Episode 8 — Aptitude and Reasoning / 8.5 — Ratio and Proportion
8.5.a Concepts and Formulas -- Ratio and Proportion
1. What Is a Ratio?
A ratio is a way of comparing two or more quantities of the same kind by division.
If two quantities are a and b, their ratio is written as:
a : b (read "a is to b")
This is equivalent to the fraction a/b.
Key points:
- A ratio has no units -- it is a pure number.
- The order matters:
a : bis NOT the same asb : a(unless a = b). - Both terms of a ratio must be in the same unit before comparing.
Example
If A has Rs 500 and B has Rs 300:
Ratio of A's money to B's money = 500 : 300 = 5 : 3
2. Simplifying Ratios
A ratio a : b is in its simplest form when a and b have no common factor other than 1 (i.e., HCF(a, b) = 1).
Method: Divide both terms by their HCF.
Example: 48 : 36
HCF(48, 36) = 12
Simplified = 48/12 : 36/12 = 4 : 3
Ratios Involving Fractions
Convert to a common denominator or multiply through to clear fractions.
Example: (2/3) : (4/5)
Multiply both by LCM(3, 5) = 15:
= (2/3) x 15 : (4/5) x 15
= 10 : 12
= 5 : 6
Ratios Involving Decimals
Multiply both terms by a power of 10 to remove decimals, then simplify.
Example: 0.6 : 1.5
Multiply by 10: 6 : 15
Simplify: 2 : 5
3. Types of Ratios
3.1 Compound Ratio
When two or more ratios are multiplied together:
Compound ratio of a:b and c:d = ac : bd
Example: Compound ratio of 2:3 and 4:5
= (2 x 4) : (3 x 5) = 8 : 15
3.2 Duplicate Ratio
The ratio of the squares of the terms:
Duplicate ratio of a : b = a^2 : b^2
Example: Duplicate ratio of 3 : 4 = 9 : 16
3.3 Triplicate Ratio
The ratio of the cubes of the terms:
Triplicate ratio of a : b = a^3 : b^3
Example: Triplicate ratio of 2 : 3 = 8 : 27
3.4 Sub-duplicate Ratio
The ratio of the square roots:
Sub-duplicate ratio of a : b = sqrt(a) : sqrt(b)
Example: Sub-duplicate ratio of 16 : 25 = 4 : 5
3.5 Sub-triplicate Ratio
The ratio of the cube roots:
Sub-triplicate ratio of a : b = a^(1/3) : b^(1/3)
Example: Sub-triplicate ratio of 8 : 27 = 2 : 3
3.6 Reciprocal (Inverse) Ratio
Reciprocal ratio of a : b = (1/a) : (1/b) = b : a
Example: Reciprocal ratio of 3 : 5 = 5 : 3
4. Proportion
4.1 Definition
Four quantities a, b, c, d are said to be in proportion if:
a : b = c : d
Written as: a : b :: c : d (read "a is to b as c is to d")
Here:
aanddare called extremesbandcare called means
4.2 Fundamental Rule of Proportion (Cross Product)
If a : b :: c : d, then:
a x d = b x c
(Product of extremes = Product of means)
This is the most important property and the basis for solving proportion problems.
Example: Is 3 : 5 :: 6 : 10 a true proportion?
Check: 3 x 10 = 30, 5 x 6 = 30
30 = 30 => Yes, it is a true proportion.
4.3 Fourth Proportional
If a : b :: c : x, then x is the fourth proportional to a, b, c.
a x x = b x c
x = (b x c) / a
Example: Fourth proportional to 3, 6, 4:
x = (6 x 4) / 3 = 24 / 3 = 8
Verification: 3 : 6 :: 4 : 8 => 3 x 8 = 24 = 6 x 4 (correct)
4.4 Third Proportional
If a : b :: b : x, then x is the third proportional to a and b.
a x x = b x b
x = b^2 / a
Example: Third proportional to 4 and 12:
x = 12^2 / 4 = 144 / 4 = 36
Verification: 4 : 12 :: 12 : 36 => 4 x 36 = 144 = 12 x 12 (correct)
4.5 Mean Proportional (Geometric Mean)
If a : x :: x : b, then x is the mean proportional between a and b.
x^2 = a x b
x = sqrt(a x b)
Example: Mean proportional between 4 and 9:
x = sqrt(4 x 9) = sqrt(36) = 6
Verification: 4 : 6 :: 6 : 9 => 4 x 9 = 36 = 6 x 6 (correct)
5. Direct Proportion
Two quantities are in direct proportion when an increase in one causes a proportional increase in the other (and vice versa).
