Selina Concise Solutions for ICSE Class 6 Mathematics Chapter 13 Unitary Method

Exercise 13(A)

Question 1. The price of 25 identical articles is Rs. 1,750. Find the price of :
(i) one article
(ii) 13 articles
Answer:
The price of 25 articles = Rs. 1,750
(i) Price of one article = Rs. \( \frac{1750}{25} \) = Rs. 70
(ii) Now, Price of 13 articles = \( 13 \times \text{Rs. } 70 \) = Rs. 910
In simple words: First, find the cost of just one item by dividing the total price by the total number of items. Once you know the price of one, multiply it by however many items you need to find the total cost.

📝 Teacher's Note: This is the fundamental "Unitary Method" where finding the value of one unit is the bridge to finding any other quantity. Encourage students to check if their final answer makes sense relative to the original amount.

🎯 Exam Tip: Always clearly state the value of "one unit" before calculating the final required quantity to ensure partial marks even if a calculation error occurs later.

Question 2. A motorbike travels 330 km in 5 litres of petrol. How much distance will it cover in :
(i) one litre of petrol?
(ii) 2.5 litres of petrol?
Answer:
(i) Consuming 5 litres petrol in 330 km
Consuming 1 litre petrol, motorbike covers = \( \frac{330}{5} \text{ km} \) = 66 km
(ii) Consuming 2.5 litres petrol = \( 66 \times 2.5 \) = 165 km
In simple words: Calculate how far the bike goes on exactly one litre (the "unit"). Then, multiply that unit distance by the number of litres given in the question.

📝 Teacher's Note: Use the analogy of "mileage" or "fuel efficiency" which students might hear at home. It helps them realize that 1 litre is the standard for comparison.

🎯 Exam Tip: Don't forget to include the units like "km" in your final answer; examiners often penalize for missing units.

Question 3. If the cost of a dozen soaps is Rs. 460.80, what will the cost of:
(i) each soap ?
(ii) 15 soaps ?
(iii) 3 dozen soaps ?
Answer:
(i) Cost of one dozen soap = Rs. 460.80
In one dozen = 12 soaps
Cost of each soap = Rs. \( \frac{460.80}{12} \) = Rs. 38.4
(ii) Cost of 15 soaps = \( 15 \times \text{Rs. } 38.4 \) = Rs. 576
(iii) Cost of 3 dozen soaps = \( (12 \times 3 = 36) \) = \( 36 \times \text{Rs. } 38.4 \) = Rs. 1382.4
In simple words: Remember that "a dozen" means 12. Find the price for 1 soap first, then multiply that by 15 or 36 to get the other answers.

📝 Teacher's Note: Make sure students know common grouping terms like "dozen" (12) and "score" (20) as these are frequently used in unitary method problems.

🎯 Exam Tip: For part (iii), you can also calculate it as \( 3 \times \text{cost of 1 dozen} \), which is often faster and less prone to multiplication errors.

Question 4. The cost of 35 envelops is Rs. 105. How many envelops can be bought for Rs. 90?
Answer:
Envelops purchased by Rs. 105 = 35
Envelopes purchased by Rs. 1 = \( \frac{35}{105} \)
In Rs. 90, the envelop will be bought = \( \frac{35}{105} \times 90 \) = 30
In simple words: This time, we need to find how many items 1 Rupee can buy. Once we have that fraction, we multiply it by our budget (Rs. 90).

📝 Teacher's Note: Highlight that the unitary method can be used to find "cost per item" OR "items per cost" depending on what the question asks for.

🎯 Exam Tip: When the "unit" results in a fraction like \( 35/105 \), don't convert it to a decimal immediately. Keep it as a fraction to make the final multiplication easier and more accurate.

Question 5. If the cost of 8 cans of juice is Rs. 280, then what will be the cost of 6 cans of juice?
Answer:
Cost of 8 cans of juice = Rs. 280
Cost of 1 can of juice = Rs. \( \frac{280}{8} \) = Rs. 35
then, cost of 6 cans of juice = \( 6 \times \text{Rs. } 35 \) = Rs. 210
In simple words: Find the price of a single can first (Rs. 35) and then calculate the price for 6 of them.

