Ratio and Proportion — AFCAT Numerical Ability Formulas and Traps

What a Ratio Means, and Its Named Forms

A ratio as equal parts

A ratio as parts

a : b = p : q \;\text{means}\; a = pk,\ b = qk

Simplest form of a ratio

Scaling both terms

a : b = ka : kb \quad (k \neq 0)

Working back from a duplicate or triplicate ratio

Undoing a duplicate ratio

a^2 : b^2 = p : q \;\Rightarrow\; a : b = \sqrt p : \sqrt q

Inverse, duplicate, triplicate and sub-duplicate ratios

Name	Of a : b	Example
Inverse (reciprocal)	$b : a$	$3 : 10 \to 10 : 3$
Duplicate	$a^2 : b^2$	$2 : 7 \to 4 : 49$ 'The compound ratio of a : b taken twice' is this same ratio.
Triplicate	$a^3 : b^3$	$1 : 3 \to 1 : 27$
Sub-duplicate	$\sqrt a : \sqrt b$	$49 : 64 \to 7 : 8$
Sub-triplicate	$\sqrt[3]{a} : \sqrt[3]{b}$	$8 : 125 \to 2 : 5$

Common traps

The ratio is not the amount

Two numbers in the ratio

3 : 5

adding to 40 are 15 and 25, not 3 and 5. Find one part before you answer.

Order matters

a : b

and

b : a

are different ratios. 'The ratio of X to Y' puts X first, so read which quantity the question names first.

Units first

45 minutes : 2 hours is not

45 : 2

. Convert to one unit,

45 : 120 = 3 : 8

, before simplifying.

A ratio answer keeps both terms

0.75 : 1.25

simplifies to

3 : 5

. Its value 0.6 is a fraction, and an option written as a ratio will not show it.

Duplicate is not double

The duplicate ratio of

3 : 4

is

9 : 16

, not

6 : 8

. Doubling both terms gives back the same ratio,

3 : 4

.

Sub-duplicate is not half

The sub-duplicate ratio of

16 : 25

is

4 : 5

, the square roots, not

8 : 12.5

.

One operation per name

If the sub-duplicate ratio of

a : b

is

2 : 3

, then

a : b = 4 : 9

and its duplicate is

16 : 81

. That is two squarings, so write each step down.

Roots do not always come out whole

a^2 : b^2 = 2 : 5

gives

a : b = \sqrt 2 : \sqrt 5

. Leave the surds in; an option that has 'cleaned them up' to whole numbers is wrong.

Joining, Compounding and Comparing Ratios

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Joining ratios through a common term

Joining two ratios

A : B = p : q,\ B : C = r : s \;\Rightarrow\; A : B : C = pr : qr : qs

Ratios from equal multiples

Equal multiples

pA = qB = rC \;\Rightarrow\; A : B : C = \tfrac 1p : \tfrac 1q : \tfrac 1r

Compound ratio: multiplying ratios

Compound ratio

(a : b),\ (c : d) \;\Rightarrow\; ac : bd

Finding a missing ratio from a compound ratio

The missing ratio

\dfrac ab \times \dfrac cd = \dfrac mn \;\Rightarrow\; \dfrac cd = \dfrac mn \div \dfrac ab

Comparing two ratios

Cross-multiplication test

\dfrac ab > \dfrac cd \;\text{exactly when}\; ad > bc \quad (b, d > 0)

Common traps

Scale, do not stitch

Joining

2 : 3

and

4 : 5

as

2 : 3 : 5

is wrong: the shared term is 3 in one and 4 in the other. Make it equal first, which gives

8 : 12 : 15

.

Read which term comes first

Given 'the ratio of the second to the third is

5 : 3

', the second number takes the 5. Writing it the other way round joins the wrong terms.

Equal multiples flip the ratio

3A = 5B

means

A : B = 5 : 3

, not

3 : 5

. The bigger multiplier goes with the smaller quantity.

