Questions: Proportional Reasoning-1

Proportional Reasoning-134 min readLesson 5 of 5

in hinglish

Is page ke liye Hinglish recap pehle. Ratio a : b ka matlab har a units ke liye b units. Do ratios proportion mein hain jab a : b :: c : d ho, yaani ad = bc . Simplest form ke liye HCF se divide karo, jaise 60 : 40 = 3 : 2. Missing term d = bc \div a. Sharing mein total parts m + n banao, phir x/(m + n) ko m aur n se multiply karo. Unit conversions yaad rakho: 1 acre = 43,560 sq ft, 1 litre = 1,000 mL. Speed aur time inverse hote hain, isliye wahan Rule of Three seedha nahi lagta. Cross multiplication ka use: 7 : 12 :: x : 48 se x = 28. Example 9 mein pehle 4 hours ko 240 minutes banao, phir 150 : 90 :: 240 : 144 milta hai. Sharing mein ₹1,200 ko 5 : 7 mein baantne par 12 parts hote hain, har part ₹100, isliye shares ₹500 aur ₹700 hain. Units same karna aur labels likhna sabse pehla step hai.

Revision and important questions This section gathers the formula box, the usual mistakes and ten exam-style questions before the ten bilingual questions below.

The formula box Ratio: a : b means a units of the first quantity for every b units of the second. Proportion: a : b :: c : d exactly when ad = bc. Missing fourth term: d = bc \div a. Sharing x in the ratio m : n: split x into m + n unit parts, then give m parts to the first share and n parts to the second. Rule of Three: yield = (fruit \times requisition) \div measure. Scale factor: if one term changes by a factor, the other term must change by the same factor. Unit conversions: 1 acre = 43,560 sq ft, 1 litre = 1,000 mL, 1 metre = 3.281 feet, Fahrenheit = (9/5) \times Celsius + 32.

a : b :: c : d ⟺ a d = b c

The test for a proportion

\frac{m x}{m + n} and \frac{n x}{m + n}

Sharing x in the ratio m : n

Reading the formula box The cross-product test is the fastest check on any proportion: multiply across the two ratios and compare. When a term is missing, isolate it by dividing. The sharing formula is just the unit-part method written in symbols: first find the size of one part, then multiply by the number of parts each share gets. The conversion list is used only after the quantities are understood; a conversion changes the units, not the relationship.

How to start a proportional problem First name the two quantities being compared and make sure they are measured in the same unit. Write the known ratio, then write the second ratio with the unknown in the matching position. If the two quantities grow or shrink together, cross multiply. If one grows while the other falls, use the inverse relation instead of the Rule of Three. For sharing problems, write the total number of parts before touching the amount. For unit-conversion problems, convert first and then set up the proportion. Keeping the labels above the numbers makes the position of the unknown obvious, and it also shows whether the answer should be larger or smaller than the known quantity. Common mistakes Adding the same number to both terms and expecting the ratio to stay the same. Example 6 shows why it changes: 1 : 10 becomes 4 : 13 when 3 is added to both. Comparing numbers with different units. Convert hours to minutes or grams to kilograms before writing the ratio. Treating every proportion problem as a Rule of Three problem. When one quantity goes up, the other can go down, as with speed and time; that is inverse variation, not direct proportion. Comparing ratios without reducing them to simplest form or to a common denominator. Reading a ratio backwards. Milk : coffee is not coffee : milk, and width : height is not height : width. Rounding too early in a chain of calculations. Keep fractions exact and round only the final answer. Treating a ratio as a fraction of the whole. In the ratio 2 : 3 the two shares are 2/5 and 3/5 of the whole, not 2/3.