If x is directly proportional to y:
x / y = constant
x1 / y1 = x2 / y2
Example:
If 5 pens cost Rs 60, what do 8 pens cost?
More pens => More cost (Direct proportion)
5 / 60 = 8 / x
x = (60 x 8) / 5 = 480 / 5 = Rs 96
6. Inverse Proportion
Two quantities are in inverse proportion when an increase in one causes a proportional decrease in the other.
If x is inversely proportional to y:
x x y = constant
x1 x y1 = x2 x y2
Example:
If 6 workers can finish a job in 10 days,
how many days will 15 workers take?
More workers => Fewer days (Inverse proportion)
6 x 10 = 15 x d
d = 60 / 15 = 4 days
7. Dividing a Quantity in a Given Ratio
Two-part division
To divide a quantity Q in the ratio a : b:
First part = Q x a / (a + b)
Second part = Q x b / (a + b)
Example:
Divide Rs 780 in the ratio 5 : 8.
Sum of ratio terms = 5 + 8 = 13
First part = 780 x 5/13 = Rs 300
Second part = 780 x 8/13 = Rs 480
Verification: 300 + 480 = 780, and 300:480 = 5:8 (correct)
Three-part division
To divide Q in the ratio a : b : c:
First part = Q x a / (a + b + c)
Second part = Q x b / (a + b + c)
Third part = Q x c / (a + b + c)
Example:
Divide Rs 1800 among A, B, C in the ratio 2 : 3 : 4.
Sum = 2 + 3 + 4 = 9
A = 1800 x 2/9 = Rs 400
B = 1800 x 3/9 = Rs 600
C = 1800 x 4/9 = Rs 800
Verification: 400 + 600 + 800 = 1800 (correct)
8. Combining and Comparing Ratios
8.1 Combining Two Ratios with a Common Term
When you know A : B and B : C, you can find A : B : C.
Given: A : B = 3 : 4 and B : C = 5 : 6
Make B the same in both ratios.
LCM of 4 and 5 = 20.
A : B = 3 : 4 => multiply by 5 => 15 : 20
B : C = 5 : 6 => multiply by 4 => 20 : 24
Therefore: A : B : C = 15 : 20 : 24
8.2 Comparing Ratios
To compare a : b and c : d, use cross multiplication:
Compare 3 : 7 and 5 : 11
Cross multiply:
3 x 11 = 33
7 x 5 = 35
Since 33 < 35 => 3/7 < 5/11
Therefore 3 : 7 < 5 : 11
9. Ratio of Increase or Decrease
When a quantity changes from an old value to a new value:
Ratio of change = Old : New
Example:
A salary increases from Rs 20,000 to Rs 25,000.
Ratio = 20000 : 25000 = 4 : 5
The salary increased in the ratio 4 : 5.
If you know the ratio of change and one value:
If income increases in ratio 3 : 5 and original income = Rs 12,000:
New income = 12,000 x (5/3) = Rs 20,000
10. Alligation (Mixing Ratios)
Alligation is a method to find the ratio in which two or more ingredients at different prices (or concentrations) must be mixed to produce a mixture at a desired price (or concentration).
The Alligation Rule
If two ingredients with values d1 and d2 (where d1 < d2) are mixed to get
a mixture of mean value m (where d1 < m < d2), then:
Quantity of cheaper d2 - m
-------------------- = ------
Quantity of dearer m - d1
Visual Diagram (Cross Method)
d1 (cheaper) d2 (dearer)
\ /
\ /
m (mean/mixture)
/ \
/ \
(d2 - m) (m - d1)
Ratio of cheaper : dearer = (d2 - m) : (m - d1)
Worked Example
In what ratio must rice at Rs 40/kg be mixed with rice at Rs 60/kg
so that the mixture costs Rs 45/kg?
d1 = 40, d2 = 60, m = 45
Cheaper : Dearer = (60 - 45) : (45 - 40) = 15 : 5 = 3 : 1
Answer: 3 : 1
Alligation for Concentration/Percentage Problems
A solution of 20% acid is mixed with a solution of 50% acid
to get a solution of 30% acid. Find the ratio.
d1 = 20, d2 = 50, m = 30
Ratio = (50 - 30) : (30 - 20) = 20 : 10 = 2 : 1
11. Partnership Ratios
In a business partnership, profit is shared in the ratio of Capital x Time.
Simple Partnership (Same Time Period)
When all partners invest for the same duration:
Profit ratio = Capital ratio
Example: A invests Rs 5000, B invests Rs 7000.
Profit ratio = 5000 : 7000 = 5 : 7
Compound Partnership (Different Time Periods)
When partners invest for different durations:
Profit ratio = (C1 x T1) : (C2 x T2)
Example:
A invests Rs 5000 for 12 months.