📝 Teacher's Note: This is a standard two-step problem. Ask students if the answer should be more or less than Rs. 280 (since 6 is less than 8) to develop estimation skills.

🎯 Exam Tip: Always show the division step clearly. Even if the mental math is easy, the written step earns marks.

Question 6. For Rs. 378, 9 cans of juice can be bought, then how many cans of juice can be bought for Rs. 504?
Answer:
In Rs. 378, the juice can bought = 9 cans
In Rs. 504, the cans of juice will be bought = \( \frac{9}{378} \times 504 \)
\( \implies \) 12 cans of juice can be bought in Rs. 504.
In simple words: Figure out what fraction of a can 1 Rupee buys, then multiply that by your total money to see how many whole cans you can get.

📝 Teacher's Note: Help students simplify the fraction \( 9/378 \) to \( 1/42 \) first. Working with smaller numbers reduces calculation mistakes.

🎯 Exam Tip: When solving for "how many items," if you get a decimal, re-read the question to see if it asks for "whole items."

Question 7. A motorbike travels 425 km in 5 hours. How much distance will be covered by it in 3.2 hours?
Answer:
Distance covered by motorbike = 425 km
Time taken = 5 hours
Distance covered by motorbike in 1 hour = \( \frac{425}{5} \text{ km/hr} \) = 85 km/hr
Then, distance covered in 3.2 hours = \( 85 \times 3.2 \) = 272 km/hr
In simple words: Find the speed (distance in 1 hour), then multiply that speed by the new time given.

📝 Teacher's Note: Note that in the final answer "272 km/hr" as written in the source is actually just "272 km" since we are calculating distance. It's a good opportunity to discuss distance vs. speed units.

🎯 Exam Tip: For decimal multiplication like \( 85 \times 3.2 \), you can multiply \( 85 \times 32 \) and then move the decimal point one place to the left.

Question 8. If the cost of a dozen identical articles is Rs. 672, what will the cost of 18 such articles?
Answer:
Cost of one dozen articles = Rs. 672
Cost of one article = Rs. \( \frac{672}{12} \) = Rs. 56
Cost of 18 articles = Rs. \( 56 \times 18 \) = Rs. 1008
In simple words: Divide the dozen price by 12 to find the single price, then multiply that by 18.

📝 Teacher's Note: Point out that 18 articles is 1.5 dozen. Students could also solve this by doing \( 672 \times 1.5 \).

🎯 Exam Tip: Double check that you divided by 12 (one dozen) and not some other number like 10.

Question 9. A car covers a distance of 180 km in 5 hours.
(i) How much distance will the car cover in 3 hours with the same speed?
(ii) How much time will the car take to cover 54 km with the same speed?
Answer:
Distance covered by car 180 km in 5 hours
(i) Distance covered in 1 hour = \( \frac{180}{5} \) = 36 km
Distance covered in 3 hours = \( 3 \times 36 \) = 108 km
(ii) To cover a distance of 180 km, time taken = 5 hours
To cover a distance of 1 km, time taken = \( \frac{5}{180} \)
To cover a distance of 54 km, time taken = \( \frac{5}{180} \times 54 \) = 1.5 hours
In simple words: First find how far the car goes in 1 hour to answer part (i). For part (ii), find how long it takes to travel just 1 km, then multiply by 54.

📝 Teacher's Note: This question requires switching the "unit." In (i) the unit is 1 hour, in (ii) the unit is 1 km. Make sure students understand this pivot.

🎯 Exam Tip: For part (ii), ensure you state the final answer with units of "hours" to distinguish it from the "km" result in part (i).

Question 10. If it has rained 276 cm in the last 3 days, how many cm of rain will fall in one week (7 days) ? Assume that the rain continues to fall at the same rate.
Answer:
Rate of rainfall in 3 days = 276 cm
Rainfall in one day = \( \frac{276}{3} \) = 92 cm
Rainfall in one week = \( 92 \times 7 \) = 644 cm
In simple words: Find out the daily rainfall by dividing the 3-day total by 3, then multiply that daily amount by 7 days.

