Questions: Fractions in Disguise

Q1. What does the symbol % mean, and what does 25% stand for?

Answer (English): The symbol % is read as per cent, which comes from the Latin per centum and means by the hundred. So a percentage is a fraction with denominator 100. 25% stands for 25 out of every 100, which is 25/100 = 1/4. It can describe 25 marks out of 100, ₹25 from a purse of ₹100, or 25 people out of 100. The fixed denominator is what makes percentages easy to compare.

Hinglish: % ko per cent padhte hain, jiska matlab hai sau mein se. Isliye 25% ka matlab 25/100 = 1/4 hai. Ye 100 marks mein 25 marks bhi ho sakta hai aur 100 logon mein 25 log bhi. Denominator 100 fix hone ki wajah se alag-alag cheezon ko aasani se compare kar sakte hain.

Q2. How do you convert a fraction into a percentage? Use 3/8 as an example.

Answer (English): Multiply the fraction by 100 and write the % sign. For 3/8, the calculation is 3/8 × 100 = 300/8 = 37.5. So 3/8 = 37.5%. A second method scales the fraction to denominator 100: 3/8 = 37.5/100 if we allow a decimal numerator, and 37.5 out of 100 is 37.5%. A third method sets up 3/8 = x/100 and solves x = 3 × 100/8 = 37.5. All three routes give the same answer.

Hinglish: Fraction ko 100 se multiply karo aur % lagao. 3/8 × 100 = 300/8 = 37.5, yaani 3/8 = 37.5%. Doosra tarika hai denominator ko 100 banana, aur teesra tarika hai 3/8 = x/100 likhkar x nikaalna. Teeno se wahi answer aata hai, bas calculation ka rasta alag hai.

Q3. How do you find a percentage of a quantity? Find 15% of 240.

Answer (English): Write the percentage as a fraction with denominator 100 and multiply by the quantity: y% of a = (y/100) × a. For 15% of 240, the working is (15/100) × 240 = 15 × 2.4 = 36. A mental route is to split 15% into 10% plus 5%: 10% of 240 is 24, 5% is 12, and 24 + 12 = 36. Both give the same answer.

Hinglish: Percentage ko y/100 likho aur quantity se multiply karo. 15% of 240 = 15/100 × 240 = 36. Mentally karne ke liye 15% ko 10% + 5% mein todo: 10% = 24 aur 5% = 12, dono ka jod 36. Isse calculation jaldi aur galti kam hoti hai.

Q4. Why are percentages used instead of fractions when comparing two quantities?

Answer (English): Percentages give every quantity the same denominator, 100, so the comparison becomes direct. In the chapter, sugar is 9/34 of one biscuit variety and 13/45 of another. Those fractions have different denominators and are hard to compare at a glance. Written as 26.47% and 28.88%, the second variety is clearly sweeter. The common scale removes the need for cross multiplication and also connects easily with decimals.

Hinglish: Percentage har quantity ko ek hi denominator, 100, par le aata hai. Chapter mein ek biscuit mein 9/34 sugar hai aur doosre mein 13/45. In fractions ko dekh kar compare karna mushkil hai, lekin 26.47% aur 28.88% likhne par turant pata chal jata hai ki doosra zyada meetha hai. Common scale ki wajah se comparison seedha ho jata hai.

Q5. What does a percentage greater than 100% mean? Give an example.

Answer (English): A percentage above 100% means the quantity is larger than the base whole; the percentage says how many times the whole it is. If a shop's target was ₹5000 and it sold goods worth ₹6000, the sale is 6000/5000 × 100 = 120% of the target, which is 20% more than the target. A farmer whose harvest rose from 260 kg to 650 kg now has two and a half times the earlier crop.

Hinglish: 100% se zyada ka matlab hai quantity base se badi hai, aur percentage batata hai ki base ka kitna guna hai. Agar target 5000 tha aur sale 6000 hui, to 6000/5000 × 100 = 120% hua, yaani target se 20% zyada. Isi tarah 260 kg se 650 kg ka matlab 250% hai, yaani dhai guna.

Q6. How are percentage increase and percentage decrease calculated?

