GSEB Class 6 Maths Solutions Chapter 1 Knowing Our Numbers InText Questions

Try These (Page 2)

Question 1. Can you instantly find the greatest and the smallest numbers in each row?
(i) 382, 4972, 18, 59785, 750
(ii) 1473, 89423, 100, 5000, 310
(iii) 1834, 75284, 111, 2333, 450
(iv) 2853, 7691, 9999, 12002, 124
Answer:
(i) 59785 is the greatest, and 18 is the smallest.
(ii) 89423 is the greatest, and 100 is the smallest.
(iii) 75284 is the greatest, and 111 is the smallest.
(iv) 12002 is the greatest, and 124 is the smallest.

Exam Tip: To quickly find the greatest and smallest numbers, first compare the number of digits. More digits generally mean a larger number.

Question. Was that easy? Why was it easy?
Answer: Yes, it was simple to find the greatest or the smallest number in a row. This is because a numeral containing more digits is always bigger. If the digit count is the same, we compare digits from left to right.

Exam Tip: Always remember that a number with more digits is generally larger. If the digit count is equal, compare from the leftmost digit.

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Question 1. Find the greatest and the smallest numbers:
(a) 4536, 4892, 4370, 4452
(b) 15623, 15073, 15189, 15800
(c) 25286, 25245, 25270, 25210
(d) 6895, 23787, 24569, 24659
Answer:
(a) Each of the given numbers has four digits, and their digits at the thousands place are the same. When comparing the next digit of each number, we observe that \( 8 > 5 > 4 > 3 \). Thus, the greatest number is 4892, and the smallest number is 4370.
(b) Each of the given numbers has five digits, and their two leftmost places have the same digits (i.e., 1 and 5). On comparing the third leftmost digits, we find that \( 8 > 6 > 1 > 0 \). Therefore, the greatest number is 15800, and the smallest number is 15073.
(c) Each of the given numbers has five digits, and their three leftmost places have the same digits (i.e., 2, 5, and 2). On comparing the fourth leftmost digits, we find that \( 8 > 7 > 4 > 1 \). So, the greatest number is 25286, and the smallest number is 25210.
(d) The number 6895 has four digits, so it must be the smallest number. Each of the numbers 23787, 24569, and 24659 has five digits, and their leftmost digit is the same. On comparing the second leftmost digit in 23787, 24569, 24659, we notice that \( 4 > 3 \). Again, the two leftmost places of 24569 and 24659 share the same digits (i.e., 2 and 4). On comparing their third leftmost digits, we observe that \( 6 > 5 \), which means \( 24659 > 24569 \). Thus, the greatest number is 24659, and the smallest number is 6895.

Exam Tip: When comparing numbers, always start by checking the number of digits. If they have the same number of digits, compare from the leftmost digit and move right until a difference is found.

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Question 1. Use the given digits without repetition and make the greatest and smallest 4-digit numbers.
(a) 2, 8, 7, 4
(b) 9, 7, 4, 1
(c) 4, 7, 5, 0
(d) 1, 7, 6, 2
(e) 5, 4, 0, 3
Hint: 0754 is a 3-digit number.
Answer:
(a) The ascending order of the given digits is 2, 4, 7, 8. The smallest 4-digit number is 2478. The descending order of the given digits is 8, 7, 4, 2. The greatest 4-digit number is 8742.
(b) The given digits are 9, 7, 4, and 1. The ascending order of the given digits is 1, 4, 7, 9. The smallest 4-digit number is 1479. The descending order of the given digits is 9, 7, 4, 1. The greatest 4-digit number is 9741.
(c) The given digits are 4, 7, 5, and 0. The ascending order of the given digits is 0, 4, 5, 7. Keeping 0 in the second leftmost place, the smallest number becomes 4057. (Note: A 0 in the leftmost place makes it a 3-digit number, e.g., 0457 is 457). The descending order of the given digits is 7, 5, 4, 0. The greatest 4-digit number is 7540.
(d) The given digits are 1, 7, 6, and 2. The ascending order of the given digits is 1, 2, 6, 7. The smallest 4-digit number is 1267. The descending order of the given digits is 7, 6, 2, 1. The largest 4-digit number is 7621.
(e) The given digits are 5, 4, 0, and 3. The ascending order of the given digits is 0, 3, 4, 5. Keeping 0 in the second leftmost place, the smallest 4-digit number is 3045. The descending order of the given digits is 5, 4, 3, 0. The greatest 4-digit number is 5430.

Exam Tip: To form the smallest number, arrange digits in ascending order. For the greatest, arrange in descending order. Remember, zero cannot be the leading digit for a number with a specific digit count.