Ten exam-style questions with answers Write 45 : 60 in simplest form. Answer: divide both terms by 15 to get 3 : 4. Are 9 : 12 and 15 : 20 proportional? Answer: 9 : 12 = 3 : 4 and 15 : 20 = 3 : 4, so yes, 9 : 12 :: 15 : 20. Find x if 5 : 8 :: x : 24. Answer: 5 \times 24 = 8x, so x = 15. Divide ₹900 in the ratio 2 : 7. Answer: 9 parts of ₹100, so ₹200 and ₹700. If 3 pens cost ₹45, what do 5 pens cost? Answer: one pen costs ₹15, so five pens cost ₹75. In proportion form, 3 : 45 :: 5 : 75. A map uses 1 cm for 5 km. What distance does 7 cm represent? Answer: 1 : 5 :: 7 : 35, so 35 km. A car covers 120 km in 2 hours. How long for 300 km at the same speed? Answer: 120 : 2 :: 300 : x, so x = 5 hours. A 60 cm ribbon is cut in the ratio 3 : 5. Find the longer piece. Answer: 8 parts of 7.5 cm, so the pieces are 22.5 cm and 37.5 cm; the longer piece is 37.5 cm. A recipe for 4 people needs 500 g of rice. How much for 10 people? Answer: 4 : 500 :: 10 : x, so x = 1,250 g. A car uses 6 litres of petrol for 90 km. How much petrol is needed for 150 km? Answer: 6 : 90 :: x : 150 gives x = 10 litres.

5 : 8 :: x : 24 x = \frac{5 \times 24}{8} = 15

Exam question 3: cross multiplication

4 : 500 :: 10 : x x = \frac{500 \times 10}{4} = 1250

Exam question 9: scaling a recipe

Exam strategy For a proportion question, write the two ratios with labels above each term, check that the units match, and only then cross multiply. For a sharing question, write the number of parts first; most errors happen before the multiplication. For a comparison question, put both ratios in simplest form and compare the resulting fractions. When the problem involves speed, time, workers or pipes, check whether the relationship is inverse before applying the Rule of Three.

Q1 Write 24 : 36 in simplest form and explain the step. Answer (English): The HCF of 24 and 36 is 12. Dividing both terms by 12 gives 2 : 3. Simplest form means the two terms have no common factor other than 1, so no further division is possible. Hinglish: 24 aur 36 ka HCF 12 hai. Dono terms ko 12 se divide karo to 2 : 3 milta hai. Simplest form ka matlab dono terms mein 1 ke alawa koi common factor na ho.

Q2 Check whether 3 : 5 and 21 : 35 are proportional. Answer (English): Cross multiply: 3 \times 35 = 105 and 5 \times 21 = 105. The cross products are equal, so 3 : 5 :: 21 : 35. The direct check also works: 21 and 35 are both 7 times the terms of 3 : 5. Hinglish: Cross multiply karo: 3 \times 35 = 105 aur 5 \times 21 = 105. Cross products barabar hain, isliye 3 : 5 :: 21 : 35. Seedha check bhi hai: 21 aur 35 dono 3 : 5 ke terms ke 7 guna hain.

Q3 Find x if 7 : 12 :: x : 48. Answer (English): Cross multiplication gives 7 \times 48 = 12x, so 336 = 12x and x = 28. Check: 7 : 12 = 28 : 48 because both terms of the second ratio are 4 times the first. Hinglish: Cross multiplication se 7 \times 48 = 12x, yaani 336 = 12x aur x = 28. Check: 7 : 12 = 28 : 48 kyunki doosre ratio ke dono terms pehle ke 4 guna hain.

Q4 Divide ₹1,200 between two people in the ratio 5 : 7. Answer (English): The ratio has 5 + 7 = 12 parts. Each part is 1,200 \div 12 = ₹100. The shares are 5 \times 100 = ₹500 and 7 \times 100 = ₹700. Check: 500 : 700 = 5 : 7 and 500 + 700 = 1,200. Hinglish: Ratio mein 5 + 7 = 12 parts hain. Har part 1,200 \div 12 = ₹100. Shares 5 \times 100 = ₹500 aur 7 \times 100 = ₹700. Check: 500 : 700 = 5 : 7 aur total 1,200.