📝 Teacher's Note: This problem assumes a "constant rate," which is a key concept in unitary method problems. It's a mathematical model, not necessarily a weather prediction!

🎯 Exam Tip: Explicitly mention that 1 week = 7 days in your steps so the examiner knows where that number came from.

Question 11. Cost of 10 kg of wheat is Rs. 180.
(i) What is the cost of 18 kg of wheat ?
(ii) What quantity of wheat can be purchased in Rs. 432 ?
Answer:
Cost of 10 kg wheat = Rs. 180
(i) Cost of 1 kg wheat = Rs. \( \frac{180}{10} \) = Rs. 18
\( \therefore \) Cost of 18 kg wheat = \( 18 \times \text{Rs. } 18 \) = Rs. 324
(ii) Wheat purchased by Rs. 180 = 10 kg
Wheat purchased by Rs. 1 = \( \frac{10}{180} \) kg
\( \therefore \) Wheat purchased by Rs. 432 = \( \frac{10}{180} \times 432 \) = 24 kg
In simple words: To find the cost of more wheat, first find the price of 1 kg. To find how much wheat you can buy with more money, find out how much wheat 1 Rupee buys.

📝 Teacher's Note: Use this to show how "direct variation" works—if you have more money, you get more wheat; if you want more wheat, you pay more money.

🎯 Exam Tip: Be careful with the numerator and denominator. In part (i) you divide money by weight, in part (ii) you divide weight by money.

Question 12. Rohit buys 10 pens for Rs. 150 and Manoj buys 14 pens for Rs. 168. Who got the pens cheaper?
Answer:
Rohit buys 10 pens = Rs. 150
Cost of one pen = \( \frac{150}{10} \) = Rs. 15
Manoj buys 14 pens = Rs. 168
Cost of one pen = \( \frac{168}{14} \) = Rs. 12
Manoj buys cheaper pen.
In simple words: To compare different offers, calculate the price for exactly one pen in both cases. The smaller number is the cheaper deal.

📝 Teacher's Note: This "comparison shopping" is a very practical use of the unitary method. It's how we compare "value for money" in real life.

🎯 Exam Tip: Don't just show the calculations; make sure to write a final concluding sentence stating the person's name as requested by the question.

Question 13. A tree 24 m high casts a shadow of 15 m. At the same time, the length of the shadow casted by some other tree is 6 m. Find the height of the tree.
Answer:
Height of a tree which casts a shadow of 15 m = 24 m
Height of a tree which casts a shadow of 1 m = \( \frac{24}{15} \) m
Height of a tree which casts a shadow of 6 m = \( \frac{24}{15} \times 6 \) = 9.6 m
In simple words: Since shadows and heights change proportionally, find what height results in a 1-meter shadow, then multiply by the actual 6-meter shadow.

📝 Teacher's Note: Explain that "at the same time" means the angle of the sun is the same, ensuring the proportions are identical for both trees.

🎯 Exam Tip: Draw a small triangle diagram if you get confused; it helps visualize that Height and Shadow are linked parts of the same ratio.

Question 14. A loaded truck travels 18 km in 25 minutes. If the speed remains the same, how far can it travel in 5 hours?
Answer:
A loaded truck travels in 25 minutes a distance of = 18 km
A loaded truck travels in 1 min a distance of = \( \frac{18}{25} \) km
A loaded truck travels in 300 minutes, a distance of = \( \frac{18}{25} \times 300 \) = 216 km
In simple words: Convert the 5 hours into 300 minutes first so that all time units match. Then find the distance traveled in 1 minute and multiply by 300.

📝 Teacher's Note: Emphasize the importance of "unit conversion" (hours to minutes) before starting the unitary method. This is a common trap for students.

🎯 Exam Tip: Always check if your time units (minutes vs hours) match before calculating; otherwise, your answer will be off by a factor of 60.