Compounding multiplies, it does not add

The compound ratio of

2 : 3

and

4 : 5

is

8 : 15

. Adding the terms to get

6 : 8

has no meaning.

A ratio of fractions is one division

\tfrac 12 : \tfrac 13

is

\tfrac 12 \div \tfrac 13 = 3 : 2

, not

1 : 1

(the numerators) or

2 : 3

(the denominators).

Divide, do not subtract

If the compound ratio is

6 : 5

and one ratio is

3 : 2

, the other is

\tfrac 65 \div \tfrac 32 = 4 : 5

. Subtracting the terms to get

3 : 3

is wrong.

Bigger terms are not a bigger ratio

17 : 30

has bigger terms than

2 : 3

but is the smaller ratio,

0.57

against

0.67

. Compare values, not the size of the terms.

Dividing a Total in a Ratio, and Partnership

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Dividing a total in a ratio

Share in a ratio

\text{share of } p = \dfrac{p}{p + q + r} \times T

Dividing in a ratio of fractions

Clearing fractions

\tfrac 1a : \tfrac 1b : \tfrac 1c = \tfrac La : \tfrac Lb : \tfrac Lc \quad (L = \text{LCM of } a, b, c)

Numbers in a ratio with one more condition

Numbers from a ratio

x : y : z = p : q : r \;\Rightarrow\; x = pk,\ y = qk,\ z = rk

A changed ratio, or shares with an amount taken off

Gain or loss from a changed ratio

\text{change} = \left(\dfrac{q}{p + q} - \dfrac{s}{r + s}\right) T

Partnership: profit in the ratio of capital times time

Profit ratio

P_1 : P_2 = C_1 t_1 : C_2 t_2

Common traps

Answer in the unit asked

A 20 kg load split

1 : 4

has 16 kg in the larger part, which is 16,000 g. If the options are in grams, 16 is not among them.

A share of 'the other two together'

'A gets

\tfrac 29

of what B and C get together' makes A 2 parts out of 11, not 2 out of 9.

The denominators are not the ratio

In

\tfrac 12 : \tfrac 13

the first person gets more, 3 parts to 2. Reading the denominators as

2 : 3

turns the ratio round.

A sum of squares gives k squared

If

2k, 3k, 4k

have squares adding to 116, then

29k^2 = 116

, so

k^2 = 4

and

k = 2

. Stopping at

k = 4

doubles every number.

Add the deduction back

When shares less ₹10 each are in a given ratio, the ratio describes the reduced shares. Solve for those, then add the ₹10 back before you answer.

Time counts as much as money

When partners stay for different numbers of months, the ratio of capitals alone is wrong. Multiply each capital by its months first.

Proportion: Fourth, Third and Mean Proportionals

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Four numbers in proportion, and the fourth proportional

Product of extremes equals product of means

a : b :: c : d \;\text{means}\; ad = bc, \qquad d = \dfrac{bc}{a}

Third proportional

a : b = b : c \;\Rightarrow\; c = \dfrac{b^2}{a}

Mean proportional

a : x = x : b \;\Rightarrow\; x = \sqrt{ab}

Continued proportion

a : b = b : c \;\text{means}\; b^2 = ac

Common traps

Extremes times extremes

In

a : b :: c : d

the test is

a \times d = b \times c

. Multiplying

a \times c

, the first terms of each ratio, tests nothing.

Square the second number, not the first

The third proportional to 25 and 30 is

\dfrac{30^2}{25} = 36

.

\dfrac{25^2}{30}

is the third proportional to 30 and 25, the other way round.

A square root, not an average

The mean proportional between 4 and 16 is

\sqrt{4 \times 16} = 8

. Their ordinary average, 10, is a different number.

Continued proportion is a G.P., not an A.P.

In 4, 6, 9 the middle term passes

6^2 = 4 \times 9

. The test

p + r = 2q

is for an arithmetic progression and is a different condition.

