Chapter 14: Direct and Inverse Proportions
This Class 8 Maths chapter helps students understand how two quantities vary together. In direct proportion, as one quantity increases, the other also increases; in inverse proportion, one increases while the other decreases. Class 8 Maths Notes provide clear explanations, tables, and graphs to illustrate the difference. Students learn to solve real-life problems involving speed, distance, and time using formulas and ratio concepts. Regular practice with these notes helps build a strong foundation for algebra and arithmetic topics in higher classes.
RATIO
Definition: The number of times one quantity contains another quantity of same kind is called the 'RATIO' of the two quantities. Clearly, the ratio of two quantities is equivalent to the fraction that one quantity is of the other.
Observe carefully that the two quantities must be of the same kind. There can be a ratio between Rs. 20 and Rs. 30 but cannot be between Rs. 20 and 30 mangoes.
Form of Ratio - a : b where 'a' is antecedent and 'b' is consequent
Note: Ratio is not altered by multiplying or dividing both its terms by the same number.
Thus 3 : 5 is the same as 6 : 10 and 15 : 20 is the same as 3 : 4.
Types of Ratios
1. Compound Ratio
Ratios are compound when antecedent multiplied by antecedent and consequents multiplied by consequent.
If a : b, c : d, e : f
Then compound ratio is: (a × c × e) : (b × d × f)
Example 1: Find the ratio compounded if the four ratios are 4 : 3, 9 : 13, 26 : 5 and 2 : 15.
Solution:
The compound ratio = (4 × 9 × 26 × 2) : (3 × 13 × 5 × 15)
= 1872 : 2925
= 16 : 25
2. DUPLICATE RATIO
If a : b then a² : b² is duplicate ratio.
Example 2: Find duplicate ratio of 4 : 3.
Solution:
4² : 3² = 16 : 9
3. TRIPLICATE RATIO
If a : b, then a³ : b³ is triplicate ratio.
Example 3: Find triplicate ratio of 2 : 3.
Solution:
(2)³ : (3)³ = 8 : 27
4. SUBDUPLICATE RATIO
If a : b then √a : √b is subduplicate ratio.
Example 4: Find subduplicate ratio of 3 : 4.
Solution:
√3 : √4 = √3 : 2
5. SUBTRIPLICATE RATIO
If a : b then ∛a : ∛b is subtriplicate ratio.
Example 5: Find the subtriplicate of 27 : 8.
Solution:
∛27 : ∛8 = 3 : 2
Proportions
Definition: The equality of ratios is called "proportion".
If a : b :: c : d then a, b, c and d are said to be in proportion. Then a, b, c and d are called terms.
Important Terms in Proportion
- The first and fourth (i.e. a and d) are called extremes
- The second and third term (i.e. b and c) are called mean terms
- d is called the fourth proportional
If four quantities be in proportion, the product of the extremes is equal to the product of the means.
a × d = b × c
Understanding Proportional Relationships
Proportional relationships tell us that two variables scale with each other in a predictable way:
- If you drive twice as fast, you'll go twice as far in a given time: Speed and distance are directly proportional to each other.
- If you drive twice as fast, it takes half the time to go a set distance: Speed and time are inversely proportional to each other.
To show proportionality we use a Greek alpha symbol (∝) in place of an equals sign (=).
Sometimes we use a constant k to show that we need to either multiply or divide something by our other variable.
Direct Proportionality or Direct Variation
Definition: If two quantities are linked in such a way that an increase in one quantity leads to a corresponding increase in the other and vice versa, then such a variation is called direct variation.
If two things are directly proportional to each other, they will increase and decrease in the same proportion to each other:
- Double one, double the other
- Increase one by 12%, the other increases by 12%
Real-Life Example of Direct Proportion
If we know that the cost of a taxi journey is directly proportional to the distance travelled, we can use any pair of values to find the relationship between cost and distance:
If a 300 mile journey costs £1200, we could work out:
- A 30 mile taxi ride would cost £120
- A 3 mile journey would be £12
- Meaning that the trip costs £4 a mile
We could then find the cost of any distance travelled, or how far we could get for a given price.
In this example, our constant k = 4, as the cost = 4 × the distance in miles.
