RS Aggarwal Solutions for Class 6 Chapter 9 Linear Equations in One Variable

Exercise 9.1

Question 1. (i) Ratio of number of girls to that of boys in the merit list is 2:1.
Answer: The ratio compares the quantity of girls with the quantity of boys in the merit list, which is 2:1. This means for every 2 girls, there is 1 boy listed.
In simple words: The merit list has twice as many girls as boys — 2 girls for each boy.

Exam Tip: Always read ratio notation carefully — the first number refers to the first quantity mentioned, and the second number to the second quantity.

Question 1. (ii) Ratio of number of students passing a mathematics test to that of total students appearing in test is 2:3.
Answer: The ratio shows how many students passed the math test compared to the total number of students who took it. A ratio of 2:3 indicates that for every 3 students who sat for the test, 2 passed.
In simple words: Out of every 3 students who took the math test, 2 students passed it.

Exam Tip: In a ratio comparing "passing" to "total," the second term is always the larger number (or equal) since passing students are part of the total.

Question 2. (i) The number of bad pencils produced in a factory is \( \frac{1}{9} \) of the number of good pencils produced in the factory.
Answer: If the number of bad pencils is one-ninth of the good pencils, then for every 9 good pencils made, 1 is defective. The ratio of bad pencils to good pencils is 1:9.
In simple words: For every 9 good pencils, there is 1 bad pencil made in the factory.

Exam Tip: When a quantity is described as a fraction of another, convert it to a ratio by using the fraction's numerator and denominator.

Question 2. (ii) The number of villages is 2000 times that of cities in India.
Answer: Since villages are 2000 times the number of cities, the ratio of villages to cities is 2000:1. This shows that for every city in India, there are 2000 villages.
In simple words: India has 2000 villages for every 1 city, making the ratio 2000:1.

Exam Tip: When one quantity is stated as a multiple of another, the multiplier becomes the first term in the ratio, with 1 as the second term.

Question 3. (i) Express 60:72 in simplest form.
Answer: To reduce 60:72 to its simplest form, we find the highest common factor of 60 and 72, which is 12. Dividing both terms by 12 gives us:
\[ \frac{60}{72} = \frac{60 \div 12}{72 \div 12} = \frac{5}{6} \]
Therefore, the simplest form of the ratio 60:72 is 5:6.
In simple words: Divide both numbers by their greatest common divisor to get the simplified ratio. Here, both divide by 12 to give 5:6.

Exam Tip: Always verify the HCF before dividing — using the wrong divisor will not reduce the ratio to its simplest form.

Question 3. (ii) Express 324:144 in simplest form.
Answer: To express 324:144 in its simplest form, we need the highest common factor of 324 and 144, which is 36. Dividing each term by 36:
\[ \frac{324}{144} = \frac{324 \div 36}{144 \div 36} = \frac{9}{4} \]
The simplest form of the ratio 324:144 is 9:4.
In simple words: Find the HCF of both numbers (which is 36), then divide each by it. The result is the simplest ratio 9:4.

Exam Tip: Double-check that the resulting fraction cannot be reduced further — if the numerator and denominator share no common factors, you have the simplest form.

Question 3. (iii) Express 85:391 in simplest form.
Answer: To simplify 85:391, we find the highest common factor of 85 and 391, which is 17. Dividing both terms by 17:
\[ \frac{85}{391} = \frac{85 \div 17}{391 \div 17} = \frac{5}{23} \]
The simplest form of the ratio 85:391 is 5:23.
In simple words: Both 85 and 391 are divisible by 17. When you divide both by 17, you get 5:23, which cannot be simplified further.

Exam Tip: For larger numbers, try small prime factors systematically — if neither term ends in 0 or 5, they are not divisible by 5; check divisibility by 2, 3, 7, 11, etc. methodically.

Question 4. (i) Express 75 paise to Rs 3 in simplest form.
Answer: First, convert both quantities to the same unit. Since 1 rupee equals 100 paise, Rs 3 becomes 300 paise. Now the ratio is 75 paise : 300 paise. The highest common factor of 75 and 300 is 75. Dividing both by 75 yields 1:4. [Note: 1 Rs = 100 paise]
In simple words: Change Rs 3 to 300 paise. Both 75 and 300 divide evenly by 75, giving 1:4.

Exam Tip: Always convert to the same units before comparing — paise to paise or rupees to rupees — never mix units in a ratio.