B invests Rs 6000 for 10 months.
A's share : B's share = (5000 x 12) : (6000 x 10)
= 60000 : 60000
= 1 : 1
Three or More Partners
Example:
A invests Rs 10,000 for 6 months.
B invests Rs 15,000 for 4 months.
C invests Rs 20,000 for 3 months.
Ratio = (10000 x 6) : (15000 x 4) : (20000 x 3)
= 60000 : 60000 : 60000
= 1 : 1 : 1
12. Age-Based Ratio Problems
A very common exam pattern involves ratios of ages at different points in time.
Core Idea
If the present ages of A and B are in ratio a : b,
then their ages can be written as:
A = a*k, B = b*k (for some positive constant k)
Worked Example
The ratio of ages of A and B is 4 : 5.
After 6 years, the ratio will be 5 : 6.
Find their present ages.
Let present ages = 4k and 5k.
After 6 years:
(4k + 6) / (5k + 6) = 5/6
Cross multiply:
6(4k + 6) = 5(5k + 6)
24k + 36 = 25k + 30
k = 6
Present ages: A = 4(6) = 24 years, B = 5(6) = 30 years.
13. Income-Expenditure-Savings Ratio Problems
Standard Framework
Income - Expenditure = Savings
If income ratio of A:B = a:b and expenditure ratio = c:d,
Let incomes = a*x and b*x, expenditures = c*y and d*y.
Then savings: A saves (ax - cy), B saves (bx - dy).
Use any additional information (savings amount) to find x and y.
Worked Example
Incomes of A and B are in the ratio 5 : 4.
Expenditures are in the ratio 3 : 2.
Each saves Rs 1600.
Let incomes = 5x and 4x.
Let expenditures = 3y and 2y.
5x - 3y = 1600 ... (i)
4x - 2y = 1600 ... (ii)
From (ii): 2x - y = 800 => y = 2x - 800
Substitute in (i):
5x - 3(2x - 800) = 1600
5x - 6x + 2400 = 1600
-x = -800
x = 800
y = 2(800) - 800 = 800
Income of A = 5 x 800 = Rs 4000
Income of B = 4 x 800 = Rs 3200
Expenditure of A = 3 x 800 = Rs 2400
Expenditure of B = 2 x 800 = Rs 1600
Verification: A saves 4000 - 2400 = 1600, B saves 3200 - 1600 = 1600 (correct)
14. Componendo, Dividendo, and Related Properties
If a/b = c/d, then each of the following also holds:
Componendo
(a + b) / b = (c + d) / d
Dividendo
(a - b) / b = (c - d) / d
Componendo and Dividendo (Combined)
(a + b) / (a - b) = (c + d) / (c - d)
This is extremely useful for simplifying proportion equations quickly.
Worked Example
If (a + b) / (a - b) = 5/3, find a : b.
Using componendo-dividendo in reverse:
a/b = (5 + 3) / (5 - 3) = 8/2 = 4/1
Therefore a : b = 4 : 1
15. Variation
Direct Variation
y = kx (y is directly proportional to x)
Inverse Variation
y = k/x (y is inversely proportional to x)
Joint Variation
z = kxy (z varies directly as x and y)
Combined Variation
z = kx/y (z varies directly as x and inversely as y)
Example:
If z varies directly as x and inversely as y,
and z = 12 when x = 6 and y = 2:
z = kx/y => 12 = k(6)/2 => k = 4
When x = 10, y = 5:
z = 4(10)/5 = 8
Summary of All Key Formulas
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| Concept | Formula |
+------------------------------------------+-------------------------------+
| Basic Ratio | a : b = a/b |
| Simplify Ratio | Divide both by HCF(a,b) |
| Proportion | a:b :: c:d => ad = bc |
| Fourth Proportional to a, b, c | x = bc/a |
| Third Proportional to a, b | x = b^2/a |
| Mean Proportional of a, b | x = sqrt(ab) |
| Compound Ratio of a:b and c:d | ac : bd |
| Duplicate Ratio | a^2 : b^2 |
| Triplicate Ratio | a^3 : b^3 |
| Sub-duplicate Ratio | sqrt(a) : sqrt(b) |
| Reciprocal Ratio | b : a |
| Divide Q in ratio a:b | Qa/(a+b), Qb/(a+b) |
| Direct Proportion | x1/y1 = x2/y2 |
| Inverse Proportion | x1*y1 = x2*y2 |
| Alligation | (d2-m):(m-d1) |
| Partnership (same time) | C1 : C2 |
| Partnership (different time) | C1*T1 : C2*T2 |
| Componendo-Dividendo | (a+b)/(a-b) = (c+d)/(c-d) |
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