Inverse Proportionality
Definition: Inverse proportionality is a similar idea, but one of our variables decreases as the other increases. The changes are still proportional, but "one over" each other:
- Double one, halve the other
- One gets four times smaller, the other gets four times bigger
Types of Proportion
There are four types of proportion:
- Direct Proportion
- Inverse Proportion
- Compound Proportion
- Continued Proportion
Direct Proportion - Detailed Study
Understanding Direct Proportion with Real Example
Suppose the price of one piece of soap is 20 Rs. Let's see what happens as we buy more:
| Number of Pieces | Total Price (Rs.) | Ratio (Price/Pieces) |
|---|---|---|
| 12 (1 dozen) | 240 | 20 |
| 24 (2 dozen) | 480 | 20 |
We can easily see that if the person buys more pieces, he has to pay more or he has to pay less if he buys less pieces.
In other words: If increase in one quantity causes increase in other quantity OR decrease in one quantity causes decrease in other quantity, then we say that they are related directly (They are in direct proportion).
If x and y are in direct proportion, then division of x and y will be constant.
x/y = constant
Principle of Direct Proportion
If we are dealing with quantities which are related directly (which are in direct proportion), then:
a₁ × b₂ = a₂ × b₁
Or in general form:
a₁ : a₂ :: b₁ : b₂
Solved Examples on Direct Proportion
Example 1: If 30 dozens of eggs cost 300 Rs. Find the cost of 5-dozens of eggs.
Solution:
Let x be the required price of 5 dozens eggs.
Since quantities are in direct proportion, so we use the above principle:
30 × x = 5 × 300
30x = 1500
x = 1500/30 = 50 Rs.
Answer: 50 Rs.
Example 2: A car travels 81 km in 4.5 liters. How far will it go by 20 liters of petrol?
Solution:
Let x be required distance travelled by car in 20 liters.
Since quantities are related directly, so by the above principle:
4.5 × x = 20 × 81
4.5x = 1620
x = 1620/4.5 = 360 km
Answer: 360 km
4-Step Approach to Solve Direct Proportion Problems
The Direct Proportion questions are usually of the form: "If x results in a, what will be the result if x changes to y?"
You need a simple 4 step approach:
- IDENTIFY VARIABLES: Usually this is a simpler thing to do. The variables are fairly obvious in typical proportion problems.
- UNDERSTAND THE RELATIONSHIP: Is the relationship between variables directly Proportional?
- EXPRESS THE PROPORTIONALITY: Get the problem to a form: "If x results in a, what will be the result if x changes to y?"
- SOLVE: Apply the direct proportion formula.
Examples of Direct Proportional Relationships
- The quantity of goodies money can buy. More money lets you buy more.
- The distance a car covers in a given time is directly proportional to its speed. Higher the speed, longer the car travels.
- The weight of a cylinder is directly proportional to its height. Higher the cylinder, the more it weighs.
- The length of a shadow to the height
- The length between two points on a map to the actual distance
Example 3: Stopwatch Problem
A stopwatch is defective and shows a time of 55 seconds for every minute. If this stopwatch is used in an experiment and shows a time of 3 Minutes and 40 seconds, what time has actually elapsed (1 minute = 60 seconds)?
Options:
(A) 2 Minutes 30 seconds
(B) 3 Minutes
(C) 3 Minutes 30 seconds
(D) 4 Minutes
(E) 5 Minutes
Solution:
Step 1: The two variables (the time shown on the watch and the real time) are directly proportional.
Step 2: The proportionality is 55 Seconds : 1 Minute = 3 Minutes and 40 seconds : ?
To make things simple, let's convert the minutes to seconds. We know 1 Minute = 60 Seconds.
Therefore 3 minutes and 40 seconds is equal to 3 × 60 + 40 = 220 seconds
If the actual elapsed time is x, we get:
55/60 = 220/x
55x = 60 × 220
55x = 13200
x = 13200/55 = 240 seconds
This time is in seconds. Since 1 minute = 60 seconds, the actual time elapsed is 240/60 = 4 minutes
The correct answer is (D) 4 Minutes
Inverse Proportion - Detailed Study
Definition: The concept of inverse proportion means that as the absolute value or magnitude of one variable gets bigger, the absolute value or magnitude of another gets smaller, such that their product is always the same.
