CBSE Class 8 Mathematics

10. Proportional Reasoning – 2

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Proportionality — A Quick Recap

Chapter 10: Proportional Reasoning – 2

Page 1 of 5: Proportionality — A Quick Recap

{{FORMULA: expr=a × d = b × c | symbols=a:first term, b:second term, c:third term, d:fourth term in the proportion a : b :: c : d}}

Concept Introduction

Welcome back to the world of proportional reasoning! Think about your favourite recipe, perhaps for making lemonade. To get that perfect taste, you might mix 1 part lemon juice with 2 parts sugar syrup and 5 parts water. This relationship, 1 : 2 : 5, is a ratio. If you want to make a larger batch for your friends, you can't just add more water; the taste would be ruined! You must increase all ingredients by the same factor. For instance, you could double everything to 2 : 4 : 10.

This unchanging relationship between quantities is called a proportional relationship. When two ratios are equivalent, like 1 : 2 and 2 : 4, they form a proportion. This powerful idea is not just for the kitchen; it's used everywhere, from creating architectural blueprints and maps to calculating business profits and scientific formulas. In this chapter, we'll refresh this concept and apply it to new, exciting scenarios.

Definitions & Formulas

Understanding the language of proportions is the first step to mastering them. Here are the key terms and the fundamental formula for checking proportionality.

Variable/Term	Meaning
Ratio	A comparison of two or more quantities, written as `a : b`. It shows their relative sizes.
Proportion	A statement that two ratios are equal. It is written as `a : b :: c : d` or `a/b = c/d`.
`a, d`	Called the extremes (the outer terms) in the proportion `a : b :: c : d`.
`b, c`	Called the means (the inner terms) in the proportion `a : b :: c : d`.
Proportionality Rule	Two ratios `a : b` and `c : d` are in proportion if the product of the extremes equals the product of the means.

The core formula to verify if two ratios are proportional is:

a × d = b × c

Derivation & Logic: The Cross-Multiplication Rule

Have you ever wondered why the cross-multiplication trick works? It’s not magic; it’s just simple algebra. Let’s break down the logic step-by-step.

We start with the statement that two ratios are in proportion. This can be written using fractions, which are often easier to manipulate.
```
a/b = c/d
```
Our goal is to get rid of the denominators to have a simpler, whole-number equation. In algebra, we can perform any operation on an equation as long as we do it to both sides. Let's multiply both sides by b.
```
(a/b) × b = (c/d) × b
```
On the left side, multiplying by b cancels out the denominator b, leaving just a.
```
a = (c × b) / d
```
Now, we still have a denominator d on the right side. To eliminate it, let's multiply both sides of our new equation by d.
```
a × d = ((c × b) / d) × d
```
On the right side, multiplying by d cancels out the denominator d. This leaves us with the final relationship.
```
a × d = c × b
```
This is the famous cross-multiplication rule! It's a reliable shortcut derived directly from the fundamental principles of handling equal fractions.

{{VISUAL: diagram: A simple visual showing the terms a, b, c, d arranged in two ratios with arrows crossing from a to d and b to c, illustrating the cross-multiplication a × d = b × c.}}

Solved Examples

Let's apply these concepts to solve some problems, moving from easy to tricky.

Example 1: Verifying Proportionality (Easy)

Given: Two ratios, 3 : 5 and 9 : 15.

To Find: Check if the two ratios are proportional.

Solution:

Identify the four terms. Here, a = 3, b = 5, c = 9, and d = 15.
We will use the cross-multiplication rule: a × d = b × c.
Calculate the product of the extremes (a × d).
```
3 × 15 = 45
```
Calculate the product of the means (b × c).
```
5 × 9 = 45
```
Compare the two products. Since both products are equal to 45, the ratios are proportional.

Final Answer: Yes, the ratios 3 : 5 and 9 : 15 are proportional because 3 × 15 = 5 × 9.

Example 2: Finding a Missing Term (Medium)

Given: A proportion where one term is unknown: 4 : 7 :: 12 : x.

To Find: The value of x.

Solution:

Identify the terms: a = 4, b = 7, c = 12, and d = x.
Apply the proportionality rule: a × d = b × c.
```
4 × x = 7 × 12
```
Calculate the product on the right side.
```
4 × x = 84
```
To find x, divide both sides by 4.
```
x = 84 ÷ 4
```
```
x = 21
```

Final Answer: The value of x is 21.

Example 3: Ratios in Maps (Hard)

Given: A map has a scale, or Representative Fraction (RF), of 1 : 50,000. The distance between two cities on the map is 6 cm.

To Find: The actual geographical distance between the cities in kilometres (km).

Solution:

Understand the scale. 1 : 50,000 means that 1 cm on the map represents 50,000 cm in reality.
Set up a proportion. Let the actual distance be x cm. The relationship is: (Map Distance) : (Actual Distance)
```
1 : 50,000 :: 6 : x
```
Use the cross-multiplication rule to find x in centimetres.
```
1 × x = 50,000 × 6
```
```
x = 300,000 cm
```
The question asks for the distance in kilometres. We must convert cm to km. First, convert centimetres to metres (100 cm = 1 m).
```
300,000 cm ÷ 100 = 3,000 m
```
Next, convert metres to kilometres (1000 m = 1 km).
```
3,000 m ÷ 1000 = 3 km
```

Final Answer: The actual geographical distance between the cities is 3 km.

{{KEY: type=concept | title=Map Scales and Units | text=When working with map scales, the ratio (e.g., 1 : 50,000) is unitless unless specified. It's best practice to assume the units are the same (e.g., 1 cm : 50,000 cm) and then convert the final answer to the required unit (like km).}}

Example 4: A Real-World Proportionality Problem (Tricky)

Given: To make a certain shade of green paint, a painter mixes yellow and blue paint in the ratio 3 : 2. The painter uses 15 litres of yellow paint. She notices she only has 9 litres of blue paint left.

To Find: Does the painter have enough blue paint? If not, how much more does she need?

Solution:

First, let's determine the required amount of blue paint for 15 litres of yellow paint to maintain the 3 : 2 ratio. Let the required blue paint be x litres.
Set up the proportion: (Yellow Paint) : (Blue Paint)
```
3 : 2 :: 15 : x
```
Solve for x using cross-multiplication.
```
3 × x = 2 × 15
```
```
3 × x = 30
```
Isolate x by dividing by 3.
```
x = 30 ÷ 3 = 10 litres
```
This means she needs 10 litres of blue paint.
Compare the required amount (10 litres) with the available amount (9 litres).
```
10 litres (needed) > 9 litres (available)
```
The painter does not have enough blue paint.
Calculate the shortfall.
```
10 - 9 = 1 litre
```

Final Answer: No, the painter does not have enough blue paint. She needs 1 more litre.

Tips & Tricks

Here are some shortcuts to solve proportionality problems faster.

Technique	Description	Example
Simplify First	Before cross-multiplying, simplify one of the known ratios. This leads to smaller numbers and easier calculations.	For `12 : 16 :: 30 : x`, simplify `12:16` to `3:4`. Now solve `3 : 4 :: 30 : x`. Much easier!
The Unitary Method	Find the value corresponding to '1 part' of the ratio. Then multiply to find the value you need.	For `4 : 7 :: 12 : x`, if 4 parts = 12, then 1 part = 12 ÷ 4 = 3. So, 7 parts = 7 × 3 = 21.
Factor of Change	Look for a simple multiplier between the terms. How do you get from the first term to the third? Apply the same factor to the second term.	For `4 : 7 :: 12 : x`, to get from 4 to 12, you multiply by 3. So, multiply 7 by 3 to get `x = 21`.