Question 4. (ii) Express 35 minutes to 45 minutes in simplest form.
Answer: The ratio is 35 minutes : 45 minutes. The highest common factor of 35 and 45 is 5. Dividing both terms by 5 gives:
\[ \frac{35}{45} = \frac{7}{9} \]
The simplest form is 7:9.
In simple words: Both 35 and 45 are divisible by 5. When divided, the result is 7:9.

Exam Tip: Since both quantities share the same unit (minutes), they cancel out — you can work directly with the numbers.

Question 4. (iii) Express 8 kg to 400 gm in simplest form.
Answer: Convert to a common unit. Since 1 kg equals 1000 gm, 8 kg becomes 8000 gm. The ratio is now 8000 gm : 400 gm. The highest common factor of 8000 and 400 is 400. Dividing both by 400:
\[ \frac{8000}{400} = \frac{20}{1} \]
The simplest form is 20:1.
In simple words: Convert 8 kg to 8000 gm. Divide both 8000 and 400 by 400 to get 20:1.

Exam Tip: When converting units (kg to gm, m to cm, etc.), multiply by the appropriate conversion factor — 1 kg = 1000 gm, 1 m = 100 cm.

Question 4. (iv) Express 48 minutes to 1 hour in simplest form.
Answer: Convert both to the same unit. Since 1 hour equals 60 minutes, the ratio becomes 48 minutes : 60 minutes. The highest common factor of 48 and 60 is 12. Dividing both by 12:
\[ \frac{48}{60} = \frac{4}{5} \]
The simplest form is 4:5.
In simple words: Change 1 hour to 60 minutes. Divide both 48 and 60 by their HCF (12) to get 4:5.

Exam Tip: Remember: 1 hour = 60 minutes. Always convert before simplifying when units differ.

Question 4. (v) Express 2 meters to 35 cm in simplest form.
Answer: First convert to a common unit. Since 1 m equals 100 cm, 2 m becomes 200 cm. The ratio is 200 cm : 35 cm. The highest common factor of 200 and 35 is 5. Dividing both by 5:
\[ \frac{200}{35} = \frac{40}{7} \]
The simplest form is 40:7.
In simple words: Convert 2 m to 200 cm. Both 200 and 35 divide by 5, yielding the ratio 40:7.

Exam Tip: 1 m = 100 cm — use this conversion before comparing lengths in different units.

Question 4. (vi) Express 35 minutes to 45 seconds in simplest form.
Answer: Convert to a common unit. Since 1 minute equals 60 seconds, 35 minutes becomes 2100 seconds. The ratio is 2100 seconds : 45 seconds. The highest common factor of 2100 and 45 is 15. Dividing both by 15 gives:
\[ \frac{2100}{45} = \frac{140}{3} \]
The simplest form is 140:3.
In simple words: Change 35 minutes to 2100 seconds. Divide both 2100 and 45 by 15 to get 140:3.

Exam Tip: 1 minute = 60 seconds — convert minutes to seconds before finding the ratio when they are mixed in the problem.

Question 4. (vii) Express 2 dozen to 3 scores in simplest form.
Answer: Convert to a common unit. Since 1 dozen equals 12 and 1 score equals 20, we have 2 dozen = 24 and 3 scores = 60. The ratio is 24:60. The highest common factor of 24 and 60 is 12. Dividing both by 12 yields:
\[ \frac{24}{60} = \frac{2}{5} \]
The simplest form is 2:5.
In simple words: 2 dozen = 24 and 3 scores = 60. Divide both by 12 to get 2:5.

Exam Tip: Know common quantity names: 1 dozen = 12, 1 score = 20. Convert these before finding the ratio.

Question 4. (viii) Express 3 weeks to 3 days in simplest form.
Answer: Convert to a common unit. Since 1 week equals 7 days, 3 weeks becomes 21 days. The ratio is 21 days : 3 days. The highest common factor of 21 and 3 is 3. Dividing both by 3:
\[ \frac{21}{3} = \frac{7}{1} \]
The simplest form is 7:1.
In simple words: 3 weeks equals 21 days. Divide both 21 and 3 by 3 to get 7:1.

Exam Tip: 1 week = 7 days — always convert weeks to days when comparing time periods.

Question 4. (ix) Express 48 minutes to 2 hours 40 minutes in simplest form.
Answer: Convert both to the same unit. 2 hours 40 minutes equals 160 minutes (since 2 hours = 120 minutes, plus 40 minutes). The ratio is 48 minutes : 160 minutes. The highest common factor of 48 and 160 is 16. Dividing both by 16 gives:
\[ \frac{48}{160} = \frac{3}{10} \]
The simplest form is 3:10.
In simple words: Convert 2 hours 40 minutes to 160 minutes total. Divide both 48 and 160 by 16 to get 3:10.