Question 2. Now make the greatest and the smallest four-digit numbers by using any one digit twice.
(a) 3, 8, 7
(c) 0, 4, 9
(d) 8, 5, 1
Hint: Think in each case which digit you will use twice.
Answer:
(a) Given digits are 3, 8, and 7. Ascending order of given digits is 3, 7, 8. To make the smallest 4-digit number, the smallest digit (3) is used twice, resulting in 3378. Descending order of given digits is 8, 7, 3. To make the greatest 4-digit number, the greatest digit (8) is used twice, resulting in 8873.
(b) Given digits are 9, 0, and 5. Ascending order is 0, 5, 9. To make the smallest 4-digit number, keeping 5 at the leftmost place and taking the smallest digit 0 twice, we get 5009. Descending order is 9, 5, 0. To make the greatest 4-digit number, the greatest digit 9 is repeated, resulting in 9950.
(c) Given digits are 0, 4, and 9. Ascending order is 0, 4, 9. To make the smallest 4-digit number, keeping 4 at the leftmost place and repeating 0, we get 4009. Descending order is 9, 4, 0. To make the greatest 4-digit number, the greatest digit 9 is used twice, resulting in 9940.
(d) Given digits are 8, 5, and 1. Ascending order is 1, 5, 8. To make the smallest 4-digit number, the smallest digit (1) is used twice, resulting in 1158. Descending order is 8, 5, 1. To make the greatest 4-digit number, the greatest digit (8) is used twice, resulting in 8851.

Exam Tip: When repeating a digit, use the smallest digit twice for the smallest number and the largest digit twice for the greatest number, ensuring the number of digits is correct.

Question 3. Make the greatest and the smallest 4-digit numbers using any four different digits with conditions as given.
(a) Digit 7 is always at ones place
(b) Digit 4 is always at tens place
(c) Digit 9 is always at hundreds place
(d) Digit 1 is always at thousands place
Answer: Digits in ascending order are: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9; however, 0 can never be placed at the leftmost position. Digits in descending order are: 9, 8, 7, 6, 5, 4, 3, 2, 1, and 0.
(a) Keeping the digit '7' at the ones place, we have: The greatest 4-digit number is 9867. The smallest 4-digit number is 1027.
(b) Keeping the digit 4 at the tens place, we have: The greatest 4-digit number is 9847. The smallest 4-digit number is 1042.
(c) Keeping the digit 9 at the hundreds place, we have: The greatest 4-digit number is 8976. The smallest 4-digit number is 1902.
(d) Keeping the digit 1 at the thousands place, we have: The greatest 4-digit number is 1987. The smallest 4-digit number is 1023.

Exam Tip: When specific digit positions are fixed, fill the remaining positions with the largest available digits for the greatest number and the smallest available digits for the smallest number, taking care with zero.

Question 4. Take two digits, say 2 and 3. Make 4-digit numbers, using both the digits equal number of times.
Which is the greatest number?
Which is the smallest number?
How many different numbers can you make in all?
Answer: The given digits are 2 and 3. For making 4-digit numbers, we will repeat the given digits an equal number of times. The possible 4-digit numbers are 3322, 2233, 2323, 3232, 3223, and 2332. Among these, we find:
(i) The greatest number is 3322.
(ii) The smallest number is 2233.
(iii) We can make six different 4-digit numbers.

Exam Tip: To create numbers with repeated digits, list all possible permutations and then identify the greatest and smallest from the generated list.

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Question 1. Think of five more situations where you compare three or more quantities.
Answer: Here are five situations where you compare three or more quantities:
1. Comparing the heights of students in a classroom to find the tallest and shortest.
2. Comparing the weights of different fruits on a scale to find the heaviest and lightest.
3. Comparing the prices of various brands of cereal at a grocery store to find the cheapest and most expensive.
4. Comparing the running times of athletes in a race to determine the fastest and slowest.
5. Comparing the number of pages in different storybooks to find the thickest and thinnest.

Exam Tip: Comparison questions require listing multiple items and then applying a specific metric (like height, weight, price, speed, quantity) to rank or identify extremes among them.

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Question 1. Arrange the following numbers in ascending order:
(a) 847, 9754, 820, 571
(b) 9801, 25751, 36501, 38802
Answer: Arranging in ascending order means rewriting in increasing order.
(a) Given numbers are 847, 9754, 8320, and 571. We write them in ascending order as: 571, 847, 8320, 9754.
(b) Given numbers are 9801, 25751, 36501, and 38802. We write them in ascending order as: 9801, 25751, 36501, 38802.

Exam Tip: Ascending order means from smallest to largest. Start by comparing the number of digits, then compare digits from left to right for numbers with the same count.

