
What it is
The identity (a + b)^2 = a^2 + 2ab + b^2 holds for all values of a and b.
It speeds up squaring binomials and mental calculation.
💡 Memory trick
Square of a sum = first squared + twice the product + last squared.
MathematicsIdentities (a - b)^2 and (a + b)(a - b)medium
Evaluate 98 x 102 using a suitable identity.
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What it is
Two key identities are (a - b)^2 = a^2 - 2ab + b^2 and (a + b)(a - b) = a^2 - b^2.
Answer
Write 98 = 100 - 2 and 102 = 100 + 2, so 98 x 102 = (100 - 2)(100 + 2). Using (a - b)(a + b) = a^2 - b^2 with a = 100 and b = 2: = 100^2 - 2^2 = 10000 - 4 = 9996.
(a + b)(a - b) = a^2 - b^2
•(a - b)^2 = a^2 - 2ab + b^2
•(a + b)(a - b) = a^2 - b^2
•98 x 102 = 100^2 - 2^2
•= 9996
Why learn this
They make squaring differences and products of conjugates quick.
💡 Memory trick
(a+b)(a-b) = a^2 - b^2: the 'difference of two squares'.
Mensuration18 questions
MathematicsArea of a trapeziummedium
Find the area of a trapezium whose parallel sides are 12 cm and 8 cm and whose height is 5 cm.
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What it is
A trapezium's area is the average of its two parallel sides times the height.
Area = ½ (a + b) h = ½ (8 + 12) × 5 = 50 cm²
Trapezium — change the sides and height; the area follows ½ (a + b) h
Answer
Area of a trapezium = (1/2) x (sum of parallel sides) x height = (1/2) x (12 + 8) x 5 = (1/2) x 20 x 5 = 50 square cm.
Area = (1/2)(a + b) x h
•Area = (1/2)(a + b) h
•= (1/2)(12 + 8)(5)
•= 50 cm^2
Why learn this
It measures real shapes - fields, plots and cross-sections that are not perfect rectangles.
💡 Memory trick
Half the sum of the parallel sides, times the height: (1/2)(a + b) h.
MathematicsArea of a circleeasy
Find the area of a circle of radius 7 units (take pi = 3.14159). Slide the radius to explore.
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What it is
The area of a circle grows with the square of its radius.
⭕Circle area = π × r²
Area153.9 units²
Answer
Area = pi x r^2 = 3.14159 x 7^2 = 3.14159 x 49 = 153.94 square units. Because the radius is squared, doubling it makes the area four times as large.
A = pi x r^2
•Area = pi x r^2
•Unit: square units
•Double the radius -> four times the area
Why learn this
It sizes anything round - plots, pipes, plates and wheels.
💡 Memory trick
Area = pi x r^2. Double r -> four times the area.
MathematicsVolume of a cuboideasy
Find the volume of a cuboid 3 x 4 x 5 units. Slide the dimensions to explore.
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What it is
The volume of a cuboid is the product of its length, breadth and height.
📦Cuboid volume = l × b × h
Volume60 units³
Answer
Volume = length x breadth x height = 3 x 4 x 5 = 60 cubic units. Increasing any one of the three dimensions increases the volume in proportion.
V = l x b x h
•Volume = l x b x h
•Unit: cubic units
•Change any dimension -> volume scales with it
Why learn this
It's how we measure the space in boxes, rooms and tanks.
💡 Memory trick
Volume = l x b x h. All three dimensions multiply.
MathematicsArea of a triangleeasy
Find the area of a triangle with base 10 and height 8 units. Slide to explore.
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What it is
The area of a triangle is half the product of its base and height.
📐Triangle area = ½ × base × height
Area40.0 units²
Answer
Area = 1/2 x base x height = 1/2 x 10 x 8 = 40 square units. A triangle covers exactly half of a rectangle with the same base and height, which is where the 1/2 comes from.
A = 1/2 x base x height
•Area = 1/2 x base x height
•Unit: square units
•Half of the matching rectangle
Why learn this
Triangles tile every roof, ramp and truss we build.
💡 Memory trick
Area = 1/2 x base x height - half of the matching rectangle.
MathematicsArea of a rectangleeasy
Find the area of a rectangle 5 units long and 4 units wide. Slide to explore.
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What it is
The area of a rectangle is its length multiplied by its breadth.