Exam Tip: When given compound time (hours and minutes), add them together first — convert hours to minutes and combine before finding the ratio.

Question 4. (x) Express 3 m 5 cm to 35 cm in simplest form.
Answer: Convert to a common unit. 3 m 5 cm equals 305 cm (since 3 m = 300 cm, plus 5 cm). The ratio is 305 cm : 35 cm. The highest common factor of 305 and 35 is 5. Dividing both by 5 yields:
\[ \frac{305}{35} = \frac{61}{7} \]
The simplest form is 61:7.
In simple words: Convert 3 m 5 cm to 305 cm. Divide both 305 and 35 by 5 to get 61:7.

Exam Tip: For compound measurements (meters and centimeters), convert the meter part to centimeters and add to get the total, then simplify.

Question 5. (i) The number of boys is 1168.
The number of girls is 1095.
Find the ratio of boys to girls in simplest form.
Answer: The ratio of boys to girls is 1168:1095. To reduce this to simplest form, we find the highest common factor of 1168 and 1095, which is 73. Dividing both terms by 73 yields:
\[ \frac{1168}{1095} = \frac{16}{15} \]
The simplest form of the ratio is 16:15.
In simple words: For every 15 girls in the group, there are 16 boys. The ratio 16:15 cannot be reduced further.

Exam Tip: When given specific numbers for objects, always divide both by their HCF to find the simplest ratio form.

Question 6. (i) Avinash salary is Rs 12,000 per month. Wife salary is Rs 15,000 per month. Find Avinash's income to wife's income.
Answer: Avinash's income to wife's income is 12,000:15,000. To simplify, we find the highest common factor of 12,000 and 15,000, which is 3,000. Dividing both by 3,000:
\[ \frac{12,000}{15,000} = \frac{4}{5} \]
The ratio is 4:5.
In simple words: For every Rs 5 the wife earns, Avinash earns Rs 4. The simplified ratio is 4:5.

Exam Tip: When comparing salaries, drop the trailing zeros and focus on finding the HCF of the simplified numbers first — this saves time.

Question 6. (ii) Total income is Rs 27,000 per month. Avinash's income to total income.
Answer: Avinash's income is Rs 12,000 and the total income is Rs 27,000. The ratio of Avinash's income to total income is 12,000:27,000. The highest common factor is 3,000. Dividing both by 3,000 gives:
\[ \frac{12,000}{27,000} = \frac{4}{9} \]
The ratio is 4:9.
In simple words: Out of every Rs 9 of combined income, Avinash earns Rs 4. The simplified ratio is 4:9.

Exam Tip: When comparing part to whole (a person's income to total income), ensure the "whole" is indeed the sum of all parts before calculating the ratio.

Question 7. Total no.of workers is 72. Women is 28. Men to women.
Answer: Since the total number of workers is 72 and there are 28 women, the number of men is 72 - 28 = 44. The ratio of men to women is 44:28. The highest common factor of 44 and 28 is 4. Dividing both by 4:
\[ \frac{44}{28} = \frac{11}{7} \]
The ratio of men to women is 11:7.
In simple words: Out of the group, for every 7 women, there are 11 men. The simplified ratio is 11:7.

Exam Tip: When given total and one part, always subtract to find the missing part before expressing the ratio — don't assume the second quantity is provided directly.

Question 8. Men to that total no. of persons.
Answer: From the previous problem, there are 44 men and 72 total workers. The ratio of men to total persons is 44:72. The highest common factor of 44 and 72 is 4. Dividing both by 4 yields:
\[ \frac{44}{72} = \frac{11}{18} \]
The ratio of men to total persons is 11:18.
In simple words: For every 18 total workers, 11 are men. The simplified ratio is 11:18.

Exam Tip: When finding "part to total," always ensure the total includes all groups — here, 72 is the sum of both men and women.

Question 9. Persons to women.
Answer: The ratio of total persons to women is 72:28. The highest common factor of 72 and 28 is 4. Dividing both by 4 gives:
\[ \frac{72}{28} = \frac{18}{7} \]
The ratio of persons to women is 18:7.
In simple words: For every 7 women, there are 18 total workers. The simplified ratio is 18:7.

Exam Tip: When comparing total to a part, the total number always goes in the numerator, and the part in the denominator.