Question 2. Arrange the following numbers in descending order:
(a) 5000, 7500, 85400, 7861
(b) 1971, 45321, 88715, 92547
Make ten such examples of ascending/descending order and solve them.
Answer: Arranging in descending order means rewriting in decreasing order.
(a) Given numbers are 5000, 7500, 85400, and 7861. We write them in descending order as: 85400, 7861, 7500, 5000.
(b) Given numbers are 1971, 45321, 88715, and 92547. We write them in descending order as: 92547, 88715, 45321, 1971.

Exam Tip: Descending order means from largest to smallest. Begin by comparing the count of digits, then compare digits from left to right for numbers with an equal count.

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Question 1. Read and expand the numbers wherever there are blanks.
Answer: Let's read and expand the numbers as requested. The first table shows some values with blanks for number names and expansions, while the solution provides the complete information.

The numbers with their names and expansions are shown below:

Now, we consider the following 5 numbers and read and expand them:

Note: A 1-digit number 9 and \( 9 + 1 = 10 \), the smallest 2-digit number.

Exam Tip: Expanding numbers helps understand place value. Remember to assign each digit its proper place value (tens, hundreds, thousands, etc.) when writing its expanded form.

General Notes on Number Systems:

The greatest 2-digit number is 99, and \( 99 + 1 = 100 \), which is the smallest 3-digit number.

The greatest 3-digit number is 999, and \( 999 + 1 = 1000 \), which is the smallest 4-digit number.

The greatest 4-digit number is 9999, and \( 9999 + 1 = 10000 \), which is the smallest 5-digit number.

Thus we conclude that:

[The greatest 1-digit number] + 1 is the smallest 2-digit number.

[The greatest 2-digit number] + 1 is the smallest 3-digit number.

[The greatest 3-digit number] + 1 is the smallest 4-digit number.

[The greatest 4-digit number] + 1 is the smallest 5-digit number.

The greatest 5-digit number is 99999, and \( 99999 + 1 = 100000 \). 100000 is the smallest 6-digit number, and its number name is one lakh.

Also \( 100000 = 100 \times 1000 \), meaning one lakh is 100 thousands.

The smallest 7-digit number is 1000000 (ten lakh).

The smallest 8-digit number is 10000000 (one crore).

The smallest 9-digit number is 100000000 (ten crore).

1 hundred is 10 tens

1 thousand is 10 hundreds

1 lakh is 100 thousands (1000 hundreds)

1 crore is 100 lakhs (10000 thousands)

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Question 1. Read and expand the numbers wherever there are blanks:

Exam Tip: When expanding numbers, multiply each digit by its place value (ones, tens, hundreds, thousands, etc.) and then add the products together.

Question 2. Complete the following pattern:
\( 9 + 1 = 10 \)
\( 99 + 1 = 100 \)
\( 999 + 1 = \)
\( 9,999 + 1 = \)
\( 99,999 + 1 = \)
\( 9,99,999 + 1 = \)
\( 99,99,999 + 1 = 1,00,00,000 \)
Answer: Here is the completed pattern:
\( 9 + 1 = 10 \)
\( 99 + 1 = 100 \)
\( 999 + 1 = 1000 \)
\( 9,999 + 1 = 10,000 \)
\( 99,999 + 1 = 1,00,000 \)
\( 9,99,999 + 1 = 10,00,000 \)
\( 99,99,999 + 1 = 1,00,00,000 \)

Exam Tip: Recognize patterns involving powers of ten and adding one. Adding one to a number consisting of all nines always results in the smallest number with one more digit (1 followed by zeros).

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Question 1.
1. What is \( 10 - 1 = \)?
2. What is \( 100 - 1 = \)?
3. What is \( 10,000 - 1 = \)?
4. What is \( 1,00,000 - 1 = \)?
5. What is \( 1,00,00,000 - 1 = \)?
Answer: Using the pattern, we have:
1. \( 10 - 1 = 9 \)
2. \( 100 - 1 = 99 \)
3. \( 1,000 - 1 = 999 \)
4. \( 10,000 - 1 = 9,999 \)
5. \( 1,00,000 - 1 = 99,999 \)

Exam Tip: Subtracting one from a power of ten always results in a number consisting of all nines with one less digit than the original power of ten.

Question 1. Give five examples where the number of things counted would be more than a 6-digit number.
Answer: The least 6-digit number is 100000 (one lakh). Here are some examples of things that would be counted as more than 1 lakh:
(i) The total number of stars visible in the night sky.
(ii) The total number of cars registered in a large city like Delhi.
(iii) The total number of children living in a big city.
(iv) The number of individual grains in a sack full of grains.
(v) The total number of pages in all notebooks used by students in a large town.

