Questions: Proportional Reasoning-2

Q1. When are two ratios proportional?

Answer (English): The pair a : b and c : d is in proportion when the cross products agree, that is a × d = b × c, which is the same as a/c = b/d. For 6 : 3 and 4 : 2, the cross products are 6 × 2 = 12 and 3 × 4 = 12, so the ratios are proportional and the mixtures taste the same.

Hinglish: Do ratios a : b aur c : d proportional tab hote hain jab cross products barabar hon, yaani a × d = b × c. Iska doosra roop a/c = b/d hai. 6 : 3 aur 4 : 2 mein 6 × 2 = 12 aur 3 × 4 = 12, isliye dono proportional hain aur mixture ek jaisa hoga.

Q2. How do you divide a quantity in a given ratio?

Answer (English): Add the terms of the ratio, divide the quantity by that sum to find one part, and multiply one part by each term. To divide 240 in the ratio 2 : 3 : 5, the sum is 10, one part is 24, and the shares are 48, 72 and 120. In fraction form each share is the quantity times its term over the sum of the terms.

Hinglish: Ratio ke terms jodo, quantity ko sum se divide karke ek part nikaalo, phir har term se multiply karo. 240 ko 2 : 3 : 5 mein baantna ho to sum 10 hai, ek part 24, aur shares 48, 72 aur 120 banenge. Fraction wale tarike mein har share = quantity × term/sum hota hai.

Q3. What is a Representative Fraction on a map, and how is it used?

Answer (English): A Representative Fraction (RF) compares a measured map length with the real ground length. An RF of 1 : 60,00,000 means each centimetre on the map stands for 60,00,000 cm on the ground, which is 60 km. Measure the map distance with a ruler and multiply by the scale; remember that this is geographical distance, not road distance.

Hinglish: Representative Fraction map par naapi gayi lambai aur zameen ki asli lambai ka anupat hai. 1 : 60,00,000 ka matlab 1 cm map par 60,00,000 cm zameen par, yaani 60 km. Map par ruler se doori naapo aur scale se multiply karo. Yaad rakho ki ye seedhi geographical distance hai, road distance nahi.

Q4. How do you turn counts into a pie chart?

Answer (English): Find the total of all counts. Multiply each count by 360 degrees and divide by the total; that gives the angle of its slice. Draw a circle, mark a radius, and measure the angles one after another in the same direction. The angles must add to 360 degrees. If the data are percentages, multiply each percentage by 3.6 instead.

Hinglish: Pehle saare counts ka total nikaalo. Har count ko 360 se multiply karke total se divide karo, wahi us slice ka angle hoga. Circle banao, ek radius mark karo, aur angles ek hi direction mein naapate jao. Sabhi angles ka sum 360° aana chahiye. Agar data percentage mein hai to har percentage ko 3.6 se multiply karo.

Q5. What is inverse proportion? Give an example.

Answer (English): Two quantities are inversely proportional when one increases by a factor and the other decreases by the same factor. Their product stays constant. For example, 2 pumps fill a tank in 18 hours and 4 pumps take 9 hours: 2 × 18 = 4 × 9 = 36. Doubling the pumps halves the time.

Hinglish: Inverse proportion mein ek quantity kisi factor se badhti hai to doosri usi factor se ghatti hai, aur unka product constant rehta hai. Jaise 2 pumps tank ko 18 ghante mein bharte hain aur 4 pumps 9 ghante mein: 2 × 18 = 4 × 9 = 36. Pumps double karne se time aadha ho jata hai.

Q6. How do you solve a problem when x and y are inversely proportional?

Answer (English): Use xy = k. Compute k from a complete pair of values, then use it to find the missing value. For example, if x = 16 when y = 9, then k = 144. When x = 12, y = 144/12 = 12, and when y = 48, x = 144/48 = 3. Alternatively use x1 y1 = x2 y2 directly.

Hinglish: x × y = k use karo. Kisi poore pair se k nikaalo, phir missing value nikaalne ke liye use karo. Jaise x = 16 aur y = 9 ho to k = 144. x = 12 par y = 12, aur y = 48 par x = 3. Seedha x1 y1 = x2 y2 bhi likh sakte ho.

Q7. How do you decide whether a relationship is direct or inverse?

Answer (English): Ask what happens when one quantity is multiplied by a factor. If the other quantity is multiplied by the same factor, the relationship is direct and the quotient stays constant. If the other is divided by that factor, the relationship is inverse and the product stays constant. Fixed totals, such as a fixed distance or a fixed amount of work, usually create inverse proportions.

Hinglish: Poochho ki ek quantity ko factor se multiply karne par doosri ka kya hota hai. Agar doosri bhi usi factor se multiply ho to direct hai aur quotient constant rehta hai. Agar doosri usi factor se divide ho to inverse hai aur product constant rehta hai. Fixed total wale sawaal, jaise fixed doori ya fixed kaam, aksar inverse hote hain.

Q8. Why do we add rates and not times when people work together?

Answer (English): Each worker completes a fraction of the job in one hour. Ram does 1 unit in 1 hour, so his rate is 1; Shyam does 1 unit in 1.5 hours, so his rate is 2/3. Together their rate is 5/3 units per hour, and the time for one unit is 3/5 hour. Adding the times, 1 + 1.5 = 2.5, would overstate the time because they work at the same time rather than one after another.

Hinglish: Har worker ek ghante mein kaam ka kuch hissa karta hai. Ram 1 ghante mein pura kaam karta hai, isliye rate 1; Shyam 1.5 ghante mein, isliye rate 2/3. Dono milakar rate 5/3 per hour, aur ek unit ka time 3/5 ghanta. Times jodna galat hoga kyunki dono saath kaam karte hain, ek ke baad ek nahi.

Q9. How do you scale a ratio that has more than two terms?

Answer (English): Multiply or divide every term by the same non-zero number. The spice mix 8 : 4 : 2 : 1 becomes 4 : 2 : 1 : 0.5 when every term is halved, and the mixture tastes the same. The test for two multi-term ratios is that all corresponding quotients are equal: 8/4 = 4/2 = 2/1 = 1/0.5 = 2.

Hinglish: Har term ko same number se multiply ya divide karo. Spice mix 8 : 4 : 2 : 1 ko aadha karne par 4 : 2 : 1 : 0.5 milta hai aur taste wahi rehta hai. Do multi-term ratios check karne ke liye corresponding quotients barabar hone chahiye: 8/4 = 4/2 = 2/1 = 1/0.5 = 2.

Q10. Can a triangle have sides in the ratio 1 : 3 : 5?

Answer (English): No. The triangle inequality requires any two sides to add to more than the third. With sides k, 3k and 5k, the two shorter sides add to 4k, which is less than 5k. The ends of the two shorter sides cannot meet, so no triangle can be drawn with this ratio.

Hinglish: Nahi. Triangle inequality kehti hai ki kisi bhi do sides ka jod teesri side se zyada hona chahiye. Sides k, 3k aur 5k ho to chhoti sides ka jod 4k hai jo 5k se kam hai. Chhoti sides ke ends mil hi nahi payenge, isliye is ratio wala triangle nahi banta.

Quick check

You have revised ratios, pie charts, maps and inverse proportion. The quiz below has five multiple choice questions on these ideas. Answer without looking back at the notes, then read the explanation for each question. Three or more correct answers is a good signal before the next chapter.