Ratios in Algebra: Substitution and Componendo-Dividendo

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Evaluating an expression from a ratio

Substituting the ratio

x : y = p : q \;\Rightarrow\; \dfrac{ax + by}{cx + dy} = \dfrac{ap + bq}{cp + dq}

Finding x : y from an equation

Collecting x and y

ax + by = cx + dy \;\Rightarrow\; \dfrac xy = \dfrac{d - b}{a - c}

Componendo and dividendo

\dfrac ab = \dfrac cd \;\Rightarrow\; \dfrac{a + b}{a - b} = \dfrac{c + d}{c - d}

Equal ratios: sums and the k method

Sum of equal ratios

\dfrac ab = \dfrac cd = \dfrac ef = k \;\Rightarrow\; \dfrac{a + c + e}{b + d + f} = k

Common traps

Keep the minus sign

With

x : y = 3 : 7

,

(x + y) : (x - y) = 10 : -4 = -5 : 2

. Dropping the sign gives

5 : 2

, which is often one of the options.

Only when every term has the same degree

x : y = 2 : 3

does not fix

(x + 1) : y

, because the 1 carries no

k

. Put

x = 2k

,

y = 3k

and the

k

stays.

Collect before you read off the ratio

From

9x = 4y

,

x : y = 4 : 9

. Reading

9 : 4

off the coefficients turns the ratio round.

Do the same on both sides

From

\dfrac{x + y}{x - y} = \dfrac 73

, add and subtract on the left and on the right:

\dfrac{2x}{2y} = \dfrac{10}{4}

. Doing it on one side only changes the equation.

The multipliers must match

\dfrac{2a + c}{b + 2d}

is not

k

in general, because

a

and

b

carry different multipliers. The rule needs the same multiplier on each top-and-bottom pair.

When a Ratio Changes: Additions, Ages and Incomes

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Adding or subtracting the same number from both terms

A changed ratio

\dfrac{pk + x}{qk + x} = \dfrac rs

Age problems with ratios

Ages after n years

\dfrac{pk + n}{qk + n} = \dfrac rs

When each term changes by a percentage

Terms scaled by percentages

p : q \;\to\; p\left(1 + \tfrac{a}{100}\right) : q\left(1 + \tfrac{b}{100}\right)

Income, expenditure and savings

Savings

\text{savings} = \text{income} - \text{expenditure}

Common traps

Add to the quantities, not to the ratio

If 5 is added to two numbers in the ratio

2 : 3

, the new numbers are

2k + 5

and

3k + 5

. Writing

7 : 8

assumes

k = 1

.

Both people age

Six years later both ages rise by 6. Adding 6 to one age only is the usual slip.

'Ago' subtracts

'Four years ago' means

pk - 4

and

qk - 4

. Writing

+4

by habit still gives a neat number, so check it in the stem's ratio.

A percentage multiplies the term

Raising

3 : 4

by 20% and 50% gives

3.6 : 6

. Adding the percentages to the terms,

23 : 54

, has no meaning.

Two ratios need two letters

Incomes

4 : 3

and expenditures

5 : 4

need

4x, 3x

and

5y, 4y

. Using one

k

for both makes spending a fixed share of income, which the question never says.

Mixtures and Coins

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A mixture as a ratio of two quantities

Milk in a mixture

\text{milk} = \dfrac{m}{m + w} \times T

Adding one liquid: the other stays fixed

Water to add

\text{water needed} = \dfrac sr \times \text{milk}, \qquad \text{add} = \text{water needed} - \text{water now}

Replacing part of a mixture, and mixing two mixtures

Mixing two vessels

\text{milk} = \dfrac{m_1}{m_1 + w_1}V_1 + \dfrac{m_2}{m_2 + w_2}V_2

Coins: a ratio of how many coins there are

Total value

\text{value} = \sum (\text{number of coins} \times \text{face value})

Coins when the total value is given

Number of sets

k = \dfrac{\text{total value}}{au + bv + cw}

Common traps

The ratio is of the parts, not of the total

Milk : water

= 5 : 2

makes the milk

\tfrac 57

of the mixture. It is not

\tfrac 52

of the mixture.