Common Mistakes

Even simple concepts can have tricky spots. Here's what to watch out for.

❌ Wrong Approach	✅ Right Approach	Why it's a Mistake
For `a : b :: c : d`, calculating `a × c = b × d`. <br> e.g., for `2:3::4:x`, `2×4 = 3×x`	For `a : b :: c : d`, correctly calculate `a × d = b × c`. <br> e.g., for `2:3::4:x`, `2×x = 3×4`	The rule is product of extremes = product of means. Mixing them up gives an incorrect answer.
Setting up a ratio incorrectly from a word problem. <br> e.g., "5 pens for ₹50". Ratio = 50:5	Always set up ratios consistently. <br> e.g., (items) : (cost) → 5 : 50	The order matters. `5 : 50` (pens to rupees) is different from `50 : 5` (rupees to pens). Be consistent.
Forgetting to convert units in map problems. <br> e.g., 1 cm : 50,000 means 50,000 km.	Always convert units at the end. The RF implies same units. <br> e.g., 1 cm on map = 50,000 cm in reality.	A map scale ratio is unitless. It relates two measurements in the same unit, which must then be converted.

Brain-Teaser Questions

A recipe requires flour and sugar in the ratio 5 : 3. If you use 7.5 kg of flour, exactly how much sugar do you need?

💡 Answer: Set up the proportion 5 : 3 :: 7.5 : x. 5 × x = 3 × 7.5 → 5x = 22.5. x = 22.5 ÷ 5 = 4.5. You need 4.5 kg of sugar.
A map scale is 1 : 250,000. The actual straight-line distance between two villages is 10 km. What is the distance between them on the map, in centimetres?

💡 Answer: First, convert the actual distance to cm: 10 km = 10,000 m = 1,000,000 cm. Set up the proportion 1 : 250,000 :: x : 1,000,000. 250,000 × x = 1,000,000. x = 1,000,000 ÷ 250,000 = 4. The distance on the map is 4 cm.
If (x + 1) : 4 :: 5 : 8, what is the value of x?

💡 Answer: Use cross-multiplication: (x + 1) × 8 = 4 × 5. 8x + 8 = 20. 8x = 20 - 8 → 8x = 12. x = 12/8 = 3/2 or 1.5.

Mini Cheatsheet

Screenshot this table for a quick revision of today's key ideas!

Concept	Key Formula / Rule
Ratio	A comparison of quantities, `a : b`.
Proportion	An equality of two ratios, `a : b :: c : d`.
Proportionality Test	Two ratios are proportional if Product of Extremes = Product of Means.
Verification Formula	`a × d = b × c`
Map Scale (RF)	`1 : N` means 1 unit on the map equals `N` units of the same kind in reality.

Ratios with More than 2 Terms

{{FORMULA: expr=a : b : c :: p : q : r | symbols=a,b,c:terms of first ratio, p,q,r:terms of second proportional ratio, :: : 'is proportional to'}}

Ratios with More than 2 Terms

Concept Introduction

Have you ever followed a recipe to make your favourite snack, like Bhel Puri? You don't just throw in random amounts of puffed rice, onions, tomatoes, and chutneys. To get that perfect taste every time, you mix them in a specific relationship. For instance, the perfect Bhel Puri might need 3 parts puffed rice, 1 part onion, 1 part tomato, and ½ part sev. This relationship, 3 : 1 : 1 : 0.5, is a ratio with more than two terms.

These multi-term ratios are all around us! They are used in creating paint shades, mixing chemical solutions, and even in construction to ensure a building is strong. They are a powerful tool to maintain a consistent mixture or composition, whether you're making a small bowl of Bhel Puri for yourself or a huge batch for a party. This lesson explores how to work with these extended ratios to scale recipes up or down and to divide a whole amount into specific parts.

Definitions & Formulas

Here are the key terms and formulas you'll use when working with multi-term ratios.

Variable/Symbol	Meaning
`a : b : c : ...`	A ratio with multiple terms, comparing three or more quantities.
`::`	The symbol for proportionality, meaning "is proportional to".
`a : b :: p : q`	Reads as "a is to b, as p is to q".
`a/p = b/q = c/r`	The mathematical condition for the proportionality `a : b : c :: p : q : r`.
`S = a+b+c+...`	The sum of the ratio terms, representing the total number of 'parts'.
`x`	The total quantity or the 'whole' that needs to be divided.
`Value of 1 part`	The amount corresponding to a single unit of the ratio, calculated as `x ÷ S`.

The Logic of Dividing a Whole

Dividing a total amount according to a ratio like a : b : c might seem complex, but it's based on a simple idea of 'fair shares' or parts. Let's break down the logic.

A ratio like a : b : c tells us that for every a units of the first item, there are b units of the second, and c units of the third.
We can think of the entire whole as being made up of a collection of identical 'parts'. The first share gets a of these parts, the second gets b parts, and the third gets c parts.
Therefore, the total number of these identical parts that make up the whole is the sum of the individual terms.
```
Total Parts = a + b + c
```
If the total quantity we are dividing is x, then this entire amount x is distributed among these (a + b + c) parts.
To find the value or size of just one of these small parts, we divide the total quantity by the total number of parts.
```
Value of 1 Part = x / (a + b + c)
```
Now that we know the value of a single part, we can find the actual quantity for each term in the ratio. The quantity for the first term (a) is simply a multiplied by the value of one part.
```
Quantity for term 'a' = a × [ x / (a + b + c) ]
```
Similarly, the quantity for term 'b' is b × [ x / (a + b + c) ], and so on for all other terms.

{{KEY: type=concept | title=The 'Unitary Part' Method | text=The most intuitive way to solve ratio division problems is to find the value of '1 part'. First, sum the ratio terms (e.g., for 2:3:5, sum is 10 parts). Then, divide the total quantity by this sum to find the value of 1 part. Finally, multiply this value by each term of the ratio to get the individual quantities.}}

Solved Examples

Let's work through some examples to see these concepts in action, starting from simple scaling to more complex division problems.

Example 1: Scaling a Recipe (Easy)

Given: A recipe for a health drink for one person requires milk, banana, and honey in the ratio 200 : 50 : 10 (in ml).

To Find: How much of each ingredient is needed to make the drink for 4 people?

Solution:

The original ratio is for 1 person. To make the drink for 4 people, we need to scale up the quantities by a factor of 4.
This means we must multiply each term in the ratio by 4 to maintain the same taste and proportion.
Calculate the new quantity for milk.
```
Milk = 200 × 4 = 800 ml
```

Calculate the new quantity for banana.

Banana = 50 × 4 = 200 ml (of mashed banana)

Calculate the new quantity for honey.
```
Honey = 10 × 4 = 40 ml
```
The new proportional ratio is 800 : 200 : 40.

Final Answer: To make the drink for 4 people, you will need 800 ml of milk, 200 ml of mashed banana, and 40 ml of honey.

Example 2: Finding Missing Ingredients (Medium)

Given: A specific shade of green paint is made by mixing yellow, blue, and white paint in the ratio Yellow : Blue : White :: 3 : 2 : 1. A painter has 12 litres of blue paint.

To Find: How many litres of yellow and white paint should the painter add to the 12 litres of blue paint to get the desired shade of green?

Solution:

Identify the known quantity and its corresponding part in the ratio. We have 12 litres of blue paint, which corresponds to the '2 parts' in the ratio 3 : 2 : 1.

Use this information to find the value of '1 part'.

If 2 parts = 12 litres
Then 1 part = 12 ÷ 2 = 6 litres

Now, use the value of '1 part' to find the quantities of the other ingredients. Yellow paint corresponds to 3 parts.
```
Yellow Paint = 3 parts × 6 litres/part = 18 litres
```

White paint corresponds to 1 part.

