Arithmetic Reasoning — SSC CGL Reasoning Formulas and Traps

Calculation and Number Facts

Squares, cubes and the digit-sum test

Term n of a geometric progression

T_n = a\,r^{\,n-1}

$a$ the first term
$r$ the common ratio
$n$ the position of the term

LCM, HCF and remainders

HCF, LCM and remainders

\text{HCF} \times \text{LCM} = a \times b \qquad N = k \times \text{LCM} + r

$a, b$ the two numbers
$r$ the remainder left by every divisor
$k$ a whole number, 1 for the smallest such number

Common traps

Multiplying before adding the sum

"The sum of 12 and 8 multiplied by 3" is

(12 + 8) \times 3 = 60

. Doing

12 + 8 \times 3 = 36

answers a different question, and SSC prints that wrong total as an option.

Squaring each number inside a bracket

(7 - 4)^2

is

3^2 = 9

. Squaring each number first gives

49 - 16 = 33

, which is wrong: work out the bracket, then square.

Taking a square or a fourth power for a cube

625 is

25^2

and

5^4

, but it is not a cube:

8^3 = 512

and

9^3 = 729

, and nothing between them is a cube.

Using the digit sum for 7 or 11

The digit-sum test gives the remainder for 9 and for 3 only. For 7 or 11 it gives a wrong remainder, so divide.

Raising the ratio to the power n

The 6th term of a geometric progression is

a r^5

, not

a r^6

: the first term has had no multiplication yet.

Using HCF times LCM for three numbers

For 2, 4 and 6 the HCF is 2 and the LCM is 12, so HCF times LCM is 24, while the product is 48. The rule holds only for two numbers.

Forgetting to add the remainder

The LCM itself leaves remainder 0. A number leaving remainder r for every divisor is the LCM plus r.

Equations from Word Problems

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Two facts about two numbers

Two numbers from their sum and difference

\text{Larger} = \frac{S + D}{2} \qquad \text{Smaller} = \frac{S - D}{2}

$S$ the sum of the two numbers
$D$ their difference

Ages: the gap never changes

An age now and an age t years ago

x = k(x - t)

$x$ the present age
$t$ the number of years ago
$k$ how many times the old age the present age is

Fixed totals and equal shares

New share and the average of evenly spaced numbers

\text{New share} = \frac{\text{Old count} \times \text{Old share}}{\text{New count}} \qquad \text{Average} = \frac{\text{First} + \text{Last}}{2}

Common traps

Stopping at the middle number

When the question asks for the largest of three consecutive even numbers, the sum divided by 3 gives the middle one. Add 2 to answer what was asked.

Taking the digit difference as the change

Reversing a two-digit number changes it by 9 times the difference of its digits, not by the difference itself. A change of 36 means the digits differ by 4.

Trusting one number test

One set of numbers can let a wrong option pass by chance. Use numbers to rule options out, then confirm the last one by adding or subtracting the equations.

Picking numbers that break the given equations

Choose only two letters freely and work out the others from the equations. Numbers that do not satisfy the given equations prove nothing.

Ageing only one person

"In 20 years" adds 20 to every age in the question. Adding it to the father's age only breaks the equation.

Keeping the ratio fixed

A mother three times her son's age now will not be three times his age later. The difference stays the same; the ratio shrinks towards 1.

Dividing the old share instead of the total

When people leave, first rebuild the total (count times share), then divide by the people who remain. Dividing the old share by anything gives a smaller amount, which cannot be right.

Finding the last term by adding the count

Five consecutive odd numbers starting at 15 end at

15 + 2 \times 4 = 23

, not

15 + 5 = 20

. Their average is the middle one, 19.

Ratio and Mixtures

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Splitting a total in a ratio

Size of one part

\text{One part} = \frac{T}{a + b} \qquad \text{One part} = \frac{\text{Difference}}{b - a}

$a : b$ the ratio
$T$ the total

Chained ratios and profit shares

Joining ratios and sharing profit

\frac{A}{C} = \frac{A}{B} \times \frac{B}{C} \qquad \text{Share} \propto \text{Capital} \times \text{Time}

Adding to a mixture

Amount to add

\text{New amount of added part} = \text{Fixed part} \times \frac{\text{its new ratio term}}{\text{fixed part's new ratio term}}

Common traps

Dividing the total by one term

For a ratio of 3 : 5 and a total of 96, one part is

96 \div 8 = 12

, not

96 \div 3

or

96 \div 5

. Divide by the sum of the terms.

Checking only the sum

Wrong options are often built to add up to the total. A pair that adds up correctly but is not in the ratio is still wrong.

Joining ratios without matching the middle term

A : B = 2 : 3 and B : C = 4 : 5 do not give A : C = 2 : 5. Make B equal first (12 in both): 8 : 12 : 15, so A : C = 8 : 15.

Ignoring time in a partnership

When partners invest for different lengths of time, share the profit by capital times time. The capitals alone decide only when the times are equal.

Changing the part that was not added

Adding water leaves the milk exactly as it was. Scale the new ratio to the unchanged part, never to the old total.

Giving the new amount instead of the amount added

If the water must become 15 L and there is already 10 L, the answer is the 5 L poured in, not 15 L.