Adding water never changes the milk

A working that splits the new total in the new ratio has quietly changed the amount of milk. Hold the milk fixed and solve for the water.

Mix litres, not ratio terms

Equal amounts of a

2 : 3

and a

4 : 5

mixture do not give

(2 + 4) : (3 + 5)

. The first is split into 5 parts and the second into 9, so turn each into fractions of a litre first.

Count first, then value

₹5 and ₹10 coins in the ratio

2 : 3

is a ratio of counts. Their values are in the ratio

10 : 30 = 1 : 3

.

Divide by the value of a set, not the sum of the ratio

₹1, ₹2 and ₹5 coins in the ratio

2 : 1 : 1

, worth ₹90, are

90 \div 9 = 10

sets. Dividing 90 by

2 + 1 + 1 = 4

treats every coin as worth ₹1.

Direct, Inverse and Joint Variation

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Direct variation

x \propto y \;\text{means}\; x = ky, \text{ so } \dfrac{x_1}{y_1} = \dfrac{x_2}{y_2}

Inverse variation

x \propto \dfrac 1y \;\text{means}\; xy = k, \text{ so } x_1 y_1 = x_2 y_2

Joint and combined variation

Combined variation

x \propto \dfrac{y}{z^2} \;\text{means}\; x = \dfrac{ky}{z^2}

Variation in word problems: work, speed and wages

Man-days

M_1 D_1 = M_2 D_2

Common traps

A square doubles twice

If

x \propto y^2

and

y

doubles,

x

becomes four times as large, not twice.

Inverse means dividing, not subtracting

If

y

rises from 4 to 8, an inversely varying

x

halves. It does not fall by 4.

One relation, one constant

'

x

varies as

y

and inversely as

z

' is one relation,

x = \dfrac{ky}{z}

, with one

k

. Giving

y

and

z

a constant each leaves an equation you cannot solve.

More workers, fewer days

12 workers take 10 days, so 20 workers take 6 days. Setting it up as a direct proportion,

12 : 20 = 10 : x

, gives 16.7 days: longer with more people.

Ratio and Proportion formulas

What a Ratio Means, and Its Named Forms

A ratio as equal parts

Simplest form of a ratio

Working back from a duplicate or triplicate ratio

Inverse, duplicate, triplicate and sub-duplicate ratios

Common traps

Joining, Compounding and Comparing Ratios

Joining ratios through a common term

Ratios from equal multiples

Compound ratio: multiplying ratios

Finding a missing ratio from a compound ratio

Comparing two ratios

Common traps

Dividing a Total in a Ratio, and Partnership

Dividing a total in a ratio

Dividing in a ratio of fractions

Numbers in a ratio with one more condition

A changed ratio, or shares with an amount taken off

Partnership: profit in the ratio of capital times time

Common traps

Proportion: Fourth, Third and Mean Proportionals

Four numbers in proportion, and the fourth proportional

Common traps

Ratios in Algebra: Substitution and Componendo-Dividendo

Evaluating an expression from a ratio

Finding x : y from an equation

Equal ratios: sums and the k method

Common traps

When a Ratio Changes: Additions, Ages and Incomes

Adding or subtracting the same number from both terms

Age problems with ratios

When each term changes by a percentage

Income, expenditure and savings

Common traps

Mixtures and Coins

A mixture as a ratio of two quantities

Adding one liquid: the other stays fixed

Replacing part of a mixture, and mixing two mixtures

Coins: a ratio of how many coins there are

Coins when the total value is given

Common traps

Direct, Inverse and Joint Variation

Joint and combined variation

Variation in word problems: work, speed and wages

Common traps

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