Introduction to Proportion
You already know that a ratio compares two quantities. For example, if there are 4 boys and 6 girls in a group, the ratio of boys to girls is 4:6 or 2:3.
Now, what if another group has 8 boys and 12 girls? The ratio is 8:12, which also simplifies to 2:3. Since both groups have the same ratio, we say the two ratios are in proportion.
Proportion is a very useful idea. It helps us figure out unknown quantities when we know that two ratios are equal. If 3 kg of rice costs Rs 150, how much does 7 kg cost? You can use proportion to find out without calculating the price of 1 kg first!
In Class 6 Mathematics (NCERT), proportion is studied in the chapter Ratio and Proportion. You will learn what proportion means, how to check if two ratios are in proportion using cross multiplication, and how to solve word problems about cost, distance, maps, recipes, and sharing.
What is Introduction to Proportion?
Definition: Two ratios are said to be in proportion if they are equal.
If the ratio a:b is equal to the ratio c:d, we say that a, b, c, d are in proportion.
We write:
a : b :: c : d
Read as: "a is to b as c is to d"
This means:
- a/b = c/d
- The first ratio equals the second ratio.
Key terms:
- Extremes: The first and last terms (a and d) are called the extremes.
- Means: The second and third terms (b and c) are called the means.
Cross multiplication test:
- If a:b :: c:d, then a × d = b × c (product of extremes = product of means).
- This is the quickest way to check if two ratios are in proportion.
Introduction to Proportion Formula
Proportion Test:
a : b :: c : d means a × d = b × c
Product of extremes = Product of means
- Extremes: a and d (the outer terms)
- Means: b and c (the inner terms)
To check if ratios are in proportion:
- Method 1: Simplify both ratios and check if they are equal.
- Method 2: Cross multiply and check if a × d = b × c.
To find a missing term:
If a : b :: c : x, then x = (b × c) / a
Cross multiply and solve for the unknown.
Derivation and Proof
Understanding proportion:
Why cross multiplication works:
- If a/b = c/d, multiply both sides by b × d.
- Left side: (a/b) × (b × d) = a × d
- Right side: (c/d) × (b × d) = b × c
- So a × d = b × c.
- This is called cross multiplication because you multiply diagonally across the equal sign.
Example to understand:
- Is 2:3 the same as 6:9?
- Method 1 (Simplify): 2:3 is already in simplest form. 6:9 = 6÷3 : 9÷3 = 2:3. Yes, they are equal!
- Method 2 (Cross multiply): 2 × 9 = 18 and 3 × 6 = 18. Since 18 = 18, the ratios are in proportion.
Think of it like buying pencils:
- If 2 pencils cost Rs 6, then 3 pencils cost Rs 9.
- The ratio of pencils is 2:3.
- The ratio of costs is 6:9 = 2:3.
- Same ratio means they are in proportion: 2:3 :: 6:9.
- The cost per pencil is Rs 3 in both cases — this is why the ratios are equal.
Another way to think about it:
- Proportion means the relationship stays the same even when the numbers change.
- If you double the pencils (from 2 to 4), the cost also doubles (from 6 to 12).
- 2:6 :: 4:12. Check: 2 × 12 = 24, 6 × 4 = 24. Equal! ✓
- The ratio remains 1:3 (1 pencil costs Rs 3) no matter how many you buy.
Types and Properties
Types of proportion problems:
1. Checking if ratios are in proportion:
- Are 3:4 and 9:12 in proportion?
- Cross multiply: 3 × 12 = 36, 4 × 9 = 36. Equal! Yes, they are in proportion.
2. Finding a missing term:
- If 5:8 :: 15:x, find x.
- 5 × x = 8 × 15 → 5x = 120 → x = 24.
3. Continued proportion:
- Three numbers a, b, c are in continued proportion if a:b :: b:c.
- This means b² = a × c (b is the mean proportional).
4. Word problems:
- "If 4 chocolates cost Rs 20, how much do 10 chocolates cost?"
- Set up the proportion: 4:20 :: 10:x.
5. Map and scale problems:
- "On a map, 1 cm = 50 km. If two cities are 3.5 cm apart on the map, what is the real distance?"
- 1:50 :: 3.5:x → x = 175 km.
Solved Examples
Example 1: Example 1: Checking proportion by simplifying
Problem: Are 6:8 and 15:20 in proportion?
Solution:
Simplify each ratio:
- 6:8 = 6÷2 : 8÷2 = 3:4
- 15:20 = 15÷5 : 20÷5 = 3:4
Compare:
- 3:4 = 3:4 ✓
Answer: Yes, 6:8 and 15:20 are in proportion.