Percentages and Interest

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A percentage as a fraction

A part, the whole, and selling price

p\% \text{ of } N = \frac{p}{100} \times N \qquad N = \frac{A \times 100}{p} \qquad SP = CP \times \frac{100 + p}{100}

$A$ the amount that is p% of the number
$SP, CP$ selling price and cost price

Two changes one after the other

Net effect of two changes

\text{Net change} = a + b + \frac{ab}{100}

$a$ the first change in per cent (negative for a fall)
$b$ the second change in per cent (negative for a fall)

Simple and compound interest

Simple interest, k times, and compound rate

SI = \frac{PRT}{100} \qquad R \times T = 100(k - 1) \qquad R = \frac{A_{n+1} - A_n}{A_n} \times 100

$P$ the principal
$R$ the rate in per cent a year
$T$ the time in years
Aₙthe compound amount after n years

Common traps

Using the share spent when the amount given is what is left

If the number you are given is the money left over, it matches 100% minus the share spent. Dividing it by the spent share gives a wrong total.

Taking profit on the selling price

A profit percentage is on the cost price unless the question says otherwise. A 20% profit on a cost of ₹150 is ₹30, giving ₹180.

Adding the two percentages

A 30% rise followed by a 30% fall is not zero change. The fall is taken on the bigger price, so the result is a 9% fall.

Dropping the sign of a fall

In

a + b + \frac{ab}{100}

, a fall is negative, so the product term is negative too. A 10% rise then a 10% fall is

10 - 10 - 1 = -1

, a 1% fall.

Counting the whole amount as interest

A sum that doubles has earned interest equal to the principal, not twice it. A sum that becomes 2.5 times has earned 1.5 times the principal.

Dividing by the later amount

For a compound rate from two consecutive years, divide the gap by the earlier year's amount. Dividing by the later amount gives a rate that is too small.

Speed, Time and Work

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Speed, trains and average speed

Speed, units and average speed

v = \frac{d}{t} \qquad 1\ \text{km/h} = \frac{5}{18}\ \text{m/s} \qquad \bar{v} = \frac{2ab}{a + b}

$d, t$ distance and time
$a, b$ the speeds over two equal distances

Boats and streams

Boat and stream from the two speeds

b = \frac{d + u}{2} \qquad s = \frac{d - u}{2}

$d$ the downstream speed
$u$ the upstream speed
$b, s$ the speeds of the boat in still water and of the stream

Work, taps and pipes

Same job, and pipes working together

W_1 T_1 = W_2 T_2 \qquad \frac{1}{T} = \frac{1}{a} - \frac{1}{b}

$W, T$ number of workers and time taken
$a$ time for the filling pipe alone
$b$ time for the emptying pipe alone

Common traps

Using the equal-distance shortcut for equal times

\frac{2ab}{a + b}

is for two equal distances. If the two parts take equal times, the average speed is the simple average

\frac{a + b}{2}

.

Forgetting the platform's length

A train crossing a platform has crossed it only when its last coach leaves the far end, so the distance is the train's length plus the platform's.

Swapping the boat and the stream

Half the sum of the two speeds is the boat's speed in still water; half the difference is the stream's. The stream is almost always the smaller number.

Averaging distances instead of speeds

When the two trips have different distances, first divide each distance by its time. The half-sum rule works on speeds only.

Treating workers and time as direct

More workers means fewer days, not more. If 4 machines take 12 days, 8 machines take 6 days.

Adding the emptying pipe

An outlet takes water away, so its rate is subtracted from the inlet's. Adding it gives a time that is far too short.

Counting, Chance and Shapes

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Counting pairs

Pairs and diagonals

\text{Pairs} = \frac{n(n - 1)}{2} \qquad \text{Diagonals} = \frac{n(n - 3)}{2}

$n$ the number of people, teams or sides

Probability as a fraction

Probability and its complement

P(E) = \frac{\text{favourable}}{\text{total}} \qquad P(\text{not } E) = 1 - P(E)

Shape facts: volumes and diagonals

Cylinder, cone and diagonal

V_{\text{cyl}} = \pi r^2 h \qquad V_{\text{cone}} = \tfrac{1}{3}\pi r^2 h \qquad d = \sqrt{l^2 + b^2}

$r, h$ radius of the base and height
$l, b$ length and breadth of the rectangle

Common traps

Counting each pair from both ends

n(n - 1)

counts the handshake between A and B once for A and once for B. For single handshakes or single matches, halve it.

Counting the sides as diagonals

\frac{n(n - 1)}{2}

counts every line joining two corners, and that includes the n sides. A diagonal joins corners that are not next to each other, so take the sides away:

\frac{n(n - 3)}{2}

.

Dividing by the wrong total

With 3 red and 9 other balls, the chance of red is

\frac{3}{12}

, not

\frac{3}{9}

. Divide by every ball in the bag, including the ones you want.

Taking the cone as half the cylinder

With the same base and height, a cone holds one third of the cylinder, not one half.

Treating an approximate diagonal as exact

\sqrt{l^2 + b^2}

is a whole number only for triples such as 3, 4, 5. Otherwise it lies between two whole numbers, and the nearest option is an approximation.

Arithmetic Reasoning formulas

Calculation and Number Facts

Squares, cubes and the digit-sum test

LCM, HCF and remainders

Common traps

Equations from Word Problems

Two facts about two numbers

Ages: the gap never changes

Fixed totals and equal shares

Common traps

Ratio and Mixtures

Splitting a total in a ratio

Chained ratios and profit shares

Adding to a mixture

Common traps

Percentages and Interest

A percentage as a fraction

Two changes one after the other

Simple and compound interest

Common traps

Speed, Time and Work

Speed, trains and average speed

Boats and streams

Work, taps and pipes

Common traps

Counting, Chance and Shapes

Counting pairs

Probability as a fraction

Shape facts: volumes and diagonals

Common traps

More SSC CGL Reasoning formula sheets