Quick Revision

Ratio Basics and Types of Ratio

Compare like quantities in the same unit; a ratio has no unit. 40 paise : ₹2 = 40 : 200 = 1 : 5.
Compare 6 : 11 and 5 : 9 by cross-multiplying: 54 against 55, so 5 : 9 is bigger.
Compound ratio of a : b and c : d is ac : bd.
Duplicate $a^2 : b^2$ , triplicate $a^3 : b^3$ , sub-duplicate $\sqrt{a} : \sqrt{b}$ , sub-triplicate $\sqrt[3]{a} : \sqrt[3]{b}$ .
Inverse ratio of a : b is b : a.

A : B and B : C: make B equal using the LCM. 4 : 7 and 3 : 5 give 12 : 21 : 35.
Chains: $\frac{A}{D} = \frac{A}{B} \times \frac{B}{C} \times \frac{C}{D}$ .
2A = 3B = 5C gives $\frac{1}{2} : \frac{1}{3} : \frac{1}{5}$ = 15 : 10 : 6 (multiply by 30).
$\frac{A}{3} = \frac{B}{4} = \frac{C}{5}$ gives 3 : 4 : 5 directly.
$\frac{3}{5}$ of A = $\frac{2}{3}$ of B gives A : B = 10 : 9. Swap the fractions.

Dividing N in a : b : c: 1 part = N ÷ (a + b + c).
Fractions as the ratio: multiply by the LCM. Inverse proportion to 4, 5, 10: $\frac{1}{4} : \frac{1}{5} : \frac{1}{10}$ = 5 : 4 : 2.
‘A gets ₹k more than B’: the gap in their terms is ₹k.
Ratio a : b: sum (a + b) parts, difference (b − a) parts.
Sum of squares S: $(a^2 + b^2)x^2 = S$ . Product P: $abx^2 = P$ .

x added to both terms of a : b gives c : d when $\frac{a + x}{b + x} = \frac{c}{d}$ , so $x = \frac{bc - ad}{d - c}$ .
5 : 9 becomes 3 : 4 when x = 27 − 20 = 7. Check: 12 : 16 = 3 : 4.
Taking a number away: use a − x and b − x.
k added to one number and taken from the other: write ax and bx and solve.
Testing the options is often quicker.

a : b :: c : d means ad = bc (extremes multiplied = means multiplied).
Fourth proportional to a, b, c: $\frac{bc}{a}$ . For 3, 5, 12 it is 20.
Third proportional to a, b: $\frac{b^2}{a}$ . For 4, 10 it is 25.
Mean proportional: $\sqrt{ab}$ . For 9 and 25 it is 15.
Continued proportion a, b, c: $b^2 = ac$ .
Taking x from four numbers to make them proportional: solve (a − x)(d − x) = (b − x)(c − x); the $x^2$ terms cancel.

k-method: a : b = 5 : 3 means a = 5k, b = 3k; k cancels.
Componendo–dividendo: $\frac{a}{b} = \frac{c}{d}$ gives $\frac{a + b}{a - b} = \frac{c + d}{c - d}$ . (a + b) : (a − b) = 9 : 5 gives a : b = 7 : 2.
Invertendo b : a = d : c; alternendo a : c = b : d.
$\frac{2x + y}{x - y} = 5$ gives 2x + y = 5x − 5y, so x : y = 2 : 1.

Direct: $x \propto y$ means x = ky. Inverse: $x \propto \frac{1}{y}$ means xy = k.
Joint: $x \propto yz$ means x = kyz. Direct with y and inverse with z: $x = \frac{ky}{z}$ .
x varies as $y^2$ : $x = ky^2$ , so tripling y makes x nine times as big.
Find k from the given pair first.
Partly fixed, partly varying: cost = a + bn. Two pairs of values give a and b.
Men and days on a fixed job is inverse variation too.

Savings = income − expenditure. Income : expenditure = 7 : 4 means savings are 3 parts.
Incomes a : b, expenditures c : d, each saving S: ax − cy = S and bx − dy = S. With equal savings, set the two left sides equal first.
Different savings: solve the pair by making one letter’s terms match.
Income : expenditure = 5 : 4 with rises of 20% and 15%: savings go from 1 to 6 − 4.6 = 1.4, up 40%.

Ratio of numbers given: find the value of one set. ₹1, 50 paise and 25 paise coins in the ratio 3 : 2 : 4 make a set worth 3 + 1 + 1 = ₹5. Total value ÷ set value = number of sets.
Ratio of values given: split the total value, then divide each part by the coin’s value.
Write paise as rupees: 25 paise = ₹0.25.
Trap: never split the total value in a ratio of numbers.
₹10, ₹20 and ₹50 notes work the same way.

Joining or leaving: write the old numbers as ax and bx, apply the change, equate to the new ratio.
One group fixed: boys : girls = 4 : 3 becomes 1 : 1 after 12 girls join, so girls go from 3 parts to 4, 1 part = 12 and boys = 48.
Each term changes by its own percent: 5 : 4 with rises of 20% and 10% becomes 6 : 4.4 = 15 : 11.
A total changes by a weighted average of the percents.

Present ages ax and bx. After n years: (ax + n) : (bx + n). n years ago: (ax − n) : (bx − n).
The difference between two ages never changes.
For three people, n years ago the sum was less by 3n.
Ratio 3 : 4 now and 4 : 5 after 5 years: $\frac{3x + 5}{4x + 5} = \frac{4}{5}$ gives x = 5, ages 15 and 20.

Profit or loss is shared in the ratio of capital × time.
Same time: ratio of capitals. Same capital: ratio of times.
A working (active) partner runs the business; a sleeping partner only invests.
Count time in months. Joining after 3 months means 9 months in.
Capitals given as percents or fractions of one another: take one of them as 100.
To find a time or capital, set the profit ratio equal to the capital × time ratio.

Split each partner’s year into periods and add capital × months for each.
₹50,000 for 4 months, then ₹30,000 for 8 months: 2,00,000 + 2,40,000 = 4,40,000.
For a partner in all year, the months add up to 12.
Share from a difference: ratio 11 : 9 and the first gets ₹4,000 more means 2 parts = ₹4,000.
Several years with changing capitals: share each year’s profit by that year’s ratio.
Leaving early: the last period has no capital.

Take out the fixed amounts first: salary, commission, interest on capital, charity or reserve. Share only what is left.
A commission of x% of the profit leaves (100 − x)% to share.
Interest at r% a year: each partner gets r% of his own capital.
If the rest is also shared by capital, paying interest first changes nothing. It matters when the rest is shared equally or in some other ratio.
The working partner’s total = salary or commission + his share of the rest.

Rent is shared in the ratio of animals × months, just like capital × time.
Different animals: change them all into one kind first. If 3 horses eat as much as 5 cows, 1 horse = $\frac{5}{3}$ cows.
Say A grazes 7 cows for 4 months (28 cow-months) and B grazes 4 cows for 9 months (36). A rent of ₹3,200 splits 28 : 36 = 7 : 9, so A pays ₹1,400.
Rent per cow-month = total rent ÷ total cow-months.