Q5 Why is the ratio of ages not preserved when both ages increase by the same number? Answer (English): Adding the same number to both terms changes their quotient. If Neelima and her mother are 1 : 10 at ages 3 and 30, after 9 years the ages are 12 and 39 and the ratio is 4 : 13, not 1 : 10. A ratio is preserved by multiplying both terms, not by adding to them. Hinglish: Dono terms mein same number jodne se quotient badal jata hai. 3 aur 30 par ratio 1 : 10 tha; 9 saal baad 12 aur 39 ka ratio 4 : 13 hai. Ratio tab bachta hai jab dono terms ko same number se multiply karo, jodne se nahi.

Q6 A 5 m ribbon is cut in the ratio 2 : 3. How long is each piece? Answer (English): The ratio has 5 parts, so each part is 5 m \div 5 = 1 m. The pieces are 2 \times 1 = 2 m and 3 \times 1 = 3 m. Check: 2 : 3 and 2 + 3 = 5 m, the whole ribbon. Hinglish: Ratio ke 5 parts hain, isliye har part 5 m \div 5 = 1 m. Pieces 2 m aur 3 m hain. Check: 2 : 3 aur 2 + 3 = 5 m.

Q7 If 8 books cost ₹960, what is the cost of 5 books at the same rate? Answer (English): One book costs 960 \div 8 = ₹120, so five books cost 5 \times 120 = ₹600. In proportion form, 8 : 960 :: 5 : x gives x = 960 \times 5 \div 8 = 600. Both methods agree. Hinglish: Ek book ka daam 960 \div 8 = ₹120, isliye 5 books ka daam ₹600. Proportion form mein 8 : 960 :: 5 : x se x = 960 \times 5 \div 8 = 600. Dono tarike same answer dete hain.

Q8 Why can a speed and time problem not be solved by the Rule of Three directly? Answer (English): When speed increases, the time taken decreases, so the quantities change in opposite directions. That is inverse variation, not direct proportion. For a fixed distance, speed \times time is constant: if the speed becomes 4 times larger, the time becomes one quarter. The Rule of Three applies to quantities that change in the same direction. Hinglish: Speed badhne par time kam hota hai, yaani quantities opposite direction mein badalti hain. Yeh inverse variation hai, direct proportion nahi. Fixed distance par speed \times time constant hota hai: speed 4 guna ho to time chauthai ho jata hai. Rule of Three wahan lagta hai jahan dono same direction mein badlein.

Q9 Compare the price of two packets: 500 g for ₹120 and 1 kg for ₹220. Which is cheaper per kilogram? Answer (English): Convert both to the same mass. The first packet is ₹120 for 500 g, which is ₹240 per kg. The second packet is ₹220 per kg. The second packet is cheaper per kilogram by ₹20. Price per unit mass makes the comparison meaningful, which is why unit conversions matter. Hinglish: Dono ko same mass par laao. Pehla packet ₹120 per 500 g, yaani ₹240 per kg. Doosra ₹220 per kg. Doosra packet ₹20 sasta hai per kg. Per unit mass comparison hi sahi hai, isliye unit conversion zaroori hai.

Q10 Write the unit conversions for 1 acre, 1 litre and 1 metre, and explain why conversions come before ratios. Answer (English): 1 acre = 43,560 square feet, 1 litre = 1,000 millilitres and 1 metre = 3.281 feet. A ratio compares quantities of the same kind, so the two quantities must be in the same unit before the ratio is written. If one side is in acres and the other in square feet, the numbers are not comparable and the proportion would be meaningless. Hinglish: 1 acre = 43,560 square feet, 1 litre = 1,000 mL aur 1 metre = 3.281 feet. Ratio same kind ki quantities compare karta hai, isliye dono same unit mein hone chahiye. Agar ek side acre mein hai aur doosri square feet mein, numbers compare nahi hote aur proportion ka matlab nahi bachta.

Quiz pointer The five-question quiz for this chapter is separate from these answers. Take it without looking back, then redo any question you missed by writing the proportion first and cross multiplying on paper. When an option looks close, check the units before choosing.