Exam Tip: When thinking of large quantities, consider populations, natural phenomena, or large collections that exceed hundreds of thousands.

Question 2. Starting from the greatest 6-digit number write the previous five numbers in descending order.
Answer: The greatest 6-digit number is 999999.
The previous five numbers in descending order are:
1st previous number = \( 999999 - 1 = 999998 \)
2nd previous number = \( 999999 - 2 = 999997 \)
3rd previous number = \( 999999 - 3 = 999996 \)
4th previous number = \( 999999 - 4 = 999995 \)
5th previous number = \( 999999 - 5 = 999994 \)
Writing these numbers in descending order, we have: 999998, 999997, 999996, 999995, 999994.

Exam Tip: Descending order means listing numbers from largest to smallest. To find previous numbers, simply subtract 1, 2, 3, and so on from the starting number.

Question 3. Starting from the smallest 8-digit number write the next five numbers in ascending order and read them.
Answer: The smallest 8-digit number is 10000000.
The next five numbers in ascending order are:
1st next number is: \( 10000000 + 1 = 10000001 \)
2nd next number is: \( 10000000 + 2 = 10000002 \)
3rd next number is: \( 10000000 + 3 = 10000003 \)
4th next number is: \( 10000000 + 4 = 10000004 \)
5th next number is: \( 10000000 + 5 = 10000005 \)
Writing these numbers in ascending order, we have: 10000001, 10000002, 10000003, 10000004, 10000005.
Reading these numbers, we have:
10000001: one crore one
10000002: one crore two
10000003: one crore three
10000004: one crore four
10000005: one crore five

Exam Tip: Ascending order means listing numbers from smallest to largest. To find next numbers, simply add 1, 2, 3, and so on to the starting number.

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Question 1. Read these numbers. Write them using placement boxes and then write their expanded forms:
(i) 475320
(iii) 97645310
(iv) 30458094
(a) Which is the smallest number?
(b) Which is the greatest number?
(c) Arrange these numbers in ascending and descending orders.
Answer: Let's read these numbers, represent them using placement boxes, and write their expanded forms, then answer the comparison questions.
(i) 475320:
Reading: Four lakh seventy five thousand three hundred twenty.
Expanded form: \( 475320 = 4 \times 100000 + 7 \times 10000 + 5 \times 1000 + 3 \times 100 + 2 \times 10 + 0 \)

(ii) 9847215:
Reading: Ninety eight lakh forty seven thousand two hundred fifteen.
Expanded form: \( 9847215 = 9 \times 1000000 + 8 \times 100000 + 4 \times 10000 + 7 \times 1000 + 2 \times 100 + 1 \times 10 + 5 \times 1 \)

(iii) 97645310:
Reading: Nine crore seventy six lakh forty five thousand three hundred ten.
Expanded form: \( 97645310 = 9 \times 10000000 + 7 \times 1000000 + 6 \times 100000 + 4 \times 10000 + 5 \times 1000 + 3 \times 100 + 1 \times 10 + 0 \)

(iv) 30458094:
Reading: Three crore four lakh fifty eight thousand ninety four.
Expanded form: \( 30458094 = 3 \times 10000000 + 0 \times 1000000 + 4 \times 100000 + 5 \times 10000 + 8 \times 1000 + 0 \times 100 + 9 \times 10 + 4 \times 1 \)

(a) The smallest number is 475320.
(b) The greatest number is 97645310.
(c) Ascending order: 475320, 9847215, 30458094, 97645310.
Descending order: 97645310, 30458094, 9847215, 475320.

Exam Tip: When dealing with large numbers, understanding place value and using a placement box helps with accurate reading, expansion, and comparison. Always remember to count the number of digits first when comparing numbers.

Question 2. Read these numbers:
(i) 527864
(ii) 95432
(iii) 18950049
(iv) 70002509
(a) Write these numbers using placement boxes and then using commas in Indian as well as International System of Numeration.
(b) Arrange these in ascending and descending orders.
Answer:
Reading these numbers, we have:
(i) 5,27,864: Five lakh twenty seven thousand eight hundred sixty four
(ii) 95,432: Ninety five thousand four hundred thirty two
(iii) 1,89,50,049: One crore eighty nine lakh fifty thousand forty nine
(iv) 7,00,02,509: Seven crore two thousand five hundred nine

(a) Using the placement boxes, we have:

Indian System of Numeration

International System of Numeration

Using commas, we can rewrite these numbers as:
(i) 5,27,864 (Indian System) = 527,864 (International System)
(ii) 95,432 (Indian System) = 95,432 (International System)
(iii) 1,89,50,049 (Indian System) = 18,950,049 (International System)
(iv) 7,00,02,509 (Indian System) = 70,002,509 (International System)

(b) Arrange these in ascending and descending orders.
Answer:
Ascending order: 95,432; 5,27,864; 1,89,50,049; 7,00,02,509
Descending order: 7,00,02,509; 1,89,50,049; 5,27,864; 95,432
In simple words: To arrange numbers, we first write them out using commas in both systems. For ascending order, we list them from smallest to largest. For descending order, we list them from largest to smallest.

