Ratio, Proportion and Variation — SSC CGL Formulas and Traps

Ratio Basics

Turning fractions, percentages and equal parts into a ratio

Scaling and equal parts

a : b = ka : kb \qquad pA = qB \Rightarrow A : B = q : p \qquad pA = qB = rC \Rightarrow A : B : C = \frac{1}{p} : \frac{1}{q} : \frac{1}{r}

$k$ any non-zero number
$p, q, r$ the multipliers in the equal statement

Sharing a quantity in a ratio with the ratio unit

Share and ratio unit

\text{share of A} = \frac{a}{a + b + c} \times \text{total} \qquad k = \frac{\text{difference}}{c - a}

$a, b, c$ the terms of the ratio
$k$ the ratio unit, one part

Coins in a ratio: number of coins against their value

Value of coins in a ratio

\text{total value} = k(av_1 + bv_2 + cv_3)

$a, b, c$ the ratio of the NUMBERS of coins
$v_1, v_2, v_3$ the value of one coin of each kind, in rupees

Value of an expression when a ratio is given

Rules for given ratios

\frac{x + y}{x - y} = \frac{p}{q} \Rightarrow \frac{x}{y} = \frac{p + q}{p - q} \qquad \frac{a}{b} = \frac{c}{d} = k \Rightarrow \frac{pa + qc}{pb + qd} = k

Common traps

Putting the multipliers straight into the ratio

From 2A = 3B the ratio A : B is 3 : 2, not 2 : 3. A quantity multiplied by a bigger number must itself be smaller to give the same product.

Adding the percentage to the ratio term

Terms 6 and 4 raised by 20% and 40% become

6 \times 1.2 = 7.2

and

4 \times 1.4 = 5.6

. Writing 26 : 44 or 6.2 : 4.4 adds the percentage instead of multiplying by it.

Dividing a gap by the total parts

If two numbers in the ratio 4 : 9 differ by 35, the 35 is the gap of 5 parts. Dividing it by 13, the total of the terms, gives a wrong unit.

Reading a part-to-whole ratio as part to part

Boys : total = 7 : 17 does not mean boys : girls = 7 : 17. The girls are the other 10 parts of the 17.

Sharing the money in the coin ratio

If coins are in the number ratio 2 : 3 : 5, the rupees are not split 2 : 3 : 5. Multiply each count term by its coin value first; only then does the total fix the unit.

Substituting when the powers differ

a : b = 1 : 2 fixes

\frac{a + b}{a - b}

, but not

\frac{a^2 + b}{a}

: with a = 1, b = 2 it is 3, and with a = 2, b = 4 it is 4. Substitution is safe only when every term has the same power.

Stopping at the given ratio

From

(x + y) : (x - y) = 7 : 3

, the ratio x : y is

(7 + 3) : (7 - 3) = 5 : 2

. The given ratio is about

x + y

and

x - y

, not about x and y.

Fourth, Third and Mean Proportionals

Learn this subtopic in the notes

Proportion and the fourth proportional

Proportion and fourth proportional

a : b :: c : d \Leftrightarrow ad = bc \qquad d = \frac{bc}{a}

$a, d$ the extremes (outer terms)
$b, c$ the means (inner terms)

Third proportional

a : b = b : c \Leftrightarrow b^2 = ac \qquad c = \frac{b^2}{a}

Mean proportional

a : m = m : b \Leftrightarrow m = \sqrt{ab} \qquad b = \frac{m^2}{a}

Common traps

Taking the numbers in the wrong order

The fourth proportional to 4, 10 and 6 comes from 4 : 10 = 6 : d, which gives 15. Starting from 10 : 4 = 6 : d gives 2.4, a different number. Keep the order printed in the stem.

Squaring the wrong term

The third proportional to a and b is

\frac{b^2}{a}

, the SECOND term squared. For 9 and 12 it is 16; squaring the first term gives

\frac{81}{12} = 6.75

, which is wrong.

Halving the sum

The mean proportional of 4 and 49 is

\sqrt{196} = 14

, not

\frac{4 + 49}{2} = 26.5

. That is the ordinary average, which belongs to adding chains, not to proportion.

Misplacing the decimal in a square root

\sqrt{0.09} = 0.3

, because

0.3 \times 0.3 = 0.09

. Writing 0.03 is a common slip; count decimal places: the root has half as many as the square.

Direct and Inverse Variation

Learn this subtopic in the notes

Direct proportion and the unitary method

Direct variation

y = kx \qquad \frac{y_2}{y_1} = \frac{x_2}{x_1}

$k$ the constant of variation, $\frac{y}{x}$

Inverse and joint variation

xy = k \qquad y = \frac{k x^a}{w^c} \qquad y_2 = y_1 \left(\frac{x_2}{x_1}\right)^a \left(\frac{w_1}{w_2}\right)^c

$a, c$ the powers stated in the stem
$w$ a quantity that y varies inversely with

Common traps

Turning a direct proportion upside down

If 4 kg of rice feeds 10 people, 25 people need

4 \times \frac{25}{10} = 10

kg, not

4 \times \frac{10}{25} = 1.6

kg. Ask first whether the answer should go up or down.

