Master the basics - and everything after gets easier

What it is

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.

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.

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.

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.

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.

What it is

A trapezium's area is the average of its two parallel sides times the height.

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.

What it is

Simple interest is the extra money paid for borrowing, or earned for saving, a sum.

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.

What it is

The area of a circle grows with the square of its radius.

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.

What it is

The volume of a cuboid is the product of its length, breadth and height.

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.

What it is

A percentage expresses a part out of a whole as a number per hundred.

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.

What it is

The area of a triangle is half the product of its base and height.

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.

What it is

The area of a rectangle is its length multiplied by its breadth.

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.

What it is

A discount is a reduction on the marked price, found as a percentage of that price.

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.

What it is

The area of a square is the length of its side multiplied by itself.

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.

What it is

The volume of a cube is the length of its edge raised to the power three.

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.

What it is

The circumference is the distance around a circle, equal to 2 x pi x radius.

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.

What it is

Compound interest is calculated on the principal plus the interest already earned, so it grows faster than simple interest.

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.

What it is

Profit percentage is the profit expressed as a percentage of the cost price.

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.

What it is

The perimeter of a rectangle is twice the sum of its length and breadth.

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.

What it is

The perimeter of a square is four times the length of one side.

Answer

Perimeter = 4 x side = 4 x 6 = 24 units. Since a square has four equal sides, we simply multiply one side by four.

What it is

The perimeter of a triangle is the sum of the lengths of its three sides.

Answer

Perimeter = a + b + c = 3 + 4 + 5 = 12 units. For any triangle, the perimeter is just the sum of its three side lengths.

What it is

The area of a parallelogram is its base multiplied by its perpendicular height.

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.

What it is

The area of a rhombus is half the product of its two diagonals.

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.

What it is

Percentage change is the increase or decrease compared with the original value, expressed as a percentage.

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.

What it is

The sum of the interior angles of a polygon depends only on its number of sides.

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.

What it is

Each exterior angle of a regular polygon is 360 degrees divided by the number of sides.

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.

What it is

The number of diagonals of a polygon is n times (n minus 3), divided by two.

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.

What it is

Loss percentage is the loss expressed as a percentage of the cost price.

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.

What it is

The amount to be repaid is the principal plus the simple interest earned on it.

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.

What it is

The selling price for a required profit is the cost price increased by that profit percentage.

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.

What it is

The circumference of a circle is pi times its diameter.

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.

What it is

A percentage of a number is that fraction out of a hundred, multiplied by the number.

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.

What it is

Each interior angle of a regular polygon is the total angle sum divided by the number of sides.

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.

What it is

Depreciation reduces an asset's value by a fixed percentage each year, like compound interest in reverse.

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.

What it is

The cost price can be recovered from the selling price and the profit percentage by dividing.

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%.

What it is

When a boat travels downstream, the stream's speed adds to the boat's own speed.

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.

What it is

When a boat travels upstream, the stream's speed is subtracted from the boat's own speed.

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.

What it is

For equal distances at two different speeds, the average speed is the harmonic mean of the two speeds.

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.

What it is

Sales tax is a percentage of the price added on top of it.

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.

What it is

The side of a square is the square root of its area.

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.

What it is

The rate of interest is found by rearranging the simple interest formula.

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.

What it is

The time is found by rearranging the simple interest formula.

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.

What it is

The principal is found by rearranging the simple interest formula.

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.

What it is

The fourth proportional completes a proportion a : b = c : d.

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.

What it is

The marked price is recovered from the selling price and the discount percentage.

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.

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.

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.

What it is

The number of workers and the days to finish a job are inversely proportional.

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.

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.

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.

What it is

The unit rate is the cost of a single item, found by dividing the total cost by the quantity.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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).

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%.

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.

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.

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.

What it is

The identity (a + b)^2 = a^2 + 2ab + b^2 holds for all values of a and b.

Answer

Using (a + b)^2 = a^2 + 2ab + b^2: (x + 5)^2 = x^2 + 2(x)(5) + 5^2 = x^2 + 10x + 25. To find 103^2, write 103 = 100 + 3: (100 + 3)^2 = 100^2 + 2(100)(3) + 3^2 = 10000 + 600 + 9 = 10609.

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.

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).

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.

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.

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.

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.

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.

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.

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.

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.

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.