Understanding Inverse Proportion with Real Example
Suppose that 20 men build a house in 6 days. Let's see what happens as we increase the number of men:
| No. of Men | No. of Days | Product (Men × Days) |
|---|---|---|
| 20 | 6 | 120 |
| 30 | 4 | 120 |
| 40 | 3 | 120 |
It can be seen that as the no. of men is increased, the time taken to build the house is decreased in the same ratio.
In other words: If increase in one quantity causes decrease in other quantity or decrease in one quantity causes increase in other quantity, then we say that both quantities are inversely related.
More explicitly: If two quantities x and y are in inverse proportion, then their product will be constant.
x × y = c (where c = constant)
Principle of Inverse Proportion
If we are dealing with quantities which are related inversely, then we can use the following rule:
a₁ × c₁ = a₂ × c₂
Or equivalently:
a₁ : a₂ = c₂ : c₁
(Note the inversion in the second ratio)
Solved Examples on Inverse Proportion
Example 1: Four pipes can fill a tank in 70 minutes. How long will it take to fill the tank by 7 pipes?
Solution:
| No. of Pipes | Time Taken | Relationship |
|---|---|---|
| 4 | 70 minutes | Increase ↑ → Decrease ↓ |
| 7 | x |
By the principle of inverse proportion, we have:
4 × 70 = 7 × x
280 = 7x
x = 280/7 = 40 minutes
Answer: 40 minutes
Example 2: Thirty-five workers can build a house in 16 days. How many days will 28 workers working at the same rate take to build the same house?
Solution:
| No. of Workers | No. of Days | Relationship |
|---|---|---|
| 35 | 16 days | Decrease ↓ → Increase ↑ |
| 28 | x |
By the principle of inverse proportion, we have:
28 × x = 35 × 16
28x = 560
x = 560/28 = 20 days
Answer: 20 days
Example 3: Imran bought 40 toys each cost Rs.14. How many toys can Imran buy at Rs.8 each from the same amount?
Solution:
| Price of Toy | No. of Toys | Relationship |
|---|---|---|
| Rs. 14 | 40 toys | Decrease ↓ → Increase ↑ |
| Rs. 8 | x |
By the principle of inverse proportion, we have:
14 × 40 = 8 × x
560 = 8x
x = 560/8 = 70 toys
Answer: 70 toys
Mathematical Notation for Proportionality
The symbol ∝ represents 'proportional to'.
Direct Proportionality: If y is directly proportional to x, we write:
y ∝ x or y = kx (where k is a constant)
Inverse Proportionality: If y is inversely proportional to x, we write:
y ∝ 1/x or y = k/x
Other Types of Proportionality:
- y is inversely proportional to x²: y ∝ 1/x² or y = k/x²
- y is proportional to x²: y ∝ x² or y = kx²
- y is proportional to √x: y ∝ √x or y = k√x
4-Step Approach for Inverse Proportion Problems
Inverse proportionality questions are of the form: "If x results in a, what will be the result if x changes to y?"
- Identify Variables: The variables are fairly obvious in typical proportion problems.
- Understand the Relationship: Is the relationship between variables inversely proportional?
- Express the Proportionality: Get the problem to a form: "If x results in a, what will be the result if x changes to y?"
- Solve: Apply the inverse proportion formula.
Examples of Inverse Proportional Relationships
- The time a car needs to travel between two towns is inversely proportional to its speed (higher the speed, less time it takes)
- The time it takes to do a job is inversely proportional to the number of people employed to do the job
- The number of workers and time to complete a task
Example 4: Car Speed Problem
A car takes 1 hour and 30 minutes to travel the distance between two cities during rush hour. If the average speed is 50% more during off-peak hours, what time will it take to cover the same distance between the two cities during off-peak hours (1 hour = 60 minutes)?
Options:
(a) 45 Minutes
(b) 1 Hour
(c) 1 Hour 15 Minutes
(d) 90 Minutes
(e) 1 Hour 45 minutes
Solution:
Step 1: The time taken to cover the distance and the speed are inversely proportional. Let the speed during rush hour be s.