Question 10. Length of steel tape is 10 m. Width is 2.4 cm. Find the ratio of its length and width in simplest form.
Answer: First, convert to a common unit. Length is 10 m = 1000 cm, and width is 2.4 cm. The ratio is 1000:2.4. To work with whole numbers, multiply both by 10 to get 10000:24. The highest common factor of 10000 and 24 is 8. Dividing both by 8 yields:
\[ \frac{10000}{24} = \frac{1250}{3} \]
The simplest form of the ratio is 1250:3. [HCF = 0.8 cm]
In simple words: Convert the meter length to 1000 cm. Then divide both 10000 and 24 by 8 to get 1250:3.

Exam Tip: When decimals are involved, multiply both numbers by 10 (or an appropriate power of 10) to convert to whole numbers before finding the HCF.

Question 11. Total period office is 9 a.m. to 5 p.m., which is 8 hours. Lunch interval is 30 minutes. Find the ratio.
Answer: The office operates for 8 hours total. Lunch interval is 30 minutes, which is 0.5 hours. Converting to a common unit: 8 hours = 480 minutes and lunch = 30 minutes. The ratio is 480:30. Dividing both by 30 gives:
\[ \frac{480}{30} = 16 \]
The simplest form of the ratio is 16:1. [This shows the office time is 16 times the lunch break.]
In simple words: The 8-hour workday contains 16 lunch breaks. So the ratio of working time to lunch time is 16:1.

Exam Tip: When mixing units (hours and minutes), convert everything to the smaller unit (minutes) for easier calculation before simplifying.

Exercise 9.2

Question 1. Write the ratios 3:4 and 9:16 as fractions.
Answer: Express 3:4 in fractional form as \( \frac{3}{4} \) and 9:16 as \( \frac{9}{16} \).
In simple words: A ratio written as a fraction has the first number on top and the second number below. So 3:4 becomes three-fourths, and 9:16 becomes nine-sixteenths.

Exam Tip: Always remember that a ratio a:b transforms directly into the fraction a/b — the first term is the numerator and the second term is the denominator.

Question 2. Determine if 15:16 and 24:25 are equivalent ratios.
Answer: Write both ratios as fractions by finding a common denominator. We have \( \frac{15}{16} \) and \( \frac{24}{25} \). The LCM of 16 and 25 is 400. Convert to equivalent fractions: \( \frac{15 \times 25}{16 \times 25} = \frac{375}{400} \) and \( \frac{24 \times 16}{25 \times 16} = \frac{384}{400} \). Since \( \frac{375}{400} \neq \frac{384}{400} \), the ratios are not equivalent. Simplifying, \( \frac{15}{16} \) and \( \frac{24}{25} \) are unequal, so \( 15 > 16 \) is false.
In simple words: Two ratios are the same (equivalent) only if they represent the same fraction when simplified. Here, 15:16 and 24:25 give different fractions, so they are not equivalent.

Exam Tip: To compare two ratios, convert them to a common denominator — if the numerators differ, the ratios are not equivalent.

Question 3. Determine if 4:7 or 5:8 is greater.
Answer: Express each as a fraction: \( \frac{4}{7} \) and \( \frac{5}{8} \). The LCM of 7 and 8 is 56. Convert both fractions: \( \frac{4 \times 8}{7 \times 8} = \frac{32}{56} \) and \( \frac{5 \times 7}{8 \times 7} = \frac{35}{56} \). Since \( \frac{32}{56} < \frac{35}{56} \), we have \( \frac{4}{7} < \frac{5}{8} \). Therefore, 5:8 is the larger ratio.
In simple words: When you convert both ratios to fractions with the same denominator, you can easily see which numerator is bigger. The fraction with the bigger numerator is the larger ratio.

Exam Tip: Always use a common denominator to compare ratios — never guess by looking at the numbers without converting.

Question 4. Write 1:2 as 13:27 and determine if the statement is true or false.
Answer: We have \( 1:2 = \frac{1}{2} \) and \( 13:27 = \frac{13}{27} \). The LCM of 2 and 27 is 54. Rewrite: \( \frac{1}{2} = \frac{1 \times 27}{2 \times 27} = \frac{27}{54} \) and \( \frac{13}{27} = \frac{13 \times 2}{27 \times 2} = \frac{26}{54} \). Since \( \frac{27}{54} \neq \frac{26}{54} \), we conclude that \( 1:2 \neq 13:27 \). The statement is false.
In simple words: When you write both ratios with a common denominator, you see that 1:2 gives 27/54, but 13:27 gives only 26/54. They are not equal.

