Maharashtra SSC Arithmetic (MSBSHSE) cheat sheet

Order of operations (BODMAS): Brackets, Of, Division and Multiplication left to right, then Addition and Subtraction left to right; 2 + 3 × 4 = 14, not 20.

Sign rules for integers: (+) × (+) = (+), (−) × (−) = (+), (+) × (−) = (−). The same rules apply to division: (−12) ÷ (−4) = 3.

Adding integers with unlike signs: subtract the smaller absolute value from the larger and keep the sign of the larger; (−9) + 4 = −5.

Subtracting a negative is adding: a − (−b) = a + b, so 7 − (−5) = 12.

A rational number is any number of the form p/q where p, q are integers and q ≠ 0; every integer and terminating decimal is rational.

To compare fractions, cross-multiply: a/b > c/d exactly when ad > bc (b, d > 0); compare 3/7 and 4/9 by 27 vs 28, so 4/9 is larger.

Multiplying decimals: multiply as whole numbers, then place the decimal point so the product has as many decimal places as the two factors combined.

Dividing by a decimal: shift the decimal point in both divisor and dividend the same number of places, e.g. 4.5 ÷ 0.15 = 450 ÷ 15 = 30.

A decimal terminates only if, in lowest terms, the denominator has no prime factors other than 2 and 5; 3/8 = 0.375 terminates, 1/3 does not.

A ratio a : b compares like quantities in the same units; it is simplified by dividing both terms by their HCF, e.g. 24 : 36 = 2 : 3.

Proportion a : b = c : d holds when ad = bc (product of extremes equals product of means); this is the cross-product test.

If a : b = 2 : 3 and b : c = 4 : 5, combine by making b common: a : b : c = 8 : 12 : 15.

Dividing a quantity Q in the ratio a : b : c gives shares Qa/(a+b+c), Qb/(a+b+c), Qc/(a+b+c).

Direct variation: y ∝ x means y = kx, so y1/x1 = y2/x2; doubling x doubles y.

Inverse variation: y ∝ 1/x means xy = k, so x1y1 = x2y2; doubling x halves y.

Mean proportional between a and b is √(ab); the third proportional to a and b satisfies a : b = b : c, so c = b²/a.

Scale in maps and models is a ratio (e.g. 1 : 50000); convert both quantities to the same unit before applying it.

Percent means per hundred: x% = x/100. To find x% of N, compute (x/100) × N; 15% of 240 = 36.

Percentage change = (change ÷ original value) × 100; always divide by the original (earlier) quantity, not the new one.

Profit = SP − CP and Loss = CP − SP; Profit% = (Profit/CP) × 100, Loss% = (Loss/CP) × 100 - always on cost price unless stated.

SP = CP × (100 + Profit%)/100 and SP = CP × (100 − Loss%)/100; invert these to recover CP from SP.

Successive discounts of a% then b% give a single discount of a + b − ab/100 percent, not a + b.

Marked price relates to selling price by SP = MP × (100 − Discount%)/100; discount is always calculated on marked price.

If an article is sold at a gain of x% and another at a loss of x% for the same SP, the net result is always a loss of x²/100 percent.

To increase a quantity by p%, multiply by (1 + p/100); to decrease by p%, multiply by (1 − p/100).

Simple interest: SI = P × R × T / 100, where P is principal, R the annual rate percent, T the time in years.

Amount under simple interest: A = P + SI = P(1 + RT/100).

Compound interest amount: A = P(1 + R/100)^n when interest is compounded annually for n years.

Compound interest: CI = A − P = P[(1 + R/100)^n − 1]; CI exceeds SI for the same P, R and n > 1.

For 2 years the difference CI − SI = P(R/100)²; use this shortcut instead of computing both fully.

Half-yearly compounding: halve the rate and double the time, A = P(1 + R/200)^(2n).

Quarterly compounding: A = P(1 + R/400)^(4n).

If the rate differs by year, multiply successive factors: A = P(1 + R1/100)(1 + R2/100)(1 + R3/100).

When time is given in months, express T in years as months/12 before substituting into SI = PRT/100.

Rectangle: area = l × b, perimeter = 2(l + b); square: area = a², perimeter = 4a.

Triangle: area = ½ × base × height; Heron's formula area = √[s(s−a)(s−b)(s−c)] with s = (a+b+c)/2.

Parallelogram area = base × height; trapezium area = ½ × (sum of parallel sides) × height.

Circle: circumference = 2πr, area = πr²; use π ≈ 22/7 or 3.14 as directed.

Cuboid: volume = lbh, total surface area = 2(lb + bh + hl), lateral surface area = 2h(l + b).

Cube: volume = a³, total surface area = 6a², lateral surface area = 4a², diagonal = a√3.

Cylinder: volume = πr²h, curved surface area = 2πrh, total surface area = 2πr(r + h).

1 m = 100 cm, so 1 m² = 10000 cm² and 1 m³ = 1000000 cm³; 1 litre = 1000 cm³.

Cone: volume = ⅓πr²h, curved surface area = πrl where slant height l = √(r² + h²).

Mean of raw data = (sum of all observations) ÷ (number of observations); the mean changes if any observation changes.

Median is the middle value of data arranged in ascending order; for n odd it is the ((n+1)/2)th term, for n even the average of the (n/2)th and (n/2+1)th terms.

Mode is the observation occurring most frequently; data may have one mode, several modes, or no mode.

For grouped data, class mark = (lower limit + upper limit)/2, and mean = Σ(fᵢxᵢ)/Σfᵢ.

Median of grouped data = l + [(n/2 − cf)/f] × h, where l = lower limit of median class, cf = cumulative frequency before it, f = its frequency, h = class width.

Mode of grouped data = l + [(f₁ − f₀)/(2f₁ − f₀ − f₂)] × h, using the modal class.

Empirical relation for moderately skewed data: Mode ≈ 3 × Median − 2 × Mean.

In a bar graph bars have equal width and gaps; in a histogram bars touch and area represents frequency.

A pie chart divides 360° in proportion to each value: sector angle = (value/total) × 360°.