Answer :
Sure, let's go through each question step-by-step with detailed explanations and solutions.
### 4. Namita travels 20 km. She travels 200 m by bus and the rest by auto. How much distance does she travel by auto?
To find out how much distance Namita travels by auto, we first need to subtract the distance traveled by bus from the total distance.
Total distance traveled by Namita: 20 km
Distance traveled by bus: 200 meters
- Convert 200 meters to kilometers: [tex]\( \frac{200}{1000} = 0.2 \)[/tex] km
Distance traveled by auto:
[tex]\[ \text{Distance traveled by auto} = \text{Total distance} - \text{Distance traveled by bus} \][/tex]
[tex]\[ \text{Distance traveled by auto} = 20 \, \text{km} - 0.2 \, \text{km} = 19.8 \, \text{km} \][/tex]
Namita travels 19.8 km by auto.
### 5. Write the number 97640053 in the Indian system as well as in the International system by inserting commas appropriately.
- Indian System: In the Indian numbering system, commas are placed after every two digits starting from the right after the first three digits.
[tex]\[ 9,76,40,053 \][/tex]
- International System: In the international system, commas are placed after every three digits starting from the right.
[tex]\[ 97,640,053 \][/tex]
### 6. To stitch a shirt, 2 m 15 cm cloth is needed. Out of 40 m cloth, how many shirts can be stitched and how much cloth will remain?
First, we convert the cloth needed for one shirt to meters:
Cloth needed for one shirt:
2 meters 15 centimeters = 2 + [tex]\( \frac{15}{100} \)[/tex] meters = 2 + 0.15 meters = 2.15 meters
Total cloth available: 40 meters
Number of shirts that can be stitched:
[tex]\[ \text{Number of shirts} = \left\lfloor \frac{40 \, \text{m}}{2.15 \, \text{m/shirt}} \right\rfloor = 18 \][/tex]
Remaining cloth:
[tex]\[ \text{Remaining cloth} = 40 \, \text{m} \mod 2.15 \, \text{m} = 1.3 \, \text{m} \][/tex]
So, Namita can stitch 18 shirts and will have 1.3 meters of cloth remaining.
### 7. Find the area of a square whose side is 12 cm.
The formula to find the area of a square is:
[tex]\[ \text{Area} = \text{side}^2 \][/tex]
Given side length: 12 cm
[tex]\[ \text{Area} = 12 \, \text{cm} \times 12 \, \text{cm} = 144 \, \text{cm}^2 \][/tex]
The area of the square is 144 square centimeters.
### 8. Write all the factors of 68.
Factors of a number are integers that can divide the number without leaving a remainder.
Factors of 68:
[tex]\[ 1, 2, 4, 17, 34, 68 \][/tex]
### 9. Write the first five multiples of 6.
Multiples of a number are obtained by multiplying the number with integers.
First five multiples of 6:
[tex]\[ 6 \times 1 = 6 \][/tex]
[tex]\[ 6 \times 2 = 12 \][/tex]
[tex]\[ 6 \times 3 = 18 \][/tex]
[tex]\[ 6 \times 4 = 24 \][/tex]
[tex]\[ 6 \times 5 = 30 \][/tex]
The first five multiples of 6 are 6, 12, 18, 24, and 30.
### 10. Find the least number which when divided by 6, 15, and 18 leaves a remainder of 5 in each case.
To solve this, we first need to find the least common multiple (LCM) of 6, 15, and 18, and then add the remainder.
LCM of 6, 15, and 18 is 90.
The least number that leaves a remainder of 5 when divided by these numbers is:
[tex]\[ \text{Least number} = \text{LCM} + 5 = 90 + 5 = 95 \][/tex]
So, the least number is 95.
### 11. Test the divisibility of the following numbers by 6.
A number is divisible by 6 if it is divisible by both 2 and 3. For divisibility by 2, the number must end in an even digit. For divisibility by 3, the sum of its digits must be divisible by 3.
- a) 71232
- Ends in 2 (even, so divisible by 2).
- Sum of digits: 7 + 1 + 2 + 3 + 2 = 15 (divisible by 3).
- Therefore, 71232 is divisible by 6.
- b) 2070
- Ends in 0 (even, so divisible by 2).
- Sum of digits: 2 + 0 + 7 + 0 = 9 (divisible by 3).
- Therefore, 2070 is divisible by 6.
- c) 46523
- Ends in 3 (odd, so not divisible by 2).
- Therefore, 46523 is not divisible by 6.
Divisibility results:
- 71232 is divisible by 6.
- 2070 is divisible by 6.
- 46523 is not divisible by 6.