🟦Rectangle area = length × breadth
Area20 units²
Answer
Area = length x breadth = 5 x 4 = 20 square units. If the length and breadth are equal, the rectangle is a square and the area becomes side x side.
A = length x breadth
•Area = length x breadth
•Unit: square units
•Square is the special case length = breadth
Why learn this
It's the first area formula, used for floors, walls and screens.
💡 Memory trick
Area = length x breadth. Square is the case length = breadth.
MathematicsArea of a squareeasy
Find the area of a square of side 6 units. Slide the side to explore.
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What it is
The area of a square is the length of its side multiplied by itself.
⬛Square area = side²
Area36 units²
Answer
Area = side x side = 6 x 6 = 36 square units. Because the side is squared, doubling it to 12 makes the area 144 - four times as large.
A = side^2
•Area = side^2
•Unit: square units
•Double the side -> four times the area
Why learn this
It's the simplest area formula, used for tiles, boards and plots.
💡 Memory trick
Area = side x side = side^2. Double the side -> four times the area.
MathematicsVolume of a cubeeasy
Find the volume of a cube of side 4 units. Slide the side to explore.
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What it is
The volume of a cube is the length of its edge raised to the power three.
🧊Cube volume = side³
Volume64 units³
Answer
Volume = side x side x side = 4^3 = 64 cubic units. Since the edge is cubed, even a small increase in the side makes the volume grow quickly.
V = side^3
•Volume = side^3
•Unit: cubic units
•Cubing the edge makes volume grow fast
Why learn this
It measures the space in cubic boxes, dice and storage cubes.
💡 Memory trick
Volume = side^3. Three equal edges multiply.
MathematicsCircumference of a circleeasy
Find the circumference of a circle of radius 7 units (pi = 3.14159). Slide to explore.
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What it is
The circumference is the distance around a circle, equal to 2 x pi x radius.
⭕Circumference = 2 × π × r
Circumference44.0 units
Answer
Circumference = 2 x pi x r = 2 x 3.14159 x 7 = 43.98 units. Unlike the area, the circumference grows in direct proportion to the radius - double the radius and you double the distance around.
C = 2 x pi x r
•Circumference = 2 x pi x r
•Unit: units of length
•Grows in proportion to the radius
Why learn this
It's how we measure wheels, pipes and running tracks.
💡 Memory trick
Circumference = 2 x pi x r. Grows in step with the radius.
MathematicsPerimeter of a rectangleeasy
Find the perimeter of a rectangle 5 units long and 3 units wide. Slide to explore.
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What it is
The perimeter of a rectangle is twice the sum of its length and breadth.
🔲Perimeter = 2 × (length + breadth)
Perimeter16 units
Answer
Perimeter = 2 x (length + breadth) = 2 x (5 + 3) = 2 x 8 = 16 units. Because a rectangle has two lengths and two breadths, we add one of each and double the total.
P = 2 x (length + breadth)
•Perimeter = 2 x (length + breadth)
•Two lengths and two breadths
•Unit: units of length
Why learn this
It's the fencing or border needed around a rectangular field or frame.
💡 Memory trick
Perimeter = 2 x (length + breadth).
MathematicsPerimeter of a squareeasy
Find the perimeter of a square of side 6 units. Slide the side to explore.
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What it is
The perimeter of a square is four times the length of one side.
⬜Square perimeter = 4 × side
Perimeter24 units
Answer
Perimeter = 4 x side = 4 x 6 = 24 units. Since a square has four equal sides, we simply multiply one side by four.
P = 4 x side
•Perimeter = 4 x side
•Four equal sides
•Unit: units of length
Why learn this
It's the border or fence needed around a square plot or tile.
💡 Memory trick
Perimeter = 4 x side - four equal sides.
MathematicsPerimeter of a triangleeasy
Find the perimeter of a triangle with sides 3, 4 and 5 units. Slide to explore.
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What it is
The perimeter of a triangle is the sum of the lengths of its three sides.
🔺Triangle perimeter = a + b + c
Perimeter12 units
Answer
Perimeter = a + b + c = 3 + 4 + 5 = 12 units. For any triangle, the perimeter is just the sum of its three side lengths.
P = a + b + c
•Perimeter = a + b + c
•Add all three sides
•Unit: units of length
Why learn this
It's the total boundary of any triangular shape.