Exam Tip: To verify if two ratios are equal, always find a common denominator and compare the resulting numerators.

Exercise 9.3

Question 1. Determine if the ratios 16:24 and 20:30 are in proportion.
Answer: Simplify \( \frac{16}{24} \) by dividing both numerator and denominator by their GCD of 8: \( \frac{16}{24} = \frac{2}{3} \). Simplify \( \frac{20}{30} \) by dividing by their GCD of 10: \( \frac{20}{30} = \frac{2}{3} \). Since both reduce to \( \frac{2}{3} \), the ratios are in proportion. Therefore, 16:24 = 20:30 is true.
In simple words: Two ratios are in proportion when they simplify to the same fraction. Both 16:24 and 20:30 simplify to 2:3, so they are in proportion.

Exam Tip: Always simplify each ratio by dividing by the greatest common divisor — if both simplify to the same fraction, they are in proportion.

Question 2. Determine if 21:6 = 35:10 is true or false.
Answer: Reduce \( \frac{21}{6} \) by dividing both by their GCD of 3: \( \frac{21}{6} = \frac{7}{2} \). Reduce \( \frac{35}{10} \) by dividing both by their GCD of 5: \( \frac{35}{10} = \frac{7}{2} \). Both ratios simplify to \( \frac{7}{2} \), so the statement 21:6 = 35:10 is true.
In simple words: When you divide the top and bottom of each fraction by their biggest common factor, both 21:6 and 35:10 give you 7:2. This proves they are equal.

Exam Tip: Use the GCD method to quickly reduce ratios — equal reduced forms confirm proportion.

Question 3. Determine if 12:18 = 28:12 is true or false.
Answer: Simplify \( \frac{12}{18} \) by dividing by their GCD of 6: \( \frac{12}{18} = \frac{2}{3} \). For \( \frac{28}{12} \), divide by their GCD of 4: \( \frac{28}{12} = \frac{7}{3} \). Since \( \frac{2}{3} \neq \frac{7}{3} \), the statement 12:18 = 28:12 is false.
In simple words: Simplifying 12:18 gives 2:3, while simplifying 28:12 gives 7:3. Since 2:3 is not the same as 7:3, these ratios are not equal.

Exam Tip: If the simplified forms differ, the ratios are not in proportion — the statement is false.

Question 4. (i) Determine if 8,16,6,12 are in proportion.
Answer: For four numbers to be in proportion, the first ratio must equal the second. Check if \( \frac{8}{16} = \frac{6}{12} \). Simplify \( \frac{8}{16} = \frac{1}{2} \) and \( \frac{6}{12} = \frac{1}{2} \). Since both equal \( \frac{1}{2} \), the numbers 8, 16, 6, 12 are in proportion.
In simple words: Four numbers are in proportion if you can split them into two pairs where each pair forms the same ratio. Here, 8:16 and 6:12 both give 1:2.

Exam Tip: For four numbers to be in proportion, reduce both ratios and check if they match.

Question 4. (ii) Determine if 6,2,4,3 are in proportion.
Answer: Check if \( \frac{6}{2} = \frac{4}{3} \). Simplify: \( \frac{6}{2} = 3 \) and \( \frac{4}{3} = \frac{4}{3} \). Since \( 3 \neq \frac{4}{3} \), the numbers are not in proportion.
In simple words: The first ratio 6:2 simplifies to 3:1, but the second ratio 4:3 stays as 4:3. Since 3:1 is not the same as 4:3, these four numbers are not in proportion.

Exam Tip: Always simplify both fractions fully before comparing — if they differ, the numbers are not in proportion.

Question 4. (iii) Determine if 150,250,200,100 are in proportion.
Answer: Check if \( \frac{150}{250} = \frac{200}{100} \). Simplify \( \frac{150}{250} = \frac{3}{5} \) (by dividing by 50) and \( \frac{200}{100} = 2 \) (by dividing by 100). Since \( \frac{3}{5} \neq 2 \), these four numbers are not in proportion.
In simple words: The ratio 150:250 reduces to 3:5, while 200:100 reduces to 2:1. Since 3:5 and 2:1 are different, the four numbers are not in proportion.

Exam Tip: When simplifying ratios, divide both terms by their GCD — different reduced ratios mean no proportion exists.

