Types of Proportional with examples

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What is Proportional ? প্ৰপৰশ্যনেল মানে কি ?

Two quantities are said to be proportional if they maintain a constant ratio. যেতিয়া দুটা পৰিমাণে একেই অনুপাত (ratio) ৰক্ষা কৰে, সেইবোৰক সমানুপাত (Proportional) বোলা হয়।

Example: 2 pencils cost ₹10, 4 pencils cost ₹20. Ratio: 2 : 4 = 10 : 20 → constant → Proportional

উদাহৰণ: 2 খণ্ড কলমৰ দাম = ₹10. 4 খণ্ড কলমৰ দাম = ₹20. অনুপাত: 2 : 4 = 10 : 20 → একেই অনুপাত → সমানুপাত

Key Idea: Direct Proportion: Both increase/decrease together. Inverse Proportion: One increases, other decreases

মূল ভাবনা: সোজা সমানুপাত (Direct): দুয়োটা পৰিমাণ একেলগে বৃদ্ধি বা হ্ৰাস কৰে। বিপৰীত সমানুপাত (Inverse): এয়ে বৃদ্ধি হ’লে আনটো হ্ৰাস হয়

First Proportional (প্ৰথমানুপাত): Formula: a²/ b.

Ex: Find 1st proportional of 16 and 32: 16²/32 = 8.

Trick: Square the first number, divide by second. ট্ৰিক: প্ৰথমটো সংখ্যা বৰ্গ কৰি দ্বিতীয়টোৰে ভাগ কৰিব

Second Proportional (দ্বিতীয়ানুপাত): Formula b (middle).

Ex: In 4 : 8 = 8 : ? → 2nd proportional = 8.

Trick: The middle number itself is the 2nd proportional. ট্ৰিক: মধ্যৰ সংখ্যা ল’ব।

Third Proportional (তৃতীয়ানুপাত): Formula ²/a.

Ex: Find 3rd proportional of 8 & 12: 12²/8 = 18.

Trick: Square the second number, divide by the first. ট্ৰিক: দ্বিতীয়টো বৰ্গ কৰি প্ৰথমটোৰে ভাগ কৰিব

Fourth Proportional (চতুৰ্থানুপাত): Formula: b×c / a.

Ex: Find 4th proportional of 16, 26, 32: 26×32/16 = 52.

Trick: Multiply 2nd & 3rd numbers, divide by the first. ট্ৰিক: দ্বিতীয় আৰু তৃতীয় সংখ্যাৰ গুণফল, প্ৰথমটোৰে ভাগ কৰিব।

Q1. Find 1st proportional of 18 and 54. Find 18 আৰু 54 ৰ 1st proportional

Formula: 1st proportional = a²/ b

Solⁿ

 a = 18, b = 54 1st proportional = 18²/54 = 32  1st proportional = 18^2 / 54 = 324 / 54 = 6

Trick / ট্ৰিক: Square the first number, divide by second.

Ans: 6 

Q2. Find 2nd proportional of 7 : 14 = 14 : ?

Solⁿ

Formula: Middle number = b

2nd proportional = Middle number = 14

Trick / ট্ৰিক: The middle number itself is the 2nd proportional.

Ans: 14

Q3. Find 3rd proportional of 5 & 10. Find5 আৰু 10 ৰ 3rd proportional

Formula: 3rd proportional = b²/a

Solⁿ

 a = 5, b = 10 a = 5, b = 10a=5,b=10 3rdproportional=102/5=100/5=203rd proportional = 10^2 / 5 = 100 / 5 = 203rdproportional=102/5=100/5=20

Trick / ট্ৰিক: Square the second number, divide by the first.

Ans: 20

Q4. Find 4th proportional of 6, 8, 12. Find 6, 8, 12 ৰ 4th proportional

Solⁿ

Formula: 4th proportional = b×c/a

 a = 6, b = 8, c = 12 , 4th proportional = 8 × 12 / 6 = 96 / 6 = 16

Trick / ট্ৰিক: Multiply 2nd & 3rd numbers, divide by first.

Ans: 16

Q5. Simple Proportional Question (Direct proportion) / সোজা সমানুপাত

Q: 3 kg sugar = ₹90, 5 kg sugar = ?

Solⁿ

  5 kg sugar = 90 × 5/3 = 450/3 = 150

Trick / ট্ৰিক: Multiply cost with new quantity, divide by old quantity.

Ans: ₹150

Q6. Simple Proportional Question (Inverse proportion) / বিপৰীত সমানুপাত

Q: 4 workers can finish a work in 12 days. How many days will 6 workers take ?

Solⁿ

Formula (Inverse): a × b = k

               4 × 12 = 48

Let days = x for 6 workers: 6 × x = 48 ⇒ x = 48/6 = 8 days

Trick / ট্ৰিক: More workers → less days.

Ans: 8 days

^{Q7. Find the 3rd Proportional of 7 & 14}

Solⁿ

^{Formula / Trick:} b²/a

3^rd =14²​/ 7 = 196/7 ​= 28​

Ans: 28

Trick: 14² ÷ 7 → 28

Q8. Find the 4th Proportional of 6, 9, 12

Solⁿ

Formula: d = b × c / a​

      d = 9×12 / 6 = 108 / 6 = 18​

Ans: 18

Trick: 9×12 ÷ 6 = 18

Q9. If 3 books cost ₹45, find the cost of 10 books

Solⁿ

Direct proportion

3:10 = 45: x 3 

x = 45 × 10 / 3 = 150​

Ans: ₹150

Trick: 45×10 ÷ 3

Q10. If 5 workers finish a job in 24 days, how many days for 8 workers ?

Solⁿ

Inverse proportion (More workers → Less days)

       5 × 24 = 8 × x

    x = 5 × 24/8 = 15 days ​

Ans: 15 days

Q11. 8 : 12 = 20 : ?

Solⁿ

4th proportional / direct ratio

   x = 12 × 20 / 8 = 30​

Ans: 30:

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