💡 Memory trick
Perimeter = a + b + c.
MathematicsArea of a parallelogrameasy
Find the area of a parallelogram with base 10 and height 6 units. Slide to explore.
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What it is
The area of a parallelogram is its base multiplied by its perpendicular height.
📐Parallelogram area = base × height
Area60 units²
Answer
Area = base x height = 10 x 6 = 60 square units. The height must be the perpendicular distance between the base and its opposite side, not the length of the slanting side.
A = base x height
•Area = base x height
•Use the perpendicular height
•Unit: square units
Why learn this
It generalises the rectangle's area to slanted shapes.
💡 Memory trick
Area = base x height (use the perpendicular height).
MathematicsArea of a rhombusmedium
Find the area of a rhombus with diagonals 8 and 6 units. Slide to explore.
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What it is
The area of a rhombus is half the product of its two diagonals.
♦️Rhombus area = ½ × d₁ × d₂
Area24.0 units²
Answer
Area = 1/2 x d1 x d2 = 1/2 x 8 x 6 = 24 square units. This works because the diagonals of a rhombus bisect each other at right angles, splitting it into four right triangles.
A = 1/2 x d1 x d2
•Area = 1/2 x d1 x d2
•Diagonals meet at right angles
•Unit: square units
Why learn this
It's used for kite-shaped and diamond-shaped figures.
💡 Memory trick
Area = 1/2 x d1 x d2 (the diagonals).
MathematicsCircumference from diametereasy
Find the circumference of a circle of diameter 7 units (pi = 3.14159). Slide to explore.
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What it is
The circumference of a circle is pi times its diameter.
⭕Circumference = π × diameter
Circumference22.0 units
Answer
Circumference = pi x diameter = 3.14159 x 7 = 21.99 units. Because the diameter is twice the radius, this is exactly the same as 2 x pi x r. In fact pi is defined as the ratio of the circumference to the diameter.
C = pi x diameter
•Circumference = pi x diameter
•Same as 2 x pi x r
•pi = circumference / diameter
Why learn this
It's the most direct way to find the distance around a circle from its width.
💡 Memory trick
Circumference = pi x diameter (= 2 pi r).
MathematicsArea of a trapeziummedium
Find the area of a trapezium whose parallel sides are 12 cm and 8 cm and the distance between them is 5 cm.
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What it is
The area of a trapezium is half the sum of its parallel sides times the distance between them.
Answer
Area of a trapezium = 1/2 x (sum of parallel sides) x height = 1/2 x (12 + 8) x 5 = 1/2 x 20 x 5 = 50 cm^2. So the area is 50 square centimetres.
Area of trapezium = 1/2 x (sum of parallel sides) x height
•Area = 1/2 x (a + b) x h
•a = 12, b = 8, h = 5
•1/2 x 20 x 5 = 50
•Area = 50 cm^2
Why learn this
Many fields and plots have a trapezium shape.
💡 Memory trick
Area = 1/2 x (sum of parallel sides) x height.
MathematicsArea of a rhombusmedium
Find the area of a rhombus whose diagonals are 16 cm and 12 cm.
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What it is
The area of a rhombus is half the product of its two diagonals.
Answer
Area of a rhombus = 1/2 x d1 x d2, where d1 and d2 are the diagonals. Here = 1/2 x 16 x 12 = 1/2 x 192 = 96 cm^2. So the area of the rhombus is 96 square centimetres.
Area of rhombus = 1/2 x d1 x d2
•Area = 1/2 x d1 x d2
•d1 = 16, d2 = 12
•1/2 x 192 = 96
•Area = 96 cm^2
Why learn this
It is a quick way to find the area of kite- and diamond-shaped figures.
💡 Memory trick
Area of a rhombus = 1/2 x d1 x d2 (half the product of the diagonals).
MathematicsSurface area of a cuboidmedium
Find the total surface area of a cuboid of length 8 cm, breadth 5 cm and height 3 cm.
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What it is
The total surface area of a cuboid is 2(lb + bh + hl).
Answer
Total surface area of a cuboid = 2(lb + bh + hl). Here lb = 8 x 5 = 40, bh = 5 x 3 = 15, hl = 3 x 8 = 24. Sum = 40 + 15 + 24 = 79. Total surface area = 2 x 79 = 158 cm^2.