Example 2: Example 2: Checking proportion by cross multiplication
Problem: Are 4:7 and 12:21 in proportion?
Solution:
Cross multiply:
- Product of extremes: 4 × 21 = 84
- Product of means: 7 × 12 = 84
Compare:
- 84 = 84 ✓
Answer: Yes, 4:7 :: 12:21. They are in proportion.
Example 3: Example 3: Ratios NOT in proportion
Problem: Are 3:5 and 7:10 in proportion?
Solution:
Cross multiply:
- Product of extremes: 3 × 10 = 30
- Product of means: 5 × 7 = 35
Compare:
- 30 ≠ 35
Answer: No, 3:5 and 7:10 are NOT in proportion.
Example 4: Example 4: Finding a missing term
Problem: If 3:5 :: 12:x, find x.
Solution:
Use cross multiplication:
- Product of extremes = Product of means
- 3 × x = 5 × 12
- 3x = 60
- x = 60 ÷ 3
- x = 20
Verify: 3:5 = 3/5. 12:20 = 12/20 = 3/5 ✓
Answer: x = 20.
Example 5: Example 5: Finding a missing first term
Problem: If x:4 :: 15:20, find x.
Solution:
Cross multiply:
- x × 20 = 4 × 15
- 20x = 60
- x = 60 ÷ 20
- x = 3
Verify: 3:4 and 15:20. Simplify 15:20 = 3:4 ✓
Answer: x = 3.
Example 6: Example 6: Cost problem using proportion
Problem: If 5 notebooks cost Rs 75, how much do 8 notebooks cost?
Solution:
Set up the proportion:
- Notebooks : Cost = Notebooks : Cost
- 5 : 75 :: 8 : x
Cross multiply:
- 5 × x = 75 × 8
- 5x = 600
- x = 600 ÷ 5
- x = Rs 120
Answer: 8 notebooks cost Rs 120.
Example 7: Example 7: Map scale problem
Problem: On a map, 2 cm represents 50 km. If two cities are 7 cm apart on the map, find the real distance.
Solution:
Set up the proportion:
- Map distance : Real distance = Map distance : Real distance
- 2 : 50 :: 7 : x
Cross multiply:
- 2 × x = 50 × 7
- 2x = 350
- x = 350 ÷ 2
- x = 175 km
Answer: The real distance is 175 km.
Example 8: Example 8: Recipe problem
Problem: A recipe for 4 people uses 3 cups of flour. How many cups are needed for 12 people?
Solution:
Set up the proportion:
- People : Flour = People : Flour
- 4 : 3 :: 12 : x
Cross multiply:
- 4 × x = 3 × 12
- 4x = 36
- x = 36 ÷ 4
- x = 9 cups
Answer: You need 9 cups of flour for 12 people.
Example 9: Example 9: Identifying extremes and means
Problem: In the proportion 5:8 :: 15:24, identify the extremes and means. Verify that the product of extremes equals the product of means.
Solution:
Identify:
- Extremes: 5 and 24 (first and last terms)
- Means: 8 and 15 (middle terms)
Verify:
- Product of extremes = 5 × 24 = 120
- Product of means = 8 × 15 = 120
- 120 = 120 ✓
Answer: Extremes are 5, 24. Means are 8, 15. Their products are both 120.
Example 10: Example 10: Sharing in a given ratio
Problem: Rs 450 is shared between Anu and Bala in the ratio 2:3. How much does each get?
Solution:
Total parts:
- 2 + 3 = 5 parts
Value of 1 part:
- 450 ÷ 5 = Rs 90
Anu's share (2 parts):
- 2 × 90 = Rs 180
Bala's share (3 parts):
- 3 × 90 = Rs 270
Verify: 180 + 270 = 450 ✓ and 180:270 = 2:3 ✓
Answer: Anu gets Rs 180 and Bala gets Rs 270.
Example 11: Example 11: Speed and distance proportion
Problem: A car travels 120 km in 2 hours at a steady speed. How far will it travel in 5 hours at the same speed?
Solution:
Set up the proportion:
- Distance : Time = Distance : Time
- 120 : 2 :: x : 5
Cross multiply:
- 120 × 5 = 2 × x
- 600 = 2x
- x = 600 ÷ 2
- x = 300 km
Answer: The car will travel 300 km in 5 hours.
Example 12: Example 12: Finding the middle term in proportion
Problem: If 4:x :: x:25, find x.
Solution:
This is a continued proportion (a:b :: b:c).