I hope these explanations are clear and helpful!
### 4. Namita travels 20 km. She travels 200 m by bus and the rest by auto. How much distance does she travel by auto?
To find out how much distance Namita travels by auto, we first need to subtract the distance traveled by bus from the total distance.
Total distance traveled by Namita: 20 km
Distance traveled by bus: 200 meters
- Convert 200 meters to kilometers: [tex]\( \frac{200}{1000} = 0.2 \)[/tex] km
Distance traveled by auto:
[tex]\[ \text{Distance traveled by auto} = \text{Total distance} - \text{Distance traveled by bus} \][/tex]
[tex]\[ \text{Distance traveled by auto} = 20 \, \text{km} - 0.2 \, \text{km} = 19.8 \, \text{km} \][/tex]
Namita travels 19.8 km by auto.
### 5. Write the number 97640053 in the Indian system as well as in the International system by inserting commas appropriately.
- Indian System: In the Indian numbering system, commas are placed after every two digits starting from the right after the first three digits.
[tex]\[ 9,76,40,053 \][/tex]
- International System: In the international system, commas are placed after every three digits starting from the right.
[tex]\[ 97,640,053 \][/tex]
### 6. To stitch a shirt, 2 m 15 cm cloth is needed. Out of 40 m cloth, how many shirts can be stitched and how much cloth will remain?
First, we convert the cloth needed for one shirt to meters:
Cloth needed for one shirt:
2 meters 15 centimeters = 2 + [tex]\( \frac{15}{100} \)[/tex] meters = 2 + 0.15 meters = 2.15 meters
Total cloth available: 40 meters
Number of shirts that can be stitched:
[tex]\[ \text{Number of shirts} = \left\lfloor \frac{40 \, \text{m}}{2.15 \, \text{m/shirt}} \right\rfloor = 18 \][/tex]
Remaining cloth:
[tex]\[ \text{Remaining cloth} = 40 \, \text{m} \mod 2.15 \, \text{m} = 1.3 \, \text{m} \][/tex]
So, Namita can stitch 18 shirts and will have 1.3 meters of cloth remaining.
### 7. Find the area of a square whose side is 12 cm.
The formula to find the area of a square is:
[tex]\[ \text{Area} = \text{side}^2 \][/tex]
Given side length: 12 cm
[tex]\[ \text{Area} = 12 \, \text{cm} \times 12 \, \text{cm} = 144 \, \text{cm}^2 \][/tex]
The area of the square is 144 square centimeters.
### 8. Write all the factors of 68.
Factors of a number are integers that can divide the number without leaving a remainder.
Factors of 68:
[tex]\[ 1, 2, 4, 17, 34, 68 \][/tex]
### 9. Write the first five multiples of 6.
Multiples of a number are obtained by multiplying the number with integers.
First five multiples of 6:
[tex]\[ 6 \times 1 = 6 \][/tex]
[tex]\[ 6 \times 2 = 12 \][/tex]
[tex]\[ 6 \times 3 = 18 \][/tex]
[tex]\[ 6 \times 4 = 24 \][/tex]
[tex]\[ 6 \times 5 = 30 \][/tex]
The first five multiples of 6 are 6, 12, 18, 24, and 30.
### 10. Find the least number which when divided by 6, 15, and 18 leaves a remainder of 5 in each case.
To solve this, we first need to find the least common multiple (LCM) of 6, 15, and 18, and then add the remainder.
LCM of 6, 15, and 18 is 90.
The least number that leaves a remainder of 5 when divided by these numbers is:
[tex]\[ \text{Least number} = \text{LCM} + 5 = 90 + 5 = 95 \][/tex]
So, the least number is 95.
### 11. Test the divisibility of the following numbers by 6.
A number is divisible by 6 if it is divisible by both 2 and 3. For divisibility by 2, the number must end in an even digit. For divisibility by 3, the sum of its digits must be divisible by 3.
- a) 71232
- Ends in 2 (even, so divisible by 2).
- Sum of digits: 7 + 1 + 2 + 3 + 2 = 15 (divisible by 3).
- Therefore, 71232 is divisible by 6.
- b) 2070
- Ends in 0 (even, so divisible by 2).
- Sum of digits: 2 + 0 + 7 + 0 = 9 (divisible by 3).
- Therefore, 2070 is divisible by 6.
- c) 46523
- Ends in 3 (odd, so not divisible by 2).
- Therefore, 46523 is not divisible by 6.
Divisibility results:
- 71232 is divisible by 6.
- 2070 is divisible by 6.
- 46523 is not divisible by 6.
I hope these explanations are clear and helpful!