Exam Tip: Always compare the number of digits first. If the number of digits is the same, compare the leftmost digit, then the next digit, and so on, to accurately determine ascending or descending order.

Question 3. Take three more groups of large numbers and do the exercise given above.
Answer: Do it yourself.
In simple words: This question asks you to make up your own large number examples and then practice writing them in both number systems and arranging them. It's a practice task.

Exam Tip: Practicing with your own examples helps solidify understanding of Indian and International number systems and order of operations.

Can You Help Me Write the Numeral (Page 11)

Question 1. Write numerals for:
(a) Forty two lakh seventy thousand eight
(b) Two crore ninety lakh fifty five thousand eight hundred
(c) Seven crore sixty thousand fifty five
Answer:
(a) Forty two lakh seventy thousand eight.

(b) Two crore ninety lakh fifty five thousand eight hundred
2,90,55,800

(c) Seven crore sixty thousand fifty five
7,00,60,055

In simple words: To write numerals, identify the place value for each part of the number name (lakh, crore, thousand, etc.) and place the digits accordingly. Fill any empty places with zeros.

Exam Tip: Be careful with the placement of zeros when converting number names to numerals, especially for missing place values. Use a place value chart to avoid errors.

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Question 1. You have the following digits 4, 5, 6, 0, 7 and 8. Using them, make five numbers each with 6 digits:
(a) Put commas for easy reading.
(b) Arrange them in ascending and descending order
Answer:
Five numbers each with 6 digits using the given digits are:
(i) 876540
(ii) 876450
(iii) 867540
(iv) 867405
(v) 876045

(a) Rewriting the above numbers using commas,
(i) 8,76,540
(ii) 8,76,450
(iii) 8,67,540
(iv) 8,67,405
(v) 8,76,045

(b) Ascending order: 8,67,405; 8,67,540; 8,76,045; 8,76,450; 8,76,540
Descending order: 8,76,540; 8,76,450; 8,76,045; 8,67,540; 8,67,405.
In simple words: First, create various 6-digit numbers from the given digits. Then, add commas to them for better readability. Finally, arrange these numbers from smallest to largest for ascending order and from largest to smallest for descending order.

Exam Tip: When arranging numbers, compare them digit by digit from the leftmost position. For commas, remember the Indian system (3, 2, 2 digits from right) and International system (3, 3, 3 digits from right).

Question 2. Take the digits 4, 5, 6, 7, 8 and 9. Make any three numbers each wiLlil digits. Put commas for easy reading.
Answer:
Three numbers each With- 8 digits using given digits are:
(i) 9,88,77,456
(ii) 9,88,77,465
(iii) 9,88,77,654
In simple words: Using the given digits, create three distinct 8-digit numbers. Place commas in these numbers according to standard numeral conventions to make them easier to read.

Exam Tip: When making numbers with a specific number of digits, ensure you use enough of the given digits, possibly repeating some to reach the required length if permitted.

Question 3. From the digits 3, 0 and 4, make five numbers each with 6 digits. Use commas.
Answer:
Five number each with 6 digits using the digits 3, 0 and 4 are:
(i) 5,44,330
(ii) 5,43,340
(iii) 5,40,340
(iv) 5,00,343
(v) 5,03,403
In simple words: Create five different 6-digit numbers using only the digits 3, 0, and 4. Remember to add commas to these numbers so they are easy to read.

Exam Tip: When creating numbers with repeated digits, systematically change the positions of the digits to generate unique combinations and ensure correct comma placement.

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Question 1. How many centimetres make a kilometre?
Answer:
1 km = 1000 x 100 cm = 1,00,000 cm
= one lakh cm
There are 1 lakh centimetres in a kilometre.
In simple words: One kilometre is equal to 1000 metres, and each metre contains 100 centimetres. So, a kilometre has 1,00,000 centimetres in total.

Exam Tip: Remember standard conversions: 1 kilometre = 1000 metres and 1 metre = 100 centimetres. Multiply these values to get the relationship between kilometres and centimetres.

Question 2. Name five large cities in India? Find their population. Also, find the distance in kilometres between each pair of these cities.
Answer: Do it yourself.
In simple words: This activity asks you to research five major Indian cities, find their populations, and then determine the distances between each pair of these cities.