Total surface area of cuboid = 2(lb + bh + hl)
•TSA = 2(lb + bh + hl)
•lb=40, bh=15, hl=24
•Sum = 79
•TSA = 158 cm^2
Why learn this
It tells how much material is needed to cover a box on all sides.
💡 Memory trick
Add the three face products lb, bh, hl and double the total.
MathematicsVolume of a cylindermedium
Find the volume of a cylinder of radius 7 cm and height 10 cm. (Take pi = 22/7.)
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What it is
The volume of a cylinder is the base area times the height, that is pi r^2 h.
Answer
Volume of a cylinder = pi r^2 h = (22/7) x 7^2 x 10 = (22/7) x 49 x 10 = 22 x 7 x 10 = 1540 cm^3. So the volume is 1540 cubic centimetres.
Volume of cylinder = pi r^2 h
•Volume = pi r^2 h
•r = 7, h = 10, pi = 22/7
•(22/7) x 49 x 10 = 1540
•Volume = 1540 cm^3
Why learn this
It tells how much a pipe, tank or can can hold.
💡 Memory trick
Volume of a cylinder = pi x r^2 x h (circle area times height).
Understanding Quadrilaterals7 questions
MathematicsInterior angle sum of a polygoneasy
Find the sum of the interior angles of a pentagon (5 sides). Slide the number of sides to explore.
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What it is
The sum of the interior angles of a polygon depends only on its number of sides.
⬟Interior angle sum = (n − 2) × 180°
Sum of interior angles540 °
Answer
Sum of interior angles = (n - 2) x 180 = (5 - 2) x 180 = 3 x 180 = 540 degrees. A polygon can be split into (n - 2) triangles, each contributing 180 degrees.
sum of interior angles = (n - 2) x 180
•Sum = (n - 2) x 180
•Triangle 180, quadrilateral 360, pentagon 540
•Splits into (n - 2) triangles
Why learn this
It's the basis for finding angles in any polygon.
💡 Memory trick
Sum = (n - 2) x 180. Each extra side adds 180.
MathematicsExterior angle of a regular polygoneasy
Find each exterior angle of a regular hexagon (6 sides). Slide the number of sides to explore.
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What it is
Each exterior angle of a regular polygon is 360 degrees divided by the number of sides.
🔻Exterior angle = 360° ÷ n
Each exterior angle60.0 °
Answer
Each exterior angle = 360 / n = 360 / 6 = 60 degrees. The exterior angles of any polygon always add up to 360 degrees - one full turn - so for a regular polygon we just divide 360 by the number of sides.
exterior angle = 360 / n
•Exterior angle = 360 / n
•All exterior angles add to 360
•Regular hexagon: 60 degrees each
Why learn this
It's the turn you make at each corner walking around the shape.
💡 Memory trick
Exterior angle = 360 / n. They add up to 360.
MathematicsNumber of diagonals of a polygonmedium
How many diagonals does a hexagon (6 sides) have? Slide the number of sides to explore.
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What it is
The number of diagonals of a polygon is n times (n minus 3), divided by two.
🔷Diagonals = n(n − 3) ÷ 2
Number of diagonals9
Answer
Diagonals = n(n - 3) / 2 = 6 x (6 - 3) / 2 = 6 x 3 / 2 = 9. From each corner you can draw a diagonal to every other corner except itself and its two neighbours, and dividing by two avoids counting each diagonal twice.
diagonals = n(n - 3) / 2
•Diagonals = n(n - 3) / 2
•Triangle has 0, hexagon has 9
•Joins non-adjacent corners
Why learn this
It counts the straight lines joining non-adjacent corners.
💡 Memory trick
Diagonals = n(n - 3) / 2.
MathematicsInterior angle of a regular polygonmedium
Find each interior angle of a regular hexagon (6 sides). Slide the number of sides to explore.
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What it is
Each interior angle of a regular polygon is the total angle sum divided by the number of sides.
⬟Interior angle = (n − 2) × 180 ÷ n
Each interior angle120.0 °
Answer
Interior angle = (n - 2) x 180 / n = (6 - 2) x 180 / 6 = 720 / 6 = 120 degrees. We share the total interior angle sum, (n - 2) x 180, equally among the n corners.
interior angle = (n - 2) x 180 / n
•Interior angle = (n - 2) x 180 / n
•Regular hexagon: 120 degrees
•Approaches 180 as n grows
Why learn this
It's needed to draw or tile regular shapes accurately.