Exercise 9.4

Question 1. The cost of 2 meters of cloth is Rs 19.50. Find the cost of 15 meters of cloth.
Answer: Let x denote the cost of 15 meters of cloth. Set up the proportion: \( \frac{2}{15} = \frac{19.50}{x} \). By cross-multiplication: \( 2x = 15 \times 19.50 \). This gives \( 2x = 292.50 \). Dividing both sides by 2: \( x = 146.25 \). Therefore, 15 meters of cloth costs Rs 146.25.
In simple words: If 2 meters costs Rs 19.50, then find what 15 meters will cost by setting up a proportion and solving for the unknown price.

Exam Tip: Always use cross-multiplication for proportion problems — multiply the outer terms and the inner terms, then solve for the unknown.

Question 2. A total of 17 chairs costs Rs 9,605. Find the cost of one chair and then calculate how many chairs can be bought for Rs 6,600.
Answer: Find the cost of one chair: \( \frac{9605}{17} = 565 \) Rs per chair. Next, determine how many chairs can be obtained for Rs 6,600: \( \frac{6600}{565} = \frac{6600}{565} \approx 11.68 \). Since we cannot purchase a fraction of a chair, approximately 10 chairs can be bought with Rs 6,600.
In simple words: First, divide the total cost by the number of chairs to get the price per chair. Then, divide the available money by the price per chair to find how many can be bought.

Exam Tip: Always divide the total cost by quantity to get the unit price — this unit price is then used to solve "how many" questions.

Question 3. Three ferries can carry 150 people. How many people can be transported by 4 ferries?
Answer: If three ferries transport 150 people, then the capacity of one ferry is \( \frac{150}{3} = 50 \) people. With 4 ferries, the total capacity becomes \( 4 \times 50 = 200 \) people.
In simple words: First figure out how many people one ferry can carry by dividing the total by 3. Then multiply that by 4 to get the capacity of four ferries.

Exam Tip: Always find the unit rate first (capacity per ferry) — then multiply by the desired quantity (4 ferries).

Question 4. A train covers 200 km in 5 hours. Find the time required for the train to cover the same distance when it travels at three times its normal speed.
Answer: The original speed is \( \frac{200}{5} = 40 \) km/h. When speed is tripled, the new speed becomes \( 3 \times 40 = 120 \) km/h. Time required to cover 200 km at 120 km/h is \( \frac{200}{120} = \frac{5}{3} \) hours, which equals approximately 1.67 hours or 1 hour 40 minutes.
In simple words: Calculate the original speed by dividing distance by time. When speed triples, find the new time by dividing distance by the new speed.

Exam Tip: Remember the relationship: Speed = Distance ÷ Time, and Time = Distance ÷ Speed. Use this to solve inverse proportion problems.

Question 5. Nine kilograms of rice cost Rs 130.60. Find the cost of 50 kilograms of rice.
Answer: Determine the cost per kilogram: \( \frac{130.60}{9} = 14.51 \) Rs approximately. For 50 kilograms, the total cost is \( 50 \times 14.51 = 725.50 \) Rs. Alternatively, use the proportion \( \frac{9}{50} = \frac{130.60}{x} \), which gives \( x = \frac{50 \times 130.60}{9} = 725.56 \) Rs.
In simple words: Calculate the price per kilogram by dividing total cost by total weight. Then multiply the unit price by the desired weight.

Exam Tip: Use unit pricing for direct proportion problems — find cost per unit, then scale up to the desired quantity.

Question 6. A train runs 200 km in 5 hours. How many times will the train run in the given time interval if it travels at a different rate?
Answer: The train's standard speed is \( \frac{200}{5} = 40 \) km/h. The number of times (or cycles) the train covers 200 km in the same 5-hour window depends on the adjusted speed. If speed remains constant, the train completes exactly 1 full journey in 5 hours. If speed increases, more journeys can be completed within 5 hours; if speed decreases, fewer journeys result.
In simple words: At a constant speed, the train covers its distance once in the given time. Faster speeds allow more trips; slower speeds mean fewer trips in the same period.

Exam Tip: Always clarify whether the total time period remains fixed — this determines whether you use direct or inverse proportion.

Question 7. A total of 6 workers can finish a job in 12 hours. How many workers are needed to finish the same job in 1 hour?
Answer: The total amount of work is constant and equals \( 6 \times 12 = 72 \) worker-hours. To complete this same work in just 1 hour, the number of workers required is \( \frac{72}{1} = 72 \) workers.
In simple words: Multiply the original number of workers by the original time to get total work units. Then divide this by the new time to find how many workers you need.

Exam Tip: This is inverse proportion — as time decreases, the number of workers must increase proportionally to maintain the same total work output.