The average speed is 50% more during off-peak hours, therefore off-peak hour speed = 1.5s
Step 2: With speed s it takes 1 hour 30 minutes = 1 × 60 + 30 = 90 minutes
Since the speed and time are inversely proportional:
s × 90 = 1.5s × x
90 = 1.5x
x = 90/1.5 = 60 Minutes or 1 Hour
The correct answer is (b) 1 Hour
Word Problems on Direct Proportion
Problem 1: Jane ran 100 meters in 15 seconds. How long did she take to run 1 meter?
Solution:
Step 1: Think of the word problem as: If 100 then 15. If 1 then how many?
Step 2: Write the proportional relationship:
100 : 15 :: 1 : x
100x = 15 × 1
x = 15/100 = 0.15 seconds
Answer: She took 0.15 seconds
Problem 2: If 4/7 of a tank can be filled in 2 minutes, how many minutes will it take to fill the whole tank?
Solution:
Step 1: Think of the word problem as: If 4/7 then 2. If 1 then how many? (Whole tank is 1)
Step 2: Write the proportional relationship:
(4/7) : 2 :: 1 : x
(4/7) × x = 2 × 1
x = 2 × (7/4) = 14/4 = 3.5 minutes
Answer: It took 3.5 minutes
Problem 3: A car travels 125 miles in 3 hours. How far would it travel in 5 hours?
Solution:
Step 1: Think of the word problem as: If 3 then 125. If 5 then how many?
Step 2: Write the proportional relationship:
3 : 125 :: 5 : x
3x = 125 × 5
x = 625/3 = 208⅓ miles
Answer: He traveled 208⅓ miles
Continued Proportion
Quantities of the same kind are said to be in 'continued' proportion when the ratio of the first to the second is equal to the ratio of the second to the third.
For example: Let the four quantities 3, 4, 9 and 12 be in proportion.
3 : 4 = 9 : 12 or 3/4 = 9/12
Important Terms in Continued Proportion
- The second quantity is called the mean proportional between the first and the third
- The third quantity is called the third proportional to the first and second
If 9, 6 and 4 are in continued proportion: 9 : 6 :: 6 : 4
Hence, 6 is the mean proportional between 9 and 4, and 4 is the third proportional to 9 and 6.
Mean Proportion Formula
If a : b :: b : c, then:
b² = a × c
Therefore: b = √(a × c)
Problem 1: Find the fourth proportional to the numbers 6, 8 and 15.
Solution:
Let x be fourth proportional
6 : 8 :: 15 : x
6 × x = 8 × 15
6x = 120
x = 120/6 = 20
Answer: 20
Problem 2: Find the third proportional to 15 and 20.
Solution:
Here, we have to find a fourth proportional to 15, 20 and 20 (it is continued proportion).
Let x be the fourth proportional.
We have:
15 : 20 :: 20 : x
15 × x = 20 × 20
15x = 400
x = 400/15 = 80/3 = 26⅔
Answer: 80/3 or 26⅔
Problem 3: Find the mean proportional between 3 and 75.
Solution:
Let x be the required mean proportion.
3 : x :: x : 75
x² = 3 × 75
x² = 225
x = √225 = 15
Answer: 15
Applications Of Direct Proportion
Common Direct Variation Relationships
- The cost of articles varies directly as number of articles.
- The distance covered by a moving object varies directly as its speed.
- The work done varies directly as the number of men at work.
Rule: Two quantities a and b are in direct variation, the ratio of any two values of a is equal to the ratio of the corresponding values of b.
a₁/a₂ = b₁/b₂
Or: a₁ × b₂ = a₂ × b₁
Or: a₁ : a₂ :: b₁ : b₂
Problem 1: If one score oranges cost Rs. 45, how many oranges can be bought for Rs. 72?
Solution:
For Rs. 45, number of oranges bought = 20
For Rs. 1, number of oranges bought = 20/45
For Rs. 72, number of oranges bought = (20/45) × 72
= (20 × 72)/45 = 1440/45 = 32
Answer: 32 oranges
Problem 2: A taxi charges a fare of Rs. 260 for a journey of 200 km. How much would it travel for Rs. 279.50?