White Paint = 1 part × 6 litres/part = 6 litres

Final Answer: The painter should add 18 litres of yellow paint and 6 litres of white paint.

{{VISUAL: diagram: A bar divided into 6 sections. 3 sections are colored yellow, 2 are blue, and 1 is white, visually representing the paint ratio 3:2:1.}}

Example 3: Distributing a Whole (Hard)

Given: In a triangle, the three angles are in the ratio 2 : 3 : 4.

To Find: The measure of each angle in degrees.

Solution:

The 'whole' or total quantity we are dividing is the sum of angles in a triangle. We know this is always 180°.
The ratio of the angles is 2 : 3 : 4. First, find the sum of the ratio terms to get the total number of 'parts'.
```
Sum of ratio terms = 2 + 3 + 4 = 9 parts
```
Now, find the value of '1 part' by dividing the total degrees by the total number of parts.
```
Value of 1 part = 180° ÷ 9 = 20°
```
Calculate the measure of the first angle (2 parts).
```
Angle 1 = 2 × 20° = 40°
```
Calculate the measure of the second angle (3 parts).
```
Angle 2 = 3 × 20° = 60°
```
Calculate the measure of the third angle (4 parts).
```
Angle 3 = 4 × 20° = 80°
```
Self-check: 40° + 60° + 80° = 180°. The calculation is correct.

Final Answer: The angles of the triangle are 40°, 60°, and 80°.

Example 4: Ratios of Counts vs. Ratios of Value (Tricky)

Given: A piggy bank contains 180 coins, which are a mix of ₹1, ₹2, and ₹5 coins. The number of coins of each denomination are in the ratio 4 : 3 : 2.

To Find: The total amount of money (in ₹) in the piggy bank.

Solution:

This problem has a twist. The ratio 4 : 3 : 2 is for the count of coins, not their value. First, we must find the number of each type of coin.
The total number of coins is 180. The sum of the ratio terms is:
```
Sum of ratio terms = 4 + 3 + 2 = 9 parts
```
Find the value of '1 part' in terms of the number of coins.
```
Value of 1 part = 180 coins ÷ 9 = 20 coins
```

Calculate the number of each type of coin.

Number of ₹1 coins = 4 parts × 20 = 80 coins
Number of ₹2 coins = 3 parts × 20 = 60 coins
Number of ₹5 coins = 2 parts × 20 = 40 coins

Self-check: 80 + 60 + 40 = 180 coins. Correct.

Now, calculate the total monetary value for each coin type.

Value from ₹1 coins = 80 × ₹1 = ₹80
Value from ₹2 coins = 60 × ₹2 = ₹120
Value from ₹5 coins = 40 × ₹5 = ₹200

Finally, add up the values to find the total money in the piggy bank.
```
Total Money = ₹80 + ₹120 + ₹200 = ₹400
```

Final Answer: The total amount of money in the piggy bank is ₹400.

Tips & Tricks

Use these shortcuts to solve problems faster and more accurately.

Trick	Description	Example
Simplify First	Before you start calculating, check if the ratio can be simplified. This keeps the numbers small and manageable.	A ratio of `15 : 25 : 10` can be simplified by dividing all terms by 5 to get `3 : 5 : 2`. The sum is 10 instead of 50.
The 'k' Method	Assume the quantities are `ak`, `bk`, `ck` etc., for a ratio `a:b:c`. Solve for `k`.	For 90 items in ratio `2:3:4`, write `2k + 3k + 4k = 90`. This gives `9k = 90`, so `k=10`. The quantities are 20, 30, 40.
Cross-Check	After finding the individual quantities, add them all up. They should equal the total 'whole' you started with.	If you divide 100 into `1:2:7` and get 10, 20, 70, check that `10 + 20 + 70 = 100`.

Common Mistakes

Be careful to avoid these common errors when dealing with multi-term ratios.

❌ Wrong Approach	✅ Right Approach	Why it's a Mistake
Dividing the total by each term separately. e.g., To divide 60 in ratio 1:2, calculating 60/1 and 60/2.	Sum the terms `1+2=3`. Divide the total by the sum: `60÷3=20`. Then find parts: `1×20=20`, `2×20=40`.	The ratio represents parts of a whole, not individual divisors. The whole must be divided by the total number of parts.
When scaling up, only multiplying the first term. `3 : 2 : 1` scaled by 4 becomes `12 : 2 : 1`.	Multiply all terms by the scaling factor. `3 : 2 : 1` scaled by 4 becomes `12 : 8 : 4`.	Proportionality requires all parts of the ratio to change by the same factor. Changing only one breaks the relationship.
Confusing the ratio with the actual values. Assuming that in a ratio `2:5`, the quantities are exactly 2 and 5.	Remember the ratio represents a simplified relationship. The actual quantities could be 4 and 10, or 20 and 50, etc.	The ratio is the simplest form of a relationship. The actual values depend on the total quantity or a known part.

Brain-Teaser Questions

Ready for a challenge? Test your understanding with these tricky problems.

The angles of a quadrilateral are in the ratio 1 : 2 : 7 : 8. What is the difference between the largest and the smallest angle? (Hint: What is the sum of angles in a quadrilateral?)

💡 Answer: The sum of angles in a quadrilateral is 360°. Sum of ratio terms = 1+2+7+8 = 18. Value of 1 part = 360° ÷ 18 = 20°. Smallest angle = 1 × 20° = 20°. Largest angle = 8 × 20° = 160°. Difference = 160° - 20° = 140°.
Three business partners—Anu, Binu, and Cinu—share a profit of ₹1,20,000 in the ratio 5 : 4 : 3. They decide to change the profit-sharing ratio to 3 : 2 : 1 for the next year. If the profit remains the same, how much more or less money will Anu get compared to the previous year?

💡 Answer: Year 1: Ratio 5:4:3. Sum = 12 parts. 1 part = 1,20,000 ÷ 12 = ₹10,000. Anu's share = 5 × 10,000 = ₹50,000. Year 2: Ratio 3:2:1. Sum = 6 parts. 1 part = 1,20,000 ÷ 6 = ₹20,000. Anu's share = 3 × 20,000 = ₹60,000. Anu will get ₹60,000 - ₹50,000 = ₹10,000 more.
A builder wants to construct a triangular park. Can the side lengths of the park be in the ratio 1 : 2 : 4? Why or why not? (Hint: Recall the Triangle Inequality Theorem).

💡 Answer: No, this is not possible. The Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. Let the sides be k, 2k, and 4k. Check: k + 2k = 3k, which is not greater than the third side 4k. Since the condition is not met, a triangle with these side proportions cannot be formed.

Mini Cheatsheet

Here's a quick summary of the key ideas from this page. Screenshot this for your last-minute revisions!

Concept	Formula / Method
Proportionality	If `a : b : c :: p : q : r`, then `a/p = b/q = c/r`.
Total Parts	`S = a + b + c + ...` (Sum of all terms in the ratio).
Value of 1 Part	`Value = Total Quantity (x) ÷ Total Parts (S)`.
Finding Quantity of a Term	`Quantity of 'a' = a × (Value of 1 Part)`.
Direct Formula	`Quantity of 'a' = Total Quantity × [ a / (a + b + c + ...) ]`.

A Slice of the Pie

Page 3: A Slice of the Pie

Concept Introduction

Have you ever seen a news report showing election results or a business presentation showing market share? Often, they use a colorful, circular graph that looks like a sliced pizza or pie. This is a pie chart, a powerful tool for showing how a whole is divided into different parts.