Exam Tip: For "Do it yourself" questions, showing an understanding of the task, like listing the cities you chose and where you might find the data, can still earn partial marks if the full data isn't available.

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Question 1. How many milligrams make one kilogram?
Answer:
1 kg = 1000 g and 1 g = 1000 mg
1 kg = 1000 x 1000 mg = 10,00,000 mg
= 10 lakh mg
There are ten-lakh milligrams in 1 kilogram.
In simple words: One kilogram is equivalent to 1000 grams, and each gram holds 1000 milligrams. This means one kilogram contains 1,000,000 milligrams.

Exam Tip: Similar to length conversions, remember the standard weight conversions: 1 kilogram = 1000 grams and 1 gram = 1000 milligrams. Multiply these to get total milligrams.

Question 2. A box contains 2,00,000 medicine tablets each weighing 20 mg. What is the total weight of all the tablets in the box in grams and in kilograms?
Answer:
Number of tablets = 2,00,000
Weight of one tablet = 20 mg
Therefore, total weight of all tablets
\( = 20 \times 200000 \) mg
\( = 4000000 \) mg
Since 1 kg = 1000 g and 1 g = 1000 mg
\( 4000000 \text{ mg} = \frac { 4000000 }{ 1000 } \text{ g} = 4000 \text{ g} \)
Thus, the total weight of all the tablets in grams \( = 4000 \text{g} \).
Also, \( 4000000 \text{ mg} = \frac { 4000000 }{ 1000 \times 1000 } = 4 \text{ kg} \)
Thus, the total weight of all the tablets in kilograms \( = 4 \text{ kg} \).
In simple words: To find the total weight, multiply the number of tablets by the weight of each. Then, convert the total milligrams to grams by dividing by 1000, and to kilograms by dividing by another 1000 (or 1,000,000 from milligrams).

Exam Tip: When converting units, ensure you use the correct conversion factors. Milligrams to grams is a division by 1000; grams to kilograms is another division by 1000.

Try These (Page 13)

Question 1. A bus started its journey and reached different places with a speed of 60 km/hour The journey is shown below.
A diagram illustrates a bus journey with points A, B, C, D, E, F, G and distances between them:
A to B: 4170 km
B to C: 3410 km
C to D: 2160 km
D to E: 8140 km
E to F: 4830 km
F to G: 2550 km
G to A: 1290 km
(i) Find the total distance covered by the bus from A to D.
(ii) Find the total distance covered by the bus from D to G.
(iii) Find the total distance covered by the bus, If it starts from A and returns back to A.
(iv) Can you find the difference of distances from C to D and D to E?
(v) Find out the time taken by the bus to reach
(a) A to B
(b) C to D
(c) E to G
(d) Total journey
Answer:
(i) Distance covered by the bus for going from A to D
\( = 4170 \text{ km} + 3410 \text{ km} + 2160 \text{ km} \)
\( = [4170 + 3410 + 2160] \text{ km} = 9740 \text{ km} \)
(ii) Total distance covered by the bus for going from D to G
\( = 8140 \text{ km} + 4830 \text{ km} + 2550 \text{ km} \)
\( = [8140 + 4830 + 2550] \text{ km} = 15520 \text{ km} \)
(iii) Total distance covered by the bus for going round from A to A
\( = \text{[Distance between A and D]} + \text{[Distance between D and G]} + \text{[Distance between G and A]} \)
\( = [9740 \text{ km}] + [15520 \text{ km}] + [1290 \text{ km}] \)
\( = [9740 + 15520 + 1290] \text{ km} = 26550 \text{ km} \)

(iv) [Distance between D and E] - [Distance between C and D]
\( = [8140 \text{ km}] - [2160 \text{ km}] \)
\( = [8140 - 2160] \text{ km} = 5980 \text{ km} \)
Since time \( = \frac{\text{distance}}{\text{speed}} \)
And speed of the bus is 60 km/hour.
(v) Find out the time taken by the bus to reach
(a) A to B
\( = \frac { 4170 \text{km} }{ 60 \text{km/hour} } = \frac { 4170 }{ 60 } \text{ hour} = 69 \frac { 1 }{ 2 } \text{ hour} \)
(b) C to D
\( = \frac { 2160 }{ 60 } \text{ hours} = 36 \text{ hours} \)
(c) E to G
\( = \frac { [4830 \text{km} + 2550 \text{ km}] }{ 60 \text{ km/hour} } = \frac { 7380 \text{km} }{ 60 \text{ km/hour} } = \frac { 7380 }{ 60 } \)
\( = 123 \text{ hours} \)
(d) Total journey
\( = \frac { 26550 \text{km} }{ 60 \text{ km/hour} } = \frac { 26550 }{ 60 } \text{ hour} \)
\( = 442 \frac { 1 }{ 2 } \text{ hours} \)
In simple words: To find the total distance, add up the distances between points. To find the difference, subtract the smaller distance from the larger. To calculate time, divide the distance traveled by the bus's speed.