💡 Memory trick
Interior angle = (n - 2) x 180 / n.
MathematicsAngle sum property of a quadrilateraleasy
Three angles of a quadrilateral are 90, 85 and 65 degrees. Find the fourth angle.
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What it is
The sum of the four interior angles of any quadrilateral is 360 degrees.
Answer
The sum of the interior angles of a quadrilateral is 360 degrees. Sum of the three given angles = 90 + 85 + 65 = 240 degrees. So the fourth angle = 360 - 240 = 120 degrees.
Sum of interior angles of a quadrilateral = 360 degrees
•Angle sum of a quadrilateral = 360 degrees
•Sum of given angles = 240 degrees
•Fourth angle = 360 - 240
•Fourth angle = 120 degrees
Why learn this
It lets you find a missing angle when three are known.
💡 Memory trick
A quadrilateral = two triangles, so 2 x 180 = 360 degrees.
MathematicsProperties of a parallelogrammedium
State any three properties of a parallelogram.
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What it is
In a parallelogram opposite sides are equal and parallel, opposite angles are equal, and diagonals bisect each other.
Answer
Three properties of a parallelogram are: opposite sides are equal and parallel; opposite angles are equal; and the diagonals bisect each other (they cut each other into two equal parts). Also, any two adjacent angles of a parallelogram are supplementary, adding up to 180 degrees.
Adjacent angles of a parallelogram sum to 180 degrees
•Opposite sides equal and parallel
•Opposite angles equal
•Diagonals bisect each other
•Adjacent angles are supplementary (180 degrees)
Why learn this
These properties are used to prove results and solve geometry problems.
Find the sum of the interior angles of a hexagon, and each interior angle of a regular hexagon.
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What it is
The sum of the interior angles of an n-sided polygon is (n - 2) x 180 degrees.
Answer
A hexagon has n = 6 sides. Sum of interior angles = (n - 2) x 180 = (6 - 2) x 180 = 4 x 180 = 720 degrees. For a regular hexagon, all angles are equal, so each interior angle = 720 / 6 = 120 degrees.
Sum of interior angles = (n - 2) x 180 degrees
•Sum = (n - 2) x 180 degrees
•Hexagon: (6 - 2) x 180 = 720 degrees
•Each angle of a regular polygon = sum / n
•Regular hexagon: 720/6 = 120 degrees
Why learn this
It works for any polygon, letting us find angles of pentagons, hexagons and more.
💡 Memory trick
Split the polygon into (n - 2) triangles; each gives 180 degrees.
Direct and Inverse Proportions9 questions
MathematicsDownstream speedeasy
A boat rows at 10 km/h in still water down a 4 km/h stream. Find its downstream speed. Slide to explore.
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What it is
When a boat travels downstream, the stream's speed adds to the boat's own speed.
🚣Downstream = boat + stream
Downstream speed14 km/h
Answer
Downstream speed = boat speed + stream speed = 10 + 4 = 14 km/h. Going with the current, the stream pushes the boat along, so the two speeds add up.
downstream = boat speed + stream speed
•Downstream = boat + stream
•The current helps
•10 + 4 = 14 km/h
Why learn this
It's the basis of boats-and-streams problems.
💡 Memory trick
Downstream = boat + stream.
MathematicsUpstream speedeasy
A boat rows at 10 km/h in still water against a 4 km/h stream. Find its upstream speed. Slide to explore.
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What it is
When a boat travels upstream, the stream's speed is subtracted from the boat's own speed.
🚣Upstream = boat − stream
Upstream speed6 km/h
Answer
Upstream speed = boat speed - stream speed = 10 - 4 = 6 km/h. Going against the current, the stream slows the boat, so we subtract the stream speed.
upstream = boat speed - stream speed
•Upstream = boat - stream
•The current opposes
•10 - 4 = 6 km/h
Why learn this
It pairs with downstream to solve river-travel problems.
💡 Memory trick
Upstream = boat - stream.
MathematicsAverage speed for a round triphard
A car goes one way at 40 km/h and back at 60 km/h. Find its average speed. Slide to explore.
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What it is
For equal distances at two different speeds, the average speed is the harmonic mean of the two speeds.