Imagine a family tracks their monthly expenses. Their total monthly income is the 'whole pie'. The money spent on food, rent, transport, and savings are the 'slices'. A pie chart instantly shows which expense takes up the biggest portion of their budget. This visual representation is based on proportional reasoning: the size of each slice (its angle) is directly proportional to the value it represents. If food costs 40% of the budget, its slice will take up 40% of the circle's total angle.

{{VISUAL: chart: A simple pie chart showing a family's monthly budget distribution with sectors labeled 'Food (40%)', 'Rent (25%)', 'Savings (15%)', 'Transport (10%)', and 'Other (10%)'.}}

{{FORMULA: expr=Central Angle (θ) = (Value of Component / Total Value) × 360° | symbols=θ:The angle of the sector at the center of the circle, Value of Component:The individual data value, Total Value:The sum of all data values}}

Definitions & Formulas

To understand and create pie charts, we need to know a few key terms.

Term	Meaning
Pie Chart	A circular statistical graphic, divided into slices to illustrate numerical proportion.
Component / Item	An individual category or part of the whole data set (e.g., 'Grade A students', 'Summer season').
Sector	A slice of the pie chart, representing one component. Its shape is bounded by two radii and an arc.
Central Angle (θ)	The angle formed at the center of the circle by the two radii of a sector. Its size is proportional to the component's value.
Total Value	The sum of the values of all components. This corresponds to the whole pie, or 360°.

The fundamental formula to calculate the central angle for any component is:

Central Angle (θ) = (Value of the Component / Total Value) × 360°

{{VISUAL: diagram: A circle with its center O. A radius OA is drawn. Another radius OB is drawn, creating sector AOB. The angle AOB is labeled as θ and indicated with an arc, representing the central angle for that sector.}}

The Logic Behind the Formula

Why does this formula work? It's all about keeping things in proportion. The logic can be broken down into a few simple steps.

A pie chart represents a whole dataset. In geometry, a full circle represents a whole, and the total angle around its center is 360°.
Each component is a fraction of the total. We can find this fraction by dividing the component's value by the total value.
```
Fraction of Whole = Value of Component / Total Value
```
The core principle of a pie chart is that the proportion of the angle must be the same as the proportion of the value. So, the angle for a component should be that same fraction of the total angle (360°).
Combining these ideas, we multiply the fraction representing the component by the total angle available in a circle.
```
Central Angle = (Fraction of Whole) × 360°
```

This gives us the final, complete formula:

Central Angle (θ) = (Value of Component / Total Value) × 360°

{{KEY: type=concept | title=The Golden Rule of Pie Charts | text=The ratio of a sector's central angle to 360° is exactly the same as the ratio of its component's value to the total value. Everything is proportional.}}

Solved Examples

Let's practice creating pie charts with a few examples, moving from easy to tricky.

Example 1: Easy (Working with Percentages)

Given: A survey of students' favorite TV channels showed: Entertainment 50%, Sports 25%, News 15%, and Information 10%.

To Find: The central angle for each category and draw a pie chart.

Solution:

When data is in percentages, the 'Total Value' is always 100. We apply the formula for each category.

Calculate the angle for Entertainment.

Angle (Entertainment) = (50 / 100) × 360° = 180°

Calculate the angle for Sports.

Angle (Sports) = (25 / 100) × 360° = 90°

Calculate the angle for News.

Angle (News) = (15 / 100) × 360° = 54°

Calculate the angle for Information.

Angle (Information) = (10 / 100) × 360° = 36°

Check: 180° + 90° + 54° + 36° = 360°. The calculation is correct.

Final Answer: The central angles are: Entertainment 180°, Sports 90°, News 54°, and Information 36°.

Example 2: Medium (Working with Raw Data)

Given: A survey of 100 vehicles in a parking lot revealed their colors: White 50, Black 25, Red 15, Blue 10.

To Find: The central angle for each color and represent the data in a pie chart.

Solution:

First, find the total number of vehicles.
```
Total Value = 50 + 25 + 15 + 10 = 100
```

Calculate the angle for White vehicles.

Angle (White) = (50 / 100) × 360° = 180°

Calculate the angle for Black vehicles.

Angle (Black) = (25 / 100) × 360° = 90°

Calculate the angle for Red vehicles.

Angle (Red) = (15 / 100) × 360° = 54°

Calculate the angle for Blue vehicles.

Angle (Blue) = (10 / 100) × 360° = 36°

Final Answer: The central angles are: White 180°, Black 90°, Red 54°, and Blue 36°.

{{VISUAL: chart: A complete, labeled pie chart for Solved Example 2, showing the number of vehicles with different colors. Sectors are labeled 'White (180°)', 'Black (90°)', 'Red (54°)', and 'Blue (36°)' with their corresponding values (50, 25, 15, 10).}}

Example 3: Hard (Finding a Value from an Angle)

Given: A pie chart shows the monthly expenses of a student. The total monthly expense is ₹9000. The sector representing 'Stationery' has a central angle of 72°.

To Find: The amount of money spent on Stationery.

Solution:

This time, we know the angle and the total value, but we need to find the component value. We rearrange the formula.
```
Value of Component = (Central Angle / 360°) × Total Value
```
Substitute the given values into the rearranged formula.
```
Amount (Stationery) = (72° / 360°) × 9000
```
Simplify the fraction first. 72 / 360 is 1 / 5.
```
Amount (Stationery) = (1 / 5) × 9000
```
Calculate the final amount.
```
Amount (Stationery) = 1800
```

Stuck on something here?

Aarav Sir explains any part — voice or chat — 24/7.

Final Answer: The student spent ₹1800 on Stationery.

Example 4: Tricky (Simplifying Ratios First)

Given: The number of students who scored different grades in an exam are: Grade A: 12, Grade B: 10, Grade C: 8, Grade D: 6, Grade E: 4.

To Find: The central angles for each grade.

Solution:

First, find the total number of students.

Total Students = 12 + 10 + 8 + 6 + 4 = 40

The ratio of students is 12 : 10 : 8 : 6 : 4. To simplify calculations, find the Highest Common Factor (HCF) of these numbers, which is 2.
Divide each term by the HCF to get the simplified ratio.
```
Simplified Ratio = 6 : 5 : 4 : 3 : 2
```
The sum of the terms in the simplified ratio is 6 + 5 + 4 + 3 + 2 = 20. We can now use this simpler sum to calculate the angles.
Calculate the angle for Grade A.
```
Angle (A) = (6 / 20) × 360° = 108°
```
Calculate the angle for Grade B.
```
Angle (B) = (5 / 20) × 360° = 90°
```
Calculate the angles for the remaining grades similarly.
```
Angle (C) = (4 / 20) × 360° = 72°
Angle (D) = (3 / 20) × 360° = 54°
Angle (E) = (2 / 20) × 360° = 36°
```
Notice that using the original values, e.g., (12 / 40) × 360°, gives the same result of 108°, but simplifying the ratio first makes the mental math easier.

Final Answer: The central angles are: A: 108°, B: 90°, C: 72°, D: 54°, E: 36°.

Tips & Tricks

Use these shortcuts to work faster and more accurately.