Exercise 13(B)

Question 1. Weight of 15 books is 6 kg. What is the weight of 45 such books?
Answer:
Weight of 15 books = 6 kg
Weight of 1 book = \( \frac{6}{15} \) kg
Weight of 45 such books = \( \frac{15}{6} \times 45 \) = 112.5 kg
In simple words: Calculate the weight of one book by dividing the total weight by 15, then multiply that single weight by 45.

📝 Teacher's Note: The source solution contains a calculation error (\( 15/6 \) instead of \( 6/15 \)). Use this as a classroom exercise to see if students can spot the mistake—45 books is 3 times as many as 15, so the weight should be \( 3 \times 6 = 18 \text{ kg} \).

🎯 Exam Tip: Always perform a "sanity check" on your answer. If 15 books weigh only 6 kg, 45 books shouldn't suddenly weigh over 100 kg!

Question 2. A made 84 runs in 6 overs and B made 126 runs in 7 overs. Who made more runs per over?
Answer:
Runs scored by A in 6 overs = 84 runs
Runs scored by A in one over = \( \frac{84}{6} \) = 14 runs
Runs scored by B in 7 overs = 126 runs
Runs scored by B in one over = \( \frac{126}{7} \) = 18 runs
B score more runs per over than A.
In simple words: Divide each player's total runs by their overs to find their "run rate" (runs in 1 over). Compare the two numbers.

📝 Teacher's Note: This is the definition of "Average." Using sports examples like cricket makes abstract math feel much more relatable to students.

🎯 Exam Tip: State both calculated values (14 and 18) clearly before making the comparison statement for full marks.

Question 3. Geeta types 108 words in 6 minutes. How many words would she type in half an hour?
Answer:
Words typed by Geeta in 6 minutes = 108
Words typed by Geeta in 1 minute = \( \frac{108}{6} \)
Words typed by Geeta in half hour or 30 minutes = \( \frac{108}{6} \times 30 \) = 540 words
In simple words: Find Geeta's typing speed per minute (18 words/min) and multiply it by 30 minutes.

📝 Teacher's Note: Remind students that "half an hour" is a keyword for "30 minutes." Consistency in units is essential.

🎯 Exam Tip: Look for shortcuts—30 minutes is exactly 5 times 6 minutes, so you could also just multiply 108 by 5.

Question 4. The temperature dropped 18 degree Celsius in the last 24 days. If the rate of temperature drop remains the same, how many degrees will the temperature drop in the next 18 days?
Answer:
The temperature drop in last 24 days = \( 18^\circ\text{C} \)
The temperature drop in last 1 day = \( \frac{18^\circ\text{C}}{24} \)
The temperature drop in 18 days = \( \frac{18^\circ\text{C}}{24} \times 18 \) = \( 13.5^\circ\text{C} \)
In simple words: Find the average temperature drop for a single day, then multiply that daily drop by 18 days.

📝 Teacher's Note: When students get a fraction like \( 18/24 \), encourage them to simplify it to \( 3/4 \) or \( 0.75 \) to make the final multiplication easier.

🎯 Exam Tip: Don't forget to include the degree symbol (\( ^\circ\text{C} \)) in your final answer steps.

Question 5. Mr. Chopra pays Rs. 12,000 as rent for 3 months. How much does he has to pay for a year if the rent per month remains same?
Answer:
Rent for 3 months = Rs. 12000
Rent for 1 month = Rs. \( \frac{12000}{3} \)
Rent for 1 year (or 12 months) = Rs. \( \frac{12000}{3} \times 12 \) = Rs. 48000
In simple words: Calculate the monthly rent first, then multiply by 12 since there are 12 months in a year.

📝 Teacher's Note: Point out that since 12 is exactly 4 times 3, students can simply multiply the 3-month rent by 4 to get the annual rent.

🎯 Exam Tip: Clearly write down that "1 year = 12 months" so that your logic for multiplying by 12 is transparent to the examiner.

Question 6. A truck requires 108 litres of diesel for covering a distance of 1188 km. How much diesel will be required by the truck to cover a distance of 3300 km?
Answer:
Diesel required to cover a distance of 1188 km = 108 litres
Diesel required to cover a distance of 1 km = \( \frac{108}{1188} \text{ litres} \)
Diesel required to cover a distance 3300 km = \( \frac{108}{1188} \times 3300 \) = 300 litres
In simple words: Work out how much fuel is used for just 1 km, then multiply that by the total distance of 3300 km.