Answer (English): Both use the original value as the base. Percentage increase is (increase/original) × 100, and percentage decrease is (decrease/original) × 100. If petrol goes from ₹60 to ₹100, the increase is ₹40 and the percentage increase is 40/60 × 100 = 66.67%. If the price falls from ₹100 to ₹60, the decrease is ₹40 and the percentage decrease is 40/100 × 100 = 40%. The two answers differ because the base is different in the two directions.

Hinglish: Dono mein base original value hoti hai. Increase ke liye (increase/original) × 100 aur decrease ke liye (decrease/original) × 100. Petrol 60 se 100 hua to 40/60 × 100 = 66.67% badha. Agar 100 se 60 hota to 40/100 × 100 = 40% ghatta. Base badalne se percentage badal jata hai, isliye base pehle likho.

Q7. What is the difference between simple interest and compound interest?

Answer (English): With simple interest the interest is paid out and the principal stays fixed, so the amount after t years is p(1 + rt). With compound interest the interest is added back to the principal, so the principal grows and later interest is earned on earlier interest. The amount after t years is p(1 + r)ᵗ. For ₹6000 at 10% for 3 years, simple interest gives ₹7800 while compounding gives ₹7986. The compound amount is larger because of interest on interest.

Hinglish: Simple interest mein interest har saal alag milta hai aur principal same rehta hai, isliye amount p(1 + rt). Compound interest mein interest principal mein jud jata hai, isliye agle saal us par bhi interest milta hai aur amount p(1 + r)ᵗ. 6000 rupees par 10% se 3 saal mein simple 7800 deta hai aur compound 7986. Compound zyada deta hai kyunki interest par bhi interest milta hai.

Q8. Why is 30% + 20% not the same as a 50% discount?

Answer (English): Percentages applied one after another are taken on different bases. A 30% discount on ₹200 gives ₹140, and the 20% applies to ₹140, giving ₹112. A single 50% discount gives ₹100. The combined multiplier is 0.7 × 0.8 = 0.56, which is a 44% discount, not 50%. The second discount is calculated on the reduced price, so its rupee value is smaller. This is why shops writing 30% plus 20% are not offering 50% off.

Hinglish: Ek ke baad ek discount alag-alag base par lagta hai. 200 rupees par pehle 30% discount se 140 bache, phir 20% usi 140 par laga to 112 bache. Seedha 50% discount hota to 100 milta. Dono multipliers 0.7 × 0.8 = 0.56 hain, yaani sirf 44% discount. Isliye 30% + 20% ka matlab 50% off nahi hota.

Q9. Why is profit percentage calculated on the cost price, while discount percentage is calculated on the marked price?

Answer (English): Profit measures what the shopkeeper gained on the money invested, so the cost price is the natural base. Discount measures a reduction from the price on the label, so the marked price is the base. The chapter's fertiliser example makes the difference clear. Manisha buys a bag for ₹500 and sells it for ₹750. The profit is ₹250, which is 50% of the cost but only 33.33% of the selling price. Both figures are correct for different questions, so the base must be named.

Hinglish: Profit batata hai ki lagayi hui paise par kitna fayda hua, isliye base CP hota hai. Discount label par likhi price se kitna kam hua, isliye base MP hota hai. Manisha 500 mein kharid kar 750 mein bechti hai: profit 250, jo CP par 50% hai aur selling price par 33.33%. Dono sahi hain, bas sawaal alag hai, isliye base likhna zaroori hai.

Q10. A number increased by 20% becomes 90. Find the number, and explain the common mistake.

Answer (English): Let the number be x. A 20% increase gives 1.2x, so 1.2x = 90 and x = 90/1.2 = 75. The common mistake is to take 20% of 90 and subtract it, giving 72. That is wrong because the 20% increase was calculated on the original number, not on the final 90. Check the correct answer: 20% of 75 is 15, and 75 + 15 = 90.

Hinglish: Number ko x maano to 20% badhane par 1.2x banta hai. 1.2x = 90 se x = 75. Aksar log 90 ka 20% nikaal kar ghatate hain, jisse 72 aata hai, lekin wo galat hai kyunki 20% original number par laga tha, final 90 par nahi. Check karo: 75 ka 20% = 15, aur 75 + 15 = 90.

Quick check

You have revised the chapter and worked through its questions. The next step is a short quiz of five multiple choice questions on percentages, profit and loss, and interest. Take it without looking back at the notes, then read the explanation for every question you miss. Aim for at least three correct answers before moving to the next chapter.