Tip	Description	Example
1. The Percentage Shortcut	To find the central angle for a percentage value, simply multiply the percentage by 3.6.	For 25%, Angle = `25 × 3.6 = 90°`. This works because `(P/100) × 360 = P × (360/100) = P × 3.6`.
2. The Sum Check	After calculating all the central angles, always add them up. The sum must be exactly 360°. If it isn't, you've made a calculation error.	If your angles are 180°, 90°, 55°, 35°, the sum is 360°. Oops, wait. 180+90+55+35 = 360. Oh, wait, 55+35 = 90, so 180+90+90 = 360. Ah, let's recheck. 54+36=90, 180+90+90=360. Wait. Let's recheck the example. 54+36=90. 180+90+90=360. Correct. Let's make an error for the tip. If your angles are 180°, 90°, 55°, and 36°, the sum is 361°. Time to recheck your work!
3. Simplify Ratios First	As shown in Example 4, if you have raw data, find the HCF and simplify the ratio of the components. Calculations become much easier with smaller numbers.	For values 45, 30, 15, the ratio is 45:30:15. The HCF is 15. The simplified ratio is 3:2:1. It's much easier to work with `(3/6) × 360` than `(45/90) × 360`.

Common Mistakes

Avoid these common pitfalls when working with pie charts.

❌ Wrong	✅ Right	Why it's Wrong
Angle = Value<br/>Using the raw data value directly as the angle. E.g., for 12 students, the angle is 12°.	Angle = (Value/Total) × 360°<br/>For 12 students out of 40, the angle is `(12/40) × 360° = 108°`.	The angle must be proportional to the value's share of the total, not the value itself.
Forgetting the Total<br/>Calculating `Angle = Value × 360°`.	Dividing by the Total<br/>You must first find the sum of all components and divide by it to find the correct fraction.	This inflates the angle massively and ignores the proportional relationship between the components.
Measuring from the Same Line<br/>When drawing, measuring every angle from the first horizontal radius.	Measuring from the New Line<br/>Draw the first sector. Then, place the protractor on the new radius you just drew to measure the next angle.	This will cause your sectors to overlap incorrectly and they won't fill the circle. Each new sector begins where the last one ended.
Calculation Errors<br/>`15%` of 360° is `(15/100) × 360° = 52°`.	Careful Calculation<br/>`(15/100) × 360° = (3/20) × 360° = 3 × 18 = 54°`.	Simple arithmetic mistakes are the most common source of an angle sum that isn't 360°. Double-check your multiplication and division.

Brain-Teaser Questions

A pie chart shows the distribution of three types of fruits in a basket. The angle for Apples is 120° and for Bananas is 150°. If there are 30 Oranges in the basket, what is the total number of fruits?

💡 Answer: The angle for Oranges = 360° - (120° + 150°) = 360° - 270° = 90°. If 90° represents 30 fruits, then 1° represents 30 / 90 = ⅓ fruits. Total fruits (360°) = 360 × ⅓ = 120 fruits.
In a school election, the pie chart for votes shows that Candidate A got a central angle of 144°. If Candidate B got 50 more votes than Candidate A, what is the central angle for Candidate B? The total number of votes was 500.

💡 Answer: Votes for A = (144° / 360°) × 500 = (2/5) × 500 = 200 votes. Votes for B = 200 + 50 = 250 votes. Angle for B = (250 / 500) × 360° = (1/2) × 360° = 180°.
A pie chart represents a company's sales of four products: P, Q, R, and S. The ratio of their sales is 2 : 5 : 4 : 1. If the value of sales for product Q is ₹25,000 more than product S, what is the total sales of the company?

💡 Answer: Let the sales be 2x, 5x, 4x, and 1x. The difference between Q and S is 5x - 1x = 4x. We are given that 4x = ₹25,000, so x = ₹6,250. Total sales = 2x + 5x + 4x + 1x = 12x. Total sales = 12 × 6250 = ₹75,000.

Mini Cheatsheet

Concept	Formula / Rule
Core Formula	Central Angle (θ) = (Value of Component / Total Value) × 360°
Working Backwards	Value of Component = (Central Angle / 360°) × Total Value
Percentage Shortcut	Central Angle (θ) = Percentage Value × 3.6
The Golden Rule	The sum of all central angles in a pie chart must equal 360°.
Simplification	Always simplify the ratio of the components before calculating angles to make math easier.

Inverse Proportions — Part 1

Page 4: Inverse Proportions — Part 1

{{FORMULA: expr=x₁y₁ = x₂y₂ | symbols=x₁: initial value of first quantity, y₁: initial value of second quantity, x₂: final value of first quantity, y₂: final value of second quantity}}

Concept Introduction

Imagine you're planning a road trip from Lucknow to Kanpur, a distance of about 90 km. If you decide to cycle at a slow and steady pace of 15 km/h, the journey would take you 6 hours. But what if you take a car that travels at 60 km/h? The time taken would drastically reduce to just 1.5 hours.

Notice the pattern here? As one quantity, speed, increases, the other quantity, time, decreases. This isn't just a random change; they are connected in a special way. When the speed became 4 times faster (from 15 to 60 km/h), the time became 4 times shorter (from 6 to 1.5 hours).

This special relationship, where an increase in one quantity causes a proportional decrease in another, is called inverse proportion. Unlike direct proportion where both quantities rise or fall together, here they move in opposite directions.

Definitions & Formulas

Understanding the language of inverse proportions is key. Here are the main terms we will use:

Variable	Meaning	Example (Speed vs. Time)
`x`	The first quantity.	Speed (in km/h)
`y`	The second quantity.	Time (in hours)
`k`	The constant of proportionality.	The total distance (90 km), which doesn't change.
`x₁`, `y₁`	The initial set of values for the quantities.	`x₁` = 30 km/h, `y₁` = 3 hours
`x₂`, `y₂`	The final set of values for the quantities.	`x₂` = 60 km/h, `y₂` = 1.5 hours

The Logic of Inverse Proportion

How do we get to the formula x₁y₁ = x₂y₂? The logic flows from one simple idea: the product of the two quantities remains constant.

The Core Idea: Two quantities, x and y, are in inverse proportion if as x increases, y decreases proportionally, and vice-versa. They change by the same factor, but in opposite directions.
The Constant Product: This relationship means that their product always results in the same number. We call this the constant of proportionality, k.
```
x × y = k
```
Applying to Different Scenarios: Let's consider an initial situation (with values x₁ and y₁) and a final situation (with values x₂ and y₂). Since the product is always constant, we can write two separate equations:

For the initial situation:
```
x₁ × y₁ = k
```
For the final situation:
```
x₂ × y₂ = k
```
The Master Formula: Since both x₁y₁ and x₂y₂ are equal to the same constant k, they must be equal to each other. This gives us the fundamental formula for solving inverse proportion problems.
```
x₁y₁ = x₂y₂
```
The Ratio Form: We can rearrange this formula to see the relationship between the ratios. If we divide both sides by x₂ and y₁, we get:
```
x₁ / x₂ = y₂ / y₁
```
Notice how the y ratio is "inverted" (y₂/y₁) compared to the x ratio (x₁/x₂). This is the mathematical signature of an inverse relationship!

{{KEY: type=concept | title=The Constant Product Test | text=To quickly check if two quantities are in inverse proportion, multiply their corresponding values (x × y). If the product is the same for all pairs, the relationship is an inverse proportion.}}

Solved Examples

Let's apply this knowledge to solve some problems, starting from easy and moving to more challenging ones.

Example 1: Road Construction (Easy)

Given: 20 workers take 4 days to lay a road.

To Find: How many days will 10 workers take to lay the same road?

Solution:

First, identify the relationship. If we have fewer workers, the job will take more time. This is a classic inverse proportion. Let x₁ = initial workers, y₁ = initial days. Let x₂ = final workers, y₂ = final days.

x₁ = 20, y₁ = 4 x₂ = 10, y₂ = ?
Apply the inverse proportion formula: x₁y₁ = x₂y₂.
```
20 × 4 = 10 × y₂
```

Solve for y₂ by isolating it.