Treating an inverse variation as direct

If each bucket holds one third as much, the tank needs 3 times as many buckets, not one third as many. Smaller buckets mean more trips.

Forgetting the power

If y varies as the square of x and x becomes 3 times as large, y becomes 9 times as large, not 3 times. Apply the stated power to the ratio of the x values.

Combining and Changing Ratios

Learn this subtopic in the notes

Joining ratios through a common term

Joining ratios

A : B : C = a_1b_2 : b_1b_2 : b_1c_2 \qquad \frac{A}{D} = \frac{A}{B} \times \frac{B}{C} \times \frac{C}{D}

$a_1 : b_1$ the ratio A : B
$b_2 : c_2$ the ratio B : C

A ratio after items are added or removed

Changed ratio

\frac{ak + x}{bk + y} = \frac{m}{n}

$a : b$ the ratio before the change
$x, y$ the numbers added (negative if removed)
$m : n$ the ratio after the change

Common traps

Writing the ratios side by side

A : B = 3 : 4 and B : C = 5 : 6 do not give A : B : C = 3 : 4 : 6. B is 4 parts in one and 5 in the other; matched at 20, the answer is 15 : 20 : 24.

Reading parts before matching the unchanged term

Girls and boys are 2 : 3, and after some girls join they are 3 : 4. The boys did not change, but their terms are 3 and 4, so first write 8 : 12 and 9 : 12. Only then does the change in parts (8 to 9) give the number who joined.

Answering for the wrong moment

Many stems ask for the number BEFORE the change, but the equation is solved for the unit k. Finish by working out the quantity the question names, then check it against both ratios.

Ratios in Income, Savings and Ages

Learn this subtopic in the notes

Income, expenditure and savings in a ratio

Income identity

I = E + S \qquad S_{\text{new}} = I(1 + r_I) - E(1 + r_E)

$I, E, S$ income, expenditure, savings
$r_I, r_E$ the fractional rises in income and expenditure

Ages in a ratio

Ages after t years

\frac{ak + t}{bk + t} = \frac{m}{n} \qquad bk - ak \text{ stays fixed}

$t$ years ahead (use $-t$ for years ago)

Common traps

Combining the percentages instead of the amounts

Income up 20% and expenditure up 10% does not move savings by 10% or 30%. Savings are a difference, so take numbers, apply each rise, and subtract again.

Confusing the saving rate with the spending rate

Saving 25% of income means spending 75% of it. When the stem gives what is saved, turn it into what is spent before comparing expenditures.

Adding the years to the ratio terms

Ages in the ratio 6 : 5 do not become 13 : 12 after 7 years. The terms are parts, not years; find the ages first, then add 7 to each.

Letting the age gap change

After any number of years both ages rise by the same amount, so the difference stays the same. If your working changes the gap, a sign or a term has been misread.

Partnership

Learn this subtopic in the notes

Profit in the ratio of capital times time

Profit sharing

P_1 : P_2 : P_3 = C_1t_1 : C_2t_2 : C_3t_3

$C$ a partner's capital
$t$ the months that capital stays in

Capital that changes during the year

Capital-months with changes

\text{capital-months} = C_1t_1 + C_2t_2 + \cdots

Charity, salary or interest taken out before sharing

Share after a deduction

\text{share} = \text{own extra} + \frac{c}{c_1 + c_2} \times (P - \text{deductions})

$c$ the partner's capital (times time)
$P$ the total profit

Rent for a shared field in animal-months

Share of the rent

\text{share} = \frac{\text{own animal-months}}{\text{total animal-months}} \times \text{rent}

Common traps

Counting the whole year for a late partner

A partner who joins after 5 months has money in for 7 months of a 12-month year. Multiplying the capital by 12 overstates the share.

Taking the year when the stem says two years

If the profit is shared after two years, the time is 24 months, and a partner who joined after 10 months was in for 14 months, not 2.

Using the final capital for the whole year

If ₹30,000 is taken out after 8 months, the lower capital counts only for the last 4 months. The first 8 months still earn on the full amount.

Reading "2 times more than"

In the bank, SSC's key reads "A invests 2 times more than B" as A = 3 times B (B's amount plus twice more). "Twice as much as B" is 2 times B. When the stem uses the first wording, test both readings against the options.

Sharing before deducting

If a managing partner takes 10% first, only the other 90% is shared by capital. Sharing the full profit and then adding 10% counts that 10% twice.

Forgetting the working partner's capital share

A working partner gets the fee AND a capital share of what remains. Answering with the fee alone, or the capital share alone, is a printed wrong option.

Adding animals of different kinds

20 horses and 30 cows are not 50 units of grazing. Convert every animal to one kind with the given rate before multiplying by months.