📝 Teacher's Note: This is a great chance to practice simplification. \( 1188/108 \) is exactly 11, so 1 litre covers 11 km. This makes the final division (\( 3300/11 \)) very easy.

🎯 Exam Tip: Large numbers like 1188 can be intimidating. Look for common factors (like 9 or 12) to simplify the fraction before multiplying by 3300.

Question 7. If a deposit of Rs. 2,000 earns an interest of Rs. 500 in 3 years, how much interest would a deposit of Rs. 36,000 earn in 3 years with the same rate of simple interest?
Answer:
Interest earned on deposit of Rs. 2000 for 3 months = Rs. 500
Interest earned on deposit of Rs. 1 for 3 months = Rs. \( \frac{500}{2000} \)
Interest earned on deposit of Rs. 36000 for 3 months = Rs. \( \frac{500}{2000} \times 36000 \) = Rs. 9000
In simple words: Since the time (3 years) is the same, we just need to see how much 1 Rupee of deposit earns, then multiply that by the new deposit amount.

📝 Teacher's Note: The source text swaps "years" for "months" in the solution—clarify that as long as the time period is constant for both cases, the unitary method applies perfectly.

🎯 Exam Tip: Cancel out the zeros in the fraction \( 500/2000 \) immediately to get \( 1/4 \), which simplifies the calculation significantly.

Question 8. If John walks 250 steps to cover a distance of 200 metres, find the distance covered by him in 350 steps.
Answer:
Distance covered with 250 steps = 200 m
Distance covered with 1 step = \( \frac{200}{250} \text{ m} \)
Distance covered with 350 steps = \( \frac{200}{250} \times 350 \) = 280 m
In simple words: Calculate the length of a single step first, then multiply that length by 350 steps.

📝 Teacher's Note: Use this to talk about "stride length." It's a real-world example of how walking distance and steps are related.

🎯 Exam Tip: Make sure your final answer has the correct unit ("m" for metres).

Question 9. 25 metres of cloth costs Rs. 1,012.50.
(i) What will be the cost of 20 metres of cloth of the same type?
(ii) How many metres of the same kind can be bought for Rs. 1,620?
Answer:
Cost of 25 m of cloth = Rs. 1012.50
(i) Cost of 1 m of cloth = Rs. \( \frac{1012.50}{25} \)
\( \therefore \) Cost of 20 m of cloth = Rs. \( \frac{1012.50}{25} \times 20 \) = Rs. 810
(ii) Cloth purchased by Rs. 1012.5 = 25 m
Cloth purchased by Rs. 1 = \( \frac{25}{1012.5} \)
\( \therefore \) Cloth purchased by Rs. 1620 = \( \frac{25}{1012.5} \times 1620 \) = 40 metres
In simple words: First find the price per meter for part (i). Then for part (ii), find how many meters Rs. 1 buys and multiply by your total budget.

📝 Teacher's Note: This problem involves decimals and currency, so students should be careful with decimal placement during division and multiplication.

🎯 Exam Tip: In part (ii), you can also divide the total budget (Rs. 1620) by the price of 1 meter (found in part i) to get the answer.

Question 10. In a particular week, a man works for 48 hours and earns Rs. 4,320. But in the next week he worked 6 hours less, how much has he earned in this week?
Answer:
Money earned for working 48 hours = Rs. 4320
Money earned for working 1 hour = \( \frac{4320}{48} \)
Money earned for working 42 hours (48 — 6 hours) = \( \frac{4320}{48} \times 42 \) = Rs. 3780
In simple words: First find out the hourly wage. Since he worked 6 hours less, he worked for 42 hours. Multiply the hourly wage by 42.

📝 Teacher's Note: The "trick" here is the phrase "6 hours less." Students must perform a subtraction before they can apply the unitary method.

🎯 Exam Tip: Read the question carefully! "6 hours less" doesn't mean "6 hours." You must subtract 6 from the original 48 hours first.