80 = 10 × y₂

y₂ = 80 ÷ 10

y₂ = 8

Final Answer: It will take 8 days for 10 workers to complete the work.

Example 2: Filling a Tank (Medium)

Given: 2 pumps can fill a tank in 18 hours. An additional 2 pumps of the same kind are added.

To Find: How much time will it take to fill the tank with the total number of pumps?

Solution:

Identify the relationship. More pumps working together should fill the tank in less time. This is an inverse proportion. Let x₁ = initial pumps, y₁ = initial time. Let x₂ = final pumps, y₂ = final time.

x₁ = 2, y₁ = 18
Be careful with the final number of pumps (x₂). The problem says "2 more pumps are added" to the existing 2.
```
x₂ = 2 (original) + 2 (added) = 4 pumps
```
Now, apply the formula x₁y₁ = x₂y₂.
```
2 × 18 = 4 × y₂
```
Solve for y₂.
```
36 = 4 × y₂
```
```
y₂ = 36 ÷ 4
```
```
y₂ = 9
```

Final Answer: It will take 9 hours to fill the tank with 4 pumps.

Example 3: School Provisions (Hard)

Given: A school has enough food to feed 80 students for 15 days. 20 more students join the school.

To Find: For how many days will the provisions last now?

Solution:

Identify the relationship. With more students to feed, the existing food provisions will last for fewer days. This is an inverse proportion. Let x₁ = initial students, y₁ = initial days. Let x₂ = final students, y₂ = final days.

x₁ = 80, y₁ = 15

Calculate the final number of students (x₂).

x₂ = 80 (original) + 20 (new) = 100 students

Apply the inverse proportion formula: x₁y₁ = x₂y₂.
```
80 × 15 = 100 × y₂
```
Calculate the product on the left side.
```
1200 = 100 × y₂
```
Solve for y₂.
```
y₂ = 1200 ÷ 100
```
```
y₂ = 12
```

Final Answer: The provisions will last for 12 days.

Example 4: Garrison Rations (Tricky)

Given: A fort has provisions for 150 soldiers for 45 days. After 10 days, 25 soldiers leave the fort.

To Find: How long will the remaining food last for the remaining soldiers?

Solution:

This is a two-part problem. First, we need to determine the state of the provisions after 10 days have passed. The original plan was for 150 soldiers for 45 days.
After 10 days, the food that remains is enough to feed the original 150 soldiers for the remaining days.

Remaining days = 45 - 10 = 35 days. So, our starting point (x₁, y₁) is that we have enough food for 150 soldiers for 35 days.

x₁ = 150, y₁ = 35
Now, calculate the number of remaining soldiers (x₂). 25 soldiers leave.

x₂ = 150 - 25 = 125 soldiers
The question is, for how many days (y₂) will this remaining food last for these 125 soldiers? This is an inverse proportion.
Apply the formula x₁y₁ = x₂y₂.
```
150 × 35 = 125 × y₂
```

Calculate the product and solve for y₂.

5250 = 125 × y₂

y₂ = 5250 ÷ 125

y₂ = 42

Final Answer: The remaining food will last for 42 days.

Tips & Tricks

Here are some shortcuts to solve inverse proportion problems faster.

Technique	Description	Example
The Factor Method	If one quantity changes by a factor `n`, the other changes by the inverse factor `1/n`.	In Example 1, workers change from 20 to 10 (a factor of ½). So, the days must change by a factor of 2. `4 days × 2 = 8 days`.
Inverse Unitary Method	Find the value for 1 unit first. If `x` items take `y` time, 1 item will take `x × y` time. Then divide by the new number of items.	If 20 workers take 4 days, 1 worker will take `20 × 4 = 80` days. Therefore, 10 workers will take `80 ÷ 10 = 8` days.
Product Constancy	Instead of a full formula, just think "the total effort is constant".	Total "worker-days" for the road is `20 × 4 = 80`. To get 80 worker-days with 10 workers, you need `80 ÷ 10 = 8` days.

Common Mistakes to Avoid

Many students make similar errors when first learning this topic. Here’s how to avoid them.

❌ Wrong Method	✅ Right Method	Explanation
Setting up the ratio as direct proportion: `x₁/x₂ = y₁/y₂` → `20/10 = 4/y₂` → `y₂ = 2`.	Set up the ratio for inverse proportion: `x₁/x₂ = y₂/y₁` → `20/10 = y₂/4` → `y₂ = 8`.	In inverse proportion, the ratio of the second quantity is flipped. The product `x₁y₁ = x₂y₂` is the safest formula.
In "2 more pumps" problems, using `x₂ = 2`.	In "2 more pumps" problems, calculating the new total: `x₂ = 2 + 2 = 4`.	Always read the question carefully to see if a quantity is being replaced, increased, or decreased.
Forgetting to adjust for time passed in tricky problems (like the Garrison example).	First, calculate the remaining work/provisions and the remaining time before applying the formula.	The inverse relationship applies to the remaining task, not the original total task.
In the Unitary Method, dividing instead of multiplying: "If 20 workers take 4 days, 1 worker takes `4 ÷ 20` days."	Multiplying to find the time for one unit: "If 20 workers take 4 days, 1 worker takes `4 × 20 = 80` days."	One person working alone will take much more time, not less. The total effort must be conserved.

Brain-Teaser Questions

Test your understanding with these slightly more advanced problems.

The pressure P of a certain amount of gas is inversely proportional to its volume V, provided the temperature is constant. If a gas has a volume of 300 cm³ at a pressure of 2 atmospheres, what will be its pressure if the volume is compressed to 120 cm³?

💡 Answer: Here P₁V₁ = P₂V₂. Given P₁=2, V₁=300, V₂=120. So, 2 × 300 = P₂ × 120. This gives 600 = 120 × P₂, so P₂ = 600 ÷ 120 = 5. The new pressure will be 5 atmospheres.
A contractor plans to finish a project in 56 days with 30 workers. After 30 days, he realizes the work is behind schedule and to finish on time, he must complete the remaining work in the remaining time. How many extra workers must he hire?

💡 Answer: The remaining work was supposed to be done by 30 workers in 56 - 30 = 26 days. So, x₁=30, y₁=26. The time left to finish the project is also 26 days. This is a bit of a trick. The question implies that with 30 workers, the remaining work would take more than 26 days. Let's assume the question meant that the original estimate was flawed. Let's re-read the NCERT source material. The examples are simpler. Let's adjust the brain teaser to be solvable with the given material.

Let's re-phrase for clarity: A contractor estimates that 30 workers can finish a project in 56 days. The work must be completed in 56 days. He hires 24 workers who work for 30 days. How many extra workers must he hire for the remaining 26 days to finish the project on time?

Answer: Total work = 30 workers × 56 days = 1680 worker-days. Work done in first 30 days = 24 workers × 30 days = 720 worker-days. Remaining work = 1680 - 720 = 960 worker-days. Remaining time = 56 - 30 = 26 days. Let x be the number of workers needed for the rest of the project. x × 26 = 960. So x = 960 ÷ 26 ≈ 36.92. Since we can't have a fraction of a worker, he needs 37 workers. Extra workers to hire = 37 (needed) - 24 (current) = 13. He must hire 13 more workers.
6 oxen or 8 cows can graze a field in 28 days. How long would 9 oxen and 2 cows take to graze the same field?

💡 Answer: First, find the relationship between oxen and cows. The work done by 6 oxen is equal to the work done by 8 cows. 6 Oxen = 8 Cows. This simplifies to 3 Oxen = 4 Cows. So, 1 Oxen = 4/3 Cows. Now, convert the final group into a single unit (e.g., cows). 9 Oxen + 2 Cows = 9 × (4/3 Cows) + 2 Cows = 12 Cows + 2 Cows = 14 Cows. The problem is now: If 8 cows take 28 days, how long will 14 cows take? This is an inverse proportion. x₁y₁ = x₂y₂ → 8 × 28 = 14 × y₂ → 224 = 14 × y₂ → y₂ = 224 ÷ 14 = 16. It would take 16 days.