Mixture Quantities and Replacement

Learn this subtopic in the notes

Adding one ingredient while the other stays fixed

Adding to a mixture

\text{new total} = \frac{\text{fixed amount}}{\text{its new fraction}} \qquad \frac{A + x}{T + x} = p

$A, T$ the amount of the ingredient being added to, and the total
$p$ the target fraction of that ingredient

Combining two mixtures

Combined amount of an ingredient

A_{\text{new}} = \frac{a_1}{a_1 + b_1}V_1 + \frac{a_2}{a_2 + b_2}V_2

$a : b$ the ratio in each mixture
$V_1, V_2$ the amounts taken from each

Taking out part of a mixture and topping it up

Amount left after a removal

\text{left} = \text{original} \times \left(1 - \frac{x}{T}\right)

$x$ the amount of mixture taken out
$T$ the total before it is taken out

Repeated replacement

A_n = A\left(1 - \frac{x}{T}\right)^n

$A$ the amount of the original liquid at the start
$n$ the number of rounds

Common traps

Working from the total instead of the fixed ingredient

When only water is added, the total and the water both change, but the milk does not. Start from the milk; a percentage of the old total is the wrong anchor.

Adding litres to a ratio term

Adding 10 litres of water to a 5 : 3 mixture does not make it 5 : 13. Turn the ratio into litres first, add the 10 litres, and then form the new ratio.

Adding the ratio terms of unequal amounts

10 litres of a 1 : 1 mixture and 30 litres of a 3 : 1 mixture do not give 4 : 2. In litres, milk is

5 + 22.5 = 27.5

and water

5 + 7.5 = 12.5

, a ratio of 11 : 5.

Taking out only one ingredient

Drawing 10 litres from a 4 : 1 milk and water mixture takes out 8 litres of milk and 2 of water, not 10 litres of milk. A removal always carries the mixture's own ratio.

Tracking the ingredient that is poured back

When pure acid is poured back, the acid falls and then rises, which needs two steps. The water only falls, so follow the water and the answer comes in one equation.

Subtracting the same amount every round

From 40 litres of juice, two draws of 4 litres leave 32.4 litres, not 32. The second draw takes out less juice than the first, because the jar is already weaker.

Mixtures and Alligation

Learn this subtopic in the notes

Mean price of a mixture

Mean price

m = \frac{q_1p_1 + q_2p_2}{q_1 + q_2} \qquad q_1(p_1 - m) + q_2(p_2 - m) = 0

$q$ the quantity of each kind
$p$ its price per unit
$m$ the mean price of the mixture

The alligation cross for a mixing ratio

Rule of alligation

\frac{q_{\text{cheaper}}}{q_{\text{dearer}}} = \frac{p_2 - m}{m - p_1}

$p_1, p_2$ the cheaper and the dearer price
$m$ the mean price, between $p_1$ and $p_2$

Mixtures sold at a profit or a loss

Cost price of the mixture

CP = \frac{100 \times SP}{100 + r} \qquad \text{gain \%} = \frac{SP - CP}{CP} \times 100

$SP$ the selling price per unit
$r$ the profit percentage

Common traps

Averaging the prices without the quantities

6 kg at ₹180 and 4 kg at ₹230 do not average ₹205. The mixture has more of the cheaper tea, so its price is ₹200, nearer ₹180.

Mixing price units

A price given per 5 kg must be turned into a price per kg before it meets prices per kg. ₹360 for 5 kg is ₹72 per kg.

Giving each kind its own distance

In the cross, the cheaper kind's share is the DEARER price minus the mean. If the mean is close to the cheaper price, the cheaper kind is the larger part.

A mean outside the two prices

The mean of a two-kind mixture must lie between the two prices. A mean outside them means a price was misread or a selling price was used where the cost price belongs.

Putting the selling price in the cross

A mixture of ₹48 and ₹68 rice sold at ₹70 cannot use 70 in the cross: it lies above both prices. The cross needs the cost price, here ₹56.

Taking the profit off the selling price

A 25% profit means SP is 1.25 times CP, so

CP = SP \div 1.25

. Taking 25% off the selling price (

SP \times 0.75

) gives a cost that is too low.

Ratio, Proportion and Variation formulas

Ratio Basics

Turning fractions, percentages and equal parts into a ratio

Sharing a quantity in a ratio with the ratio unit

Coins in a ratio: number of coins against their value

Value of an expression when a ratio is given

Common traps

Fourth, Third and Mean Proportionals

Proportion and the fourth proportional

Common traps

Direct and Inverse Variation

Direct proportion and the unitary method

Common traps

Combining and Changing Ratios

Joining ratios through a common term

A ratio after items are added or removed

Common traps

Ratios in Income, Savings and Ages

Income, expenditure and savings in a ratio

Ages in a ratio

Common traps

Partnership

Profit in the ratio of capital times time

Capital that changes during the year

Charity, salary or interest taken out before sharing

Rent for a shared field in animal-months

Common traps

Mixture Quantities and Replacement

Adding one ingredient while the other stays fixed

Combining two mixtures

Taking out part of a mixture and topping it up

Common traps

Mixtures and Alligation

Mean price of a mixture

The alligation cross for a mixing ratio

Mixtures sold at a profit or a loss

Common traps

More SSC CGL Quantitative Aptitude formula sheets