Exam Tip: Always pay attention to the units (km, hours) and ensure consistency in calculations. For time, remember the formula: Time = Distance / Speed.

Question 2. Raman's Shop

The sales during the last year

(a) can you find the total weight of apples and oranges Raman sold last year?
(b) Can you find the total money Raman got by selling apples?
(c) Can you find total money Ramati got by selling apples and oranges together?
(d) Make a table showing how much money Raman received from selling each item. Arrange the entries of amount of money received in descending order. Find the item which brought him the highest amount. How much is this amount?
Answer:
(a) Quantity of oranges and apples sold by Raman last year:
Weight of apples = 2457 kg
Weight of oranges = 3004 kg
Therefore, total weight = 2457 kg + 3004 kg
= 5461 kg
Answer: The total weight of oranges and apples = 5461 kg.

(b) Selling price of 1 kg of apples = Rs 40
Weight of apples sold = 2457 kg
Total money Raman got by selling apples = Rs \( 2457 \times 40 = \) Rs 98,280.

(c) Selling price of 1 kg of oranges = Rs 30
Weight of oranges sold = 3004 kg
Total money Raman got by selling oranges = Rs \( 3004 \times 30 = \) Rs 90,120
Total money Raman got by selling apples and oranges together
= Rs \( 98,280 + \text{Rs } 90,120 = \) Rs 1,88,400.

(d) Table showing money received from selling each item

Arranging the entries of amount of money received in descending order, we have:
2,53,670; 2,40,012; 1,60,040; 98,280; 90,120; 68,280; 38,530
Obviously, the highest amount of money is received against the item 'tooth brushes'. This highest amount of money is Rs 2,53,670.
In simple words: First, calculate the total earnings from each item by multiplying the rate by the quantity sold. Then, arrange these amounts from the largest to the smallest to identify which item generated the most revenue.

Exam Tip: For problems involving multiple calculations, organize your work clearly, step-by-step. Double-check multiplication and addition, especially with large numbers.

Try These (Page 19)

Question 1. Round these numbers to the nearest tens:
28 32 52 41 39 48 64 59 99 215 1453 2936
Answer:
Number 5 is rounded off to 10.

In simple words: To round a number to the nearest ten, look at the digit in the ones place. If it's 5 or more, round up to the next ten. If it's less than 5, round down to the current ten.

Exam Tip: Remember the rule for rounding: 5 rounds up. For example, 25 rounds to 30, but 24 rounds to 20.

Try These (Page 20)

Question 1. Round off the given numbers to the nearest tens, hundreds and thousands.
Answer:

In simple words: To round to tens, look at the ones digit. To round to hundreds, look at the tens digit. To round to thousands, look at the hundreds digit. Always round up if the determining digit is 5 or more, and down if it's less than 5.

Exam Tip: For each rounding level, focus only on the digit immediately to the right of the place you are rounding to. All digits to the right of the rounded place become zero.

Try These (Page 22)

Question 1. Estimate the following products:
(a) 87 x 313
(b) 9 x 795
(c) 898 x 785
(d) 958 x 387
Answer:
(a) 87 x 313
87 \( \rightarrow \) 90 [Rounding to tens]
313 \( \rightarrow \) 300 [Rounding to hundreds]
Estimated product \( = 90 \times 300 = 27000 \)

(b) 9 x 795
9 \( \rightarrow \) 10 [Rounding to tens]
795 \( \rightarrow \) 800 [Rounding to hundreds]
Estimated product \( = 10 \times 800 = 8000 \)

(c) 898 x 785
898 \( \rightarrow \) 900 [Rounding to hundreds]
785 \( \rightarrow \) 800 [Rounding to hundreds]
Estimated product \( = 900 \times 800 = 720000 \)

(d) 958 x 387
958 \( \rightarrow \) 1000 [Rounding to hundreds]
387 \( \rightarrow \) 400 [Rounding to hundreds]
Estimated product \( = 1000 \times 400 = 400000 \)
Note: Students are requested to make five more such problems and solve themselves.
In simple words: To estimate a product, first round each number to its nearest significant place (tens, hundreds, or thousands), then multiply the rounded numbers together to get an approximate answer.

Exam Tip: Choose an appropriate rounding level (tens, hundreds, thousands) based on the numbers and the desired level of estimation. Rounding to the highest place value often simplifies calculations most.