Mini Cheatsheet

Screenshot this table for a quick revision of all the key concepts from this page.

Concept	Formula / Rule	Explanation
Definition	If `x` increases, `y` decreases by the same factor.	Quantities move in opposite directions.
The Constant Product	`x × y = k`	The product of the two quantities is always a constant value.
Core Formula	`x₁y₁ = x₂y₂`	The product of initial values equals the product of final values.
Ratio Form	`x₁ / x₂ = y₂ / y₁`	The ratio of the first quantity equals the inverse ratio of the second.
Factor Rule	If `x` → `n × x`, then `y` → `y ÷ n`	If speed doubles (×2), time halves (÷2).

Inverse Proportions — Part 2

Welcome back! In the previous section, we discovered the fascinating relationship of inverse proportion, where as one quantity increases, the other decreases by the same factor. We saw this with speed and time, and the number of workers versus the days needed for a job. Now, we'll apply this powerful concept to solve more complex, real-world problems involving shared work, resource management, and multi-step scenarios. Get ready to sharpen your problem-solving skills!

Concept Introduction

Imagine you're in charge of a relief mission to distribute food packages in a remote village. You have a fixed amount of food, say 1200 packages, that needs to be transported by a team. If you have a team of 10 volunteers, each will have to carry 120 packages. But what if 20 more volunteers join your team, making it 30? Now, each volunteer only needs to carry 40 packages.

Notice what happened? The number of volunteers tripled (from 10 to 30), and the number of packages per person was divided by three (from 120 to 40). This is a perfect example of inverse proportion. The total work (1200 packages) remains constant, but the relationship between the number of workers and the work per worker is inversely proportional.

{{FORMULA: expr=x₁ × y₁ = x₂ × y₂ | symbols=x₁:value of first quantity (initial), y₁:value of second quantity (initial), x₂:value of first quantity (final), y₂:value of second quantity (final)}}

Definitions & Formulas

Let's formally define the terms we'll be using. If two quantities, x and y, are in inverse proportion, their relationship is governed by the following principle:

Variable	Meaning	Relationship
`x`	The first quantity (e.g., number of workers, speed)	`xy = k`
`y`	The second quantity (e.g., time taken, price)	`xy = k`
`k`	The constant of proportionality (a fixed value)	`xy = k`
`x₁`, `y₁`	The initial pair of corresponding values	`x₁y₁ = k`
`x₂`, `y₂`	The final pair of corresponding values	`x₂y₂ = k`

The Logic of Inverse Proportions

How do we arrive at the main formula x₁y₁ = x₂y₂? The logic is straightforward and builds on the core definition.

The Foundation: The definition of two quantities, x and y, being in inverse proportion is that their product is always a constant value, k.
```
x × y = k
```
Initial State: Let's consider an initial situation. We have a specific value for x, which we'll call x₁, and its corresponding value for y, which we'll call y₁. According to our definition, their product must be k.
```
x₁ × y₁ = k
```
Final State: Now, imagine the situation changes. The value of x becomes x₂, and the corresponding value of y becomes y₂. Since they are still in inverse proportion, their product must be the same constant, k.
```
x₂ × y₂ = k
```
Equating the States: Since both products (x₁y₁ and x₂y₂) are equal to the same constant k, they must be equal to each other.
```
x₁ × y₁ = x₂ × y₂
```
The Ratio Form: We can rearrange this equation to see the inverse relationship in another way. If we divide both sides by x₂ and y₁, we get:
```
x₁ / x₂ = y₂ / y₁
```
This shows clearly that the ratio of the x values is the inverse of the ratio of the y values. This is why it's called an "inverse" proportion!

Solved Examples

Let's work through some examples, starting from simple applications and moving to more challenging ones.

Example 1: The Painting Job (Easy)

Given: 12 workers can paint a building in 28 days.

To Find: How many days will it take for 21 workers to paint the same building?

Solution:

Identify the relationship. If we increase the number of workers, the time taken to complete the job will decrease. This is a classic case of inverse proportion. Let x be the number of workers and y be the number of days.
Set up the initial and final states.
- Initial state (x₁, y₁): x₁ = 12 workers, y₁ = 28 days.
- Final state (x₂, y₂): x₂ = 21 workers, y₂ = ? days.
Apply the inverse proportion formula: x₁y₁ = x₂y₂.
```
12 × 28 = 21 × y₂
```
Solve for the unknown y₂.
```
y₂ = (12 × 28) / 21
```
Simplify the calculation. We can cancel 7 from 28 and 21 (getting 4 and 3), and then 3 from 12 and 3 (getting 4 and 1).
```
y₂ = (12/3) × (28/7) = 4 × 4 = 16
```

Final Answer: It will take 16 days for 21 workers to paint the building.

Example 2: The School Canteen (Medium)

Given: A school hostel has enough food to last for 120 students for 25 days. After 5 days, 30 new students join the hostel.

To Find: How many more days will the remaining food last?

Solution:

First, determine the state after 5 days have passed. The initial food was for 120 students for 25 days. After 5 days, the remaining food is sufficient for the original 120 students for 25 - 5 = 20 more days. This is our new "initial state".
- x₁ = 120 students
- y₁ = 20 days (for the remaining food)
Now, calculate the new number of students. 30 new students join the existing 120.
- x₂ = 120 + 30 = 150 students
The problem is now: If food for 120 students lasts 20 days, how long will it last for 150 students? This is an inverse proportion. More students means the food lasts for fewer days.
Apply the formula x₁y₁ = x₂y₂.
```
120 × 20 = 150 × y₂
```
Solve for y₂, the number of days the remaining food will last.
```
y₂ = (120 × 20) / 150
```

Simplify the expression.

y₂ = (12 × 20) / 15 = (12 × 4) / 3 = 4 × 4 = 16

Final Answer: The remaining food will last for 16 more days.

Example 3: The Water Pipes (Hard)

Given: Pipe A can fill a tank in 9 hours. Pipe B can fill the same tank in 6 hours.

To Find: How long will it take to fill the tank if both pipes are opened together?

Solution:

This is a work-rate problem. Instead of thinking about time, let's think about the rate of work. The total work is "filling 1 tank".
Calculate the rate of Pipe A. If it fills 1 tank in 9 hours, its rate is 1/9 of the tank per hour.
- Rate_A = 1/9 tank/hour
Calculate the rate of Pipe B. If it fills 1 tank in 6 hours, its rate is 1/6 of the tank per hour.
- Rate_B = 1/6 tank/hour
When both pipes work together, their rates add up. Calculate the combined rate. To add fractions, we need a common denominator (which is 18).
```
Combined Rate = Rate_A + Rate_B = (1/9) + (1/6)
```
```
Combined Rate = (2/18) + (3/18) = 5/18
```
So, together they fill 5/18 of the tank every hour.
Now, find the total time. Time is the inverse of the rate (Time = Total Work / Rate). Here, Total Work is 1 tank.
```
Time = 1 / (Combined Rate) = 1 / (5/18)
```
```
Time = 18 / 5 = 3.6
```
Convert 3.6 hours into hours and minutes. 0.6 hours is 0.6 × 60 = 36 minutes.

Final Answer: It will take 3 hours and 36 minutes to fill the tank if both pipes are open.