Solution:
Let taxi travel x km for Rs. 279.50.
| Fare (Rs.) | Distance (km) |
|---|---|
| 260 | 200 |
| 279.50 | x |
Ratio of Rs = Ratio of distance
260/279.50 = 200/x
260x = 200 × 279.50
260x = 55900
x = 55900/260 = 215 km
Answer: The taxi will come 215 km
Problem 3: If 3 men and 4 women earn Rs. 480 in a day. Find how much will 7 men and 11 women earn in a day?
Solution:
Let men will earn x Rs. and women will earn y Rs.
For Men:
| Men | Rs. |
|---|---|
| 3 | 480 |
| 7 | x |
3/7 = 480/x
3x = 7 × 480
x = 3360/3 = 1120 Rs.
For Women:
| Women | Rs. |
|---|---|
| 4 | 480 |
| 11 | y |
4/11 = 480/y
4y = 11 × 480
y = 5280/4 = 1320 Rs.
Total money for 7 men and 11 women in a day:
= 1120 + 1320 = 2440 Rs.
Answer: Rs. 2440
Problem 4: If the wages of 15 workers for 6 days are Rs. 9450. Find the wages of 19 workers for 5 days.
Solution:
Let the wages be x.
First find relation between wages and workers:
| Workers | Wages |
|---|---|
| 15 | 9450 |
| 19 | x |
15/19 = 9450/x
15x = 19 × 9450
x = 179550/15 = 11970 Rs.
Now ratio between days and wages:
| Days | Wages |
|---|---|
| 6 | 11970 |
| 5 | y |
6/5 = 11970/y
6y = 5 × 11970
y = 59850/6 = 9975 Rs.
Answer: The wages of 19 workers for 5 days = Rs. 9975
Problem 5: The cost of 16 packets of salt, each weighing 900 grams is Rs. 84. What will be the cost of 27 packets of salt, each weighing 1 kg?
Solution:
Cost of 16 packets each weighing 9/10 kg = Rs. 84
Cost of 16 packets, each weighing 1 kg = Rs. (84 × 10/9)
Cost of 1 packet, each weighing 1 kg = Rs. (84 × 10)/(9 × 16)
Cost of 27 packets, each weighing 1 kg = Rs. (84 × 10 × 27)/(9 × 16)
= Rs. 22680/144 = Rs. 315/2 = Rs. 157.50
Answer: The cost of 27 packets, each weighing 1 kg is Rs. 157.50
Applications Of Inverse Proportion
Common Inverse Variation Relationships
- The time taken to finish a piece of work varies inversely as the number of men at work.
- The speed varies inversely as the time taken to cover a distance.
Rule: If two quantities a and b vary inversely as each other, and b₁, b₂ are the values of b corresponding to the values a₁, a₂ of a respectively, then:
a₁ × b₁ = k and a₂ × b₂ = k
Therefore: a₁ × b₁ = a₂ × b₂
Or: a₁ : a₂ = b₂ : b₁
Or: a₁ : a₂ :: b₂ : b₁
Notice: In inverse proportion, when we write the ratio form, the second ratio is inverted (b₂ : b₁ instead of b₁ : b₂).
Invariant Proportion
If two quantities are directly proportional then their quotient is an invariant (it does not change, it is constant). Further, if we have another pair of two quantities that are directly proportional then their quotient is also an invariant. Moreover, if you multiply these two quotients, you end up with a third invariant.
Example: Complex Proportion Problem
Given that x is directly proportional to y and to z and is inversely proportional to w, and that x = 4 when (w, y, z) = (6, 8, 5), what is x when (w, y, z) = (4, 10, 9)?
Solution:
Since x is directly proportional to y: x/y is constant
Since x is directly proportional to z: x/z is constant
Since x is inversely proportional to w: xw is constant
Thus, xw/(yz) is constant. (They just combined the constant terms)
So, xw/(yz) = (4)(6)/(8)(5) = 24/40 = 3/5
Thus, when (w, y, z) = (4, 10, 9), we find:
x = (3 × y × z)/(5 × w) = (3 × 10 × 9)/(5 × 4) = 270/20 = 27/2 = 13.5
Answer: x = 27/2 or 13.5