🔁Average = 2ab ÷ (a + b)
Average speed48.0 km/h
Answer
For equal distances, average speed = 2ab / (a + b) = 2 x 40 x 60 / (40 + 60) = 4800 / 100 = 48 km/h. It is less than the simple average of 50 because more time is spent at the slower speed.
average speed = 2ab / (a + b)
•Average = 2ab / (a + b) for equal distances
•The harmonic mean, not the simple average
•40 and 60 give 48, not 50
Why learn this
A common trap is to take the simple average, which is wrong for equal distances.
💡 Memory trick
Average = 2ab / (a + b), not (a + b) / 2.
MathematicsFourth proportionalmedium
Find d if 2 : 3 = 4 : d. Slide a, b, c to explore.
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What it is
The fourth proportional completes a proportion a : b = c : d.
🔢Fourth term = (b × c) ÷ a
Fourth proportional6.00
Answer
Since a : b = c : d means a x d = b x c, we get d = (b x c) / a = (3 x 4) / 2 = 12 / 2 = 6. So 2 : 3 = 4 : 6.
d = (b x c) / a
•a : b = c : d means a x d = b x c
•d = (b x c) / a
•2 : 3 = 4 : 6
Why learn this
It solves proportion problems in one step.
💡 Memory trick
If a : b = c : d, then d = (b x c) / a.
MathematicsTrain crossing a poleeasy
A 200 m train passes a pole in 10 s. Find its speed. Slide to explore.
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What it is
To pass a pole, a train travels a distance equal to its own length, so its speed is length divided by time.
🚂Speed = length ÷ time
Speed20.0 m/s
Answer
Speed = length / time = 200 / 10 = 20 m/s. When a train crosses a pole (a point), the distance it covers is just its own length, so the speed is the length divided by the time.
speed = length / time
•Speed = length / time
•Crossing a pole covers the train's length
•200 m in 10 s -> 20 m/s
Why learn this
It's a classic trains problem that builds speed intuition.
💡 Memory trick
Crossing a pole: speed = length / time.
MathematicsTime and workmedium
If 10 workers take 12 days, how long do 15 workers take? Slide to explore.
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What it is
The number of workers and the days to finish a job are inversely proportional.
👷Days = (workers₁ × days₁) ÷ workers₂
Days needed8.0
Answer
Since workers x days is constant, days2 = (workers1 x days1) / workers2 = (10 x 12) / 15 = 120 / 15 = 8 days. More workers share the same total work, so it finishes in fewer days.
days2 = (workers1 x days1) / workers2
•workers1 x days1 = workers2 x days2
•Workers and days are inversely proportional
•10 x 12 = 15 x 8
Why learn this
It's how we work out staffing to meet a deadline.
💡 Memory trick
workers1 x days1 = workers2 x days2.
MathematicsUnitary methodeasy
If 12 items cost 240, what is the cost of one? Slide the values to explore.
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What it is
The unit rate is the cost of a single item, found by dividing the total cost by the quantity.
💲Unit rate = total ÷ quantity
Cost per item20.00 ₹
Answer
Unit rate = total cost / quantity = 240 / 12 = 20 per item. This is the unitary method: find the value of one unit, then you can scale up to any number.
unit rate = total cost / quantity
•Unit rate = total cost / quantity
•The cost of one item
•240 for 12 -> 20 each
Why learn this
It's how we compare which pack is the better value.
💡 Memory trick
Unit rate = total cost / quantity.
MathematicsDirect proportionmedium
If 5 pens cost 60 rupees, what is the cost of 8 pens?
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What it is
In direct proportion two quantities increase or decrease together so that their ratio stays constant.
Answer
Cost is in direct proportion to the number of pens, so x1/y1 = x2/y2. Here 5/60 = 8/y2, so y2 = (8 x 60)/5 = 480/5 = 96 rupees. Therefore 8 pens cost 96 rupees.
x1/y1 = x2/y2 (direct proportion)
•Direct proportion: x/y is constant
•5/60 = 8/cost
•cost = (8 x 60)/5
•8 pens cost 96 rupees
Why learn this
It solves problems like cost of items or distance for fuel.
💡 Memory trick
Direct: x/y = constant. More of one means more of the other.
MathematicsInverse proportionmedium
If 6 workers finish a job in 12 days, how many days will 8 workers take?