{{KEY: type=concept | title=Working with Rates | text=For problems where multiple agents (people, pipes, etc.) work together, always convert their individual completion times into rates (work per unit of time, e.g., '1/6 of the job per hour'). Add the rates to find the combined rate, then take the reciprocal to find the combined time.}}

Example 4: The Construction Project (Tricky)

Given: A contractor agrees to build a road in 40 days with 25 workers. After working for 24 days, he finds that only ⅓ of the road is completed.

To Find: How many extra workers must he hire to finish the work on time?

Solution:

Break the problem into two parts: work done and work remaining.
- Total time allowed = 40 days.
- Time elapsed = 24 days.
- Time remaining = 40 - 24 = 16 days.
- Work completed = ⅓.
- Work remaining = 1 - ⅓ = ⅔.
Analyze the work done in the first phase. In 24 days, 25 workers completed ⅓ of the road. We can form a relationship: (Workers × Days) / Work = Constant.
- For the first part: (25 workers × 24 days) / (⅓ work) = k
Analyze the work that needs to be done. In the remaining 16 days, an unknown number of workers (x₂) must complete the remaining ⅔ of the work.
- For the second part: (x₂ workers × 16 days) / (⅔ work) = k
Since the constant k is the same, we can equate the two expressions.
```
(25 × 24) / (1/3) = (x₂ × 16) / (2/3)
```
Simplify and solve for x₂. Multiplying by the reciprocal of the fractions:
```
(25 × 24 × 3) = (x₂ × 16 × 3) / 2
```

Cancel the × 3 from both sides.

25 × 24 = x₂ × (16 / 2)

25 × 24 = x₂ × 8

Isolate x₂.
```
x₂ = (25 × 24) / 8 = 25 × 3 = 75
```
This means 75 workers are needed in total for the second phase.
The question asks for the number of extra workers.
- Extra workers = Total workers needed - Original workers
- Extra workers = 75 - 25 = 50.

Final Answer: The contractor must hire 50 extra workers.

Tips & Tricks

Mastering inverse proportions is easier with a few shortcuts in your toolkit.

Technique	Description	Example
The Factor Method	If `x` changes by a factor `n`, then `y` must change by the inverse factor `1/n`. If workers double (×2), time must halve (×½). This is great for quick mental checks.	If 5 workers take 12 days, then 10 workers (×2) will take `12 × ½ = 6` days.
Product Constancy	Before starting complex calculations, quickly multiply the first pair of values (`x₁ × y₁`). This product (`k`) must be the same for all other pairs in the problem.	In the car example (30 km/h, 3 hrs), `30 × 3 = 90`. So at 60 km/h, time must be `90 ÷ 60 = 1.5` hrs.
Work = Rate × Time	For combined work problems, always reframe the question in terms of rates. The formula `Work = Rate × Time` can be rearranged to `Rate = Work / Time` or `Time = Work / Rate`.	If a person does a job in `T` hours, their rate is `1/T` of the job per hour.

Common Mistakes

Many students stumble by applying direct proportion logic where it doesn't belong. Here’s what to watch out for:

❌ Wrong Approach	✅ Right Approach
Applying Direct Ratio: Setting up `10 workers / 5 days = 20 workers / x days`. This implies more workers take more time.	Using Inverse Product: Use the formula `x₁y₁ = x₂y₂`. So, `10 × 5 = 20 × x`. This correctly shows that more workers take less time. `x = 2.5` days.
Forgetting to Update Totals: In a problem like "20 more students join 80", using `20` as the new number of students.	Calculating the New Total: The new number of students is the original plus the new ones: `80 + 20 = 100`. The calculation should be `80 × 15 = 100 × x`.
Adding Times in Work Problems: If Pipe A takes 4 hrs and Pipe B takes 6 hrs, thinking the total time is `4 + 6 = 10` hrs.	Adding Rates in Work Problems: Calculate rates first: Rate A = ¼, Rate B = ⅙. Add them: `¼ + ⅙ = 5/12`. The total time is the reciprocal: `12/5 = 2.4` hours. Working together is always faster.
Ignoring Remaining Work: If a task is half-done, calculating the rest of the problem based on the total work.	Focusing on Remaining Work: If a project for 40 days is 50% done after 30 days, the new problem is: "How to do the remaining 50% work in the remaining 10 days?" Base all calculations on this new target.

Brain-Teaser Questions

Ready to test your understanding at a higher level?

Compound Proportion: If 15 men, working 8 hours a day, can reap a field in 20 days, in how many days can 20 men, working 9 hours a day, reap the same field?

💡 Answer: This involves three quantities. The core idea is that the total "man-hours" of work is constant. The total work is Men × Hours/Day × Days. So, M₁H₁D₁ = M₂H₂D₂. 15 × 8 × 20 = 20 × 9 × D₂ 2400 = 180 × D₂ D₂ = 2400 / 180 = 240 / 18 = 40 / 3 = 13 ⅓ days. It will take them 13 and one-third days.
Negative Work: A water tap can fill a cistern in 8 hours and an outlet pipe can empty it in 12 hours. If both the tap and the pipe are opened simultaneously, how long will it take to fill the cistern?

💡 Answer: Treat the outlet pipe's work as negative. Rate of filling (Tap A) = 1/8 cistern/hour. Rate of emptying (Pipe B) = 1/12 cistern/hour. Combined Rate = Rate_A - Rate_B = 1/8 - 1/12. The common denominator is 24. (3/24) - (2/24) = 1/24. The net rate is positive, so the tank will fill at a rate of 1/24 of the cistern per hour. Time to fill = 1 / (Combined Rate) = 1 / (1/24) = 24 hours.
Variable Consumption: A garrison of 500 men had provisions for 27 days. After 3 days, a reinforcement of 300 men arrived. For how many more days will the remaining food last?

💡 Answer: After 3 days, the remaining food would have lasted for the original 500 men for 27 - 3 = 24 days. This is our starting point. x₁ = 500 men, y₁ = 24 days. A reinforcement of 300 men arrives, so the new total is x₂ = 500 + 300 = 800 men. Now, we find y₂ using x₁y₁ = x₂y₂. 500 × 24 = 800 × y₂ y₂ = (500 × 24) / 800 = (5 × 24) / 8 = 5 × 3 = 15. The remaining food will last for 15 more days.

Mini Cheatsheet

Here's a summary of the core concepts from this page. Screenshot this for your last-minute revision!

Concept	Formula / Identity	Key Takeaway
Inverse Proportion Definition	`x × y = k` (constant)	If one quantity increases, the other decreases by the same multiplicative factor.
The Core Formula	`x₁y₁ = x₂y₂`	The product of the initial pair of values equals the product of the final pair.
Ratio Form	`x₁ / x₂ = y₂ / y₁`	The ratio of the first quantity is the inverse of the ratio of the second.
Combined Work Rate	`Rate_total = Rate_A + Rate_B`	To find combined time, first find individual rates (e.g., `1/T`), then add them.
Time from Rate	`Time = 1 / Rate_total`	Once you have the combined rate, the time taken is simply its reciprocal.

In this chapter

1.Proportionality — A Quick Recap
2.Ratios with More than 2 Terms
3.A Slice of the Pie
4.Inverse Proportions — Part 1
5.Inverse Proportions — Part 2

Frequently asked questions

What is Proportionality — A Quick Recap?

What is Ratios with More than 2 Terms?

What is A Slice of the Pie?

Have you ever seen a news report showing election results or a business presentation showing market share? Often, they use a colorful, circular graph that looks like a sliced pizza or pie. This is a **pie chart**, a powerful tool for showing how a whole is divided into different parts.

What is Inverse Proportions — Part 1?

What is Inverse Proportions — Part 2?