Product of extremes = Product of means:
- 4 × 25 = x × x
- 100 = x²
- x = √100
- x = 10
Verify: 4:10 = 2:5. 10:25 = 2:5. Both are equal ✓
Answer: x = 10. The number 10 is called the mean proportional of 4 and 25.
Real-World Applications
Where is proportion used in daily life?
- Cooking: If a recipe for 4 people needs 2 cups of rice, proportion helps you find how much rice is needed for 10 people. Every time you double or halve a recipe, you are using proportion.
- Maps: Map scales use proportion. If 1 cm on a map = 100 km, you can use proportion to find real distances between places. All road maps and Google Maps use this idea.
- Shopping: If 3 kg of apples cost Rs 240, proportion tells you that 5 kg will cost Rs 400. Shopkeepers use proportion every day.
- Sharing: When dividing money, sweets, or things among people in a given ratio, proportion is used. For example, sharing pocket money between siblings in the ratio 3:2.
- Construction: Architects use scale drawings. A 1:100 scale means 1 cm on paper = 100 cm (1 m) in real life. Every blueprint uses proportion.
- Medicine: Doctors calculate medicine dosages based on body weight using proportion. A child weighing 20 kg gets a different dose than an adult weighing 60 kg.
- Enlarging photos: When you resize a photo, proportion keeps the shape the same — if width doubles, height must also double. Otherwise the photo looks stretched.
- Speed and distance: If a car travels 60 km in 1 hour, how far does it go in 3.5 hours? Proportion gives 210 km.
- Currency conversion: If 1 US dollar = Rs 83, then 50 dollars = Rs 4150. Proportion is used in all currency exchanges.
Key Points to Remember
- Two ratios are in proportion if they are equal: a:b :: c:d means a/b = c/d.
- The symbol :: means "is in proportion to."
- Extremes = first and last terms. Means = middle terms.
- Product of extremes = Product of means: a × d = b × c.
- To check proportion: cross multiply and see if the products are equal.
- To find a missing term: cross multiply and solve.
- Proportion is used in scaling, sharing, cooking, maps, and many real-life situations.
- If two ratios are NOT equal, they are NOT in proportion.
- Proportion keeps the relationship between quantities the same as they increase or decrease.
- Always simplify ratios to check if they are equal.
Practice Problems
- Check if 4:6 and 10:15 are in proportion.
- Check if 5:9 and 15:28 are in proportion.
- If 2:7 :: x:21, find x.
- If 8:x :: 24:15, find x.
- If 6 pens cost Rs 90, find the cost of 10 pens using proportion.
- On a map, 3 cm represents 120 km. Find the real distance for 5 cm on the map.
- Divide Rs 600 between A and B in the ratio 3:7.
- A recipe for 6 cookies uses 2 eggs. How many eggs are needed for 15 cookies?
Frequently Asked Questions
Q1. What is proportion in maths?
Proportion means two ratios are equal. If a:b = c:d, then a, b, c, d are in proportion. We write it as a:b :: c:d and read it as 'a is to b as c is to d.'
Q2. How do you check if two ratios are in proportion?
Cross multiply: check if a x d = b x c. Or simplify both ratios and check if they are the same. If 4:6 and 6:9 both simplify to 2:3, they are in proportion.
Q3. What are extremes and means?
In the proportion a:b :: c:d, the extremes are a and d (the outer terms), and the means are b and c (the inner terms). The rule is: product of extremes = product of means.
Q4. What is the difference between ratio and proportion?
A ratio compares two quantities (like 3:5). A proportion says two ratios are equal (like 3:5 :: 6:10). Ratio is a comparison; proportion is an equality of two comparisons.
Q5. How do you find a missing number in a proportion?
Use cross multiplication. If 3:4 :: 9:x, then 3 x x = 4 x 9, so 3x = 36, and x = 12.
Q6. Can the order of terms in a proportion be changed?
The order matters. 2:3 :: 4:6 is correct (both equal 2:3). But 2:4 :: 3:6 is also correct (both equal 1:2). However, 2:3 :: 6:4 is NOT correct (2/3 is not equal to 6/4).
Q7. What does the symbol :: mean?
The symbol :: is read as 'is as' or 'is in proportion to.' In 5:8 :: 15:24, it means '5 is to 8 as 15 is to 24,' meaning the two ratios are equal.
Q8. Is proportion used in real life?
Yes, everywhere! Cooking (adjusting recipes), maps (converting map distances to real distances), shopping (comparing prices), sharing money fairly, medicine doses, and resizing photos all use proportion.