Try These (Page 23)

Question 1. Write the expressions for each of the following using brackets.
(a) Four multiplied by the sum of nine and two,
(b) Divide the difference of eight en and six by four
(c) Forty - five divided by three times the sum of three and two.
Answer:
(a) \( 4 \times (9 + 2) \)
(b) \( (18 - 6) \div 4 \)
(c) \( 45 \div [3(3 + 2)] \)
In simple words: When writing math expressions, use brackets to show which operations should be done first, like adding before multiplying or subtracting before dividing.

Exam Tip: Brackets (parentheses, square brackets) are crucial for ensuring operations are performed in the correct order, following the BODMAS/PEMDAS rule. Misplaced brackets can change the entire meaning of an expression.

Question 2. Write three different situations for (5 + 8) x 6. (One such situation is: Sohani and Reeta work for 6 days; Sohani works 5 hours a day and Reea 8 hours a day. How many hours do both of them work in a week?)
Answer:
Situation - I: Rahul pays Rs 5 for a morning tea and Rs 8 for the evening coffee daily. He takes tea and coffee for six days in a week. What is his weekly expenses for tea and coffee?
Situation - II: Prema and Shanti work for six days. Prema earns Rs 5 per day and Shanti earns Rs 8 per day. What do they earn together in six days?
Situation - III: Prateek read 5 pages of a novel in the morning and 8 pages in the evening. How many pages did Prateek read in 6 days?
In simple words: This question asks you to create three unique word problems where the mathematical expression \( (5 + 8) \times 6 \) would be used to find the solution. Each situation should involve adding two numbers, then multiplying the sum by six.

Exam Tip: When formulating word problems, make sure the scenario logically leads to the given mathematical expression. Clearly define the quantities and the action being performed (addition, then multiplication).

Question 3. Write five situations for the following where brackets would be necessary
(a) \( 7(8-3) \)
(b) \( (7 + 2)(10 - 3) \)
Answer:
(a) Situations for \( 7(8-3) \) :
Situation - I: Parti is a domestic lady. She charges Rs 8 per day to clean a house but gives back Rs 3 to the house owner for saving. What amount does she carry in hand from 7 houses?
Situation - II: What is the seven times the difference of eight and three?
Situation - III: A driver is supposed to work for 8 hours a day. But due to a certain reason, he had to go home 3 hours early daily for 7 days. What number of hours he could attend his duty on these seven days?
Situation - IV: Seven children with Rs 8 each went to market. Each of them bought pencils costing Rs 3. What total money is left with them?
Situation - V: There are seven containers having 8 litres of oil each. If 3 litres of oil is taken out of each container, then how many litres of total oil is left in the containers?
(b) Situations for \( (7 + 2)(10 - 3) \) :
Situation - I: A restaurant serves tiffin-lunch for 7 adults and 2 children in each of 10 houses of a society. On a Sunday, 3 houses were locked. How many persons were served lunch on that Sunday?
Situation - II: 7 persons hired a van up to a metro-station, for Rs 10 per head. The van-driver added 2 more passengers and reduced the fare by Rs 3 per head, What was the driver's total collection as fare?
Situation - III: Rahul gets Rs 10 per day as pocket money. He spends Rs 3 per day. In December, for one week he is given a bonus-pocket money for 2 days. What is his saving for that week of the month of December?
Situation - IV: A part-time gardener was hired for 7 days a month. He was supposed to work 10 hours daily. In a particular month he worked daily 3 hours less but extended his working for 2 more days. Find the total number of hours for which he worked in that month.
Situation - V: A team of 7 sales girls and 2 managers work for 10 hours daily in a garment shop. On Saturday, the shop was closed 3 hours early. How many total number of hours did the team work in the shop for the day?
In simple words: This task requires you to create word problems that accurately reflect the given mathematical expressions. For each expression, the situation should clearly show why addition/subtraction inside brackets would be performed before multiplication.

Exam Tip: Ensure that your created situations clearly illustrate the need for brackets. The problem statement should imply that certain operations must be grouped before others for the correct calculation.

Try These (Page 25)

Question 1. Write in Roman Numerals:
1. 73
2. 92
Answer:
1. 73 = LXXIII
2. 92 = XCII
In simple words: To convert to Roman numerals, break the number into its place values and use the corresponding Roman symbols. Remember that a smaller numeral before a larger one means subtraction (like IX for 9), and after means addition (like VI for 6).

Exam Tip: Memorize the basic Roman numeral values (I=1, V=5, X=10, L=50, C=100, D=500, M=1000) and practice the rules for addition and subtraction (e.g., IV, IX, XL, XC, CD, CM).