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What it is
In inverse proportion one quantity increases as the other decreases, so their product stays constant.
Answer
The number of workers and the number of days are in inverse proportion, so x1 y1 = x2 y2. Here 6 x 12 = 8 x d, so 72 = 8d and d = 72/8 = 9 days. Therefore 8 workers will finish the job in 9 days.
x1 y1 = x2 y2 (inverse proportion)
•Inverse proportion: product is constant
•6 x 12 = 8 x days
•72 = 8d
•days = 9
Why learn this
It solves problems like more workers finishing a job in less time.
💡 Memory trick
Inverse: x1 y1 = x2 y2. More workers, less time (product constant).
Rational Numbers2 questions
MathematicsProperties of rational numbersmedium
State whether rational numbers are commutative under addition and under subtraction, with an example.
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What it is
Rational numbers are closed, commutative and associative under addition and multiplication, but not under subtraction or division for commutativity.
Answer
Rational numbers are commutative under addition: for example, 1/2 + 1/3 = 1/3 + 1/2 = 5/6. However, they are not commutative under subtraction: for example, 1/2 - 1/3 = 1/6 but 1/3 - 1/2 = -1/6, which are not equal. Rational numbers are closed and associative under both addition and multiplication.
a + b = b + a (commutative addition)
•Closed under +, -, x (and / by non-zero)
•Commutative under + and x
•Not commutative under - and /
•Associative under + and x
Why learn this
These properties let us rearrange and group numbers freely while calculating.
💡 Memory trick
Commutative = order does not matter; Associative = grouping does not matter.
MathematicsAdditive and multiplicative inverseeasy
Find the additive inverse and the multiplicative inverse of 3/5.
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What it is
The additive inverse of a number gives 0 when added; the multiplicative inverse (reciprocal) gives 1 when multiplied.
Answer
The additive inverse of 3/5 is -3/5, because 3/5 + (-3/5) = 0. The multiplicative inverse (reciprocal) of 3/5 is 5/3, because (3/5) x (5/3) = 1. Note that 0 has no multiplicative inverse.
a + (-a) = 0 ; a x (1/a) = 1
•Additive inverse of a/b is -a/b (sum = 0)
•Multiplicative inverse of a/b is b/a (product = 1)
•Additive inverse of 3/5 is -3/5
•Reciprocal of 3/5 is 5/3
Why learn this
Inverses are used to solve equations and simplify expressions.
💡 Memory trick
Additive inverse: flip the sign. Multiplicative inverse: flip the fraction (reciprocal).
Cubes and Cube Roots1 question
MathematicsCubes and cube rootsmedium
Find the cube root of 3375 by prime factorisation.
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What it is
The cube of a number is that number multiplied by itself three times; the cube root reverses it.
Answer
Prime factorise 3375: 3375 = 3 x 3 x 3 x 5 x 5 x 5 = (3 x 3 x 3) x (5 x 5 x 5). Taking one factor from each triplet gives 3 x 5 = 15. So the cube root of 3375 is 15. Check: 15 x 15 x 15 = 3375.
cube root = product of one prime from each triplet
•Cube: n^3 = n x n x n
•3375 = 3^3 x 5^3
•Group primes in triplets
•cube root of 3375 = 15
Why learn this
Cubes and cube roots appear in volume calculations and algebra.
💡 Memory trick
Cube root by prime factorisation: make TRIPLETS of equal primes, take one from each.
Factorisation1 question
MathematicsFactorisation by common factors and groupingmedium
Factorise: 6xy + 9x + 4y + 6 by grouping.
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What it is
Factorisation writes an expression as a product of factors, using common factors or grouping of terms.
Answer
Group the terms: (6xy + 9x) + (4y + 6). Take common factors from each group: 3x(2y + 3) + 2(2y + 3). Now (2y + 3) is common, so factor it out: (2y + 3)(3x + 2). Therefore 6xy + 9x + 4y + 6 = (2y + 3)(3x + 2).
Factor by grouping and taking common factors
•Group terms in pairs
•3x(2y + 3) + 2(2y + 3)
•Common factor (2y + 3)
•Answer: (2y + 3)(3x + 2)
Why learn this
It is the reverse of expanding and is used to simplify and solve equations.
💡 Memory trick
Take out the greatest common factor first; if none, try grouping terms in pairs.
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