## Worksheet on Calculating Time | Elapsed Time Worksheets with Answers

Students need to plan how long it will take them to finish Calculating Time Problems. It is really necessary to learn math problem-solving skills to save time in the exam. If students know the exact process to solve Time Calculation in the exam, then they can use the time properly. Therefore, learn the easy way to solve problems by referring to the Time and Distance Worksheets. Check the Worksheet on Calculating Time provided here to know the different methods to solve Time Calculation Problems.

If you are seeking the Time Worksheets with Solutions, then you are at the correct place. Immediately start your practice by getting the Calculating Time Practice Worksheets PDF.

1. A car travel 50 km in 10 minutes. In how much time will it cover 20 km?

Solution:

Given that a car travels 50 km in 10 minutes.
Find the speed of the car.
Therefore, the distance = 50 km, the time = 10 minutes,
10 minutes = 10/60 hours = 1/6 hrs
We know that Speed = distance/time
Substitute the given value in the formula.
The speed = 50 km/ (1/6) hr
Speed = 50 × 6 km/hr
Speed = 300 km/hr
Now, you know the speed of the car.
If the car covers 20 km, then the time = distance/speed
Time = 20 km/300 km/hr
Time = 1/5 hr
Since 1 hour = 60 minutes
The time = (1/5) × 60 minutes = 12 minutes.

The car takes 12 minutes to cover a distance of 20 Km.

2. Sam covers 240 km by car at a speed of 80 km/hr. Find the time taken to cover this distance?

Solution:

Given that Sam covers 240 km by car at a speed of 80 km/hr.
Therefore, the distance = 240 km, the speed = 80 km/hr.
We know that Speed = distance/time.
Time = distance/speed
Time = 240 km/80 km/hr = 3 hours.

Sam takes 3 hours to cover the distance 240 km at a speed if 80 Km/hr.

3. A bus covers a distance of 72 km in 30 minutes. Find the time taken by it to cover the same distance if its speed is decreased by 18 km/hr?

Solution:

Given that a bus covers a distance of 72 km in 30 minutes.
Therefore, the distance = 72 km, the time = 30 minutes,
30 minutes = 30/60 hours = 1/2 hours
We know that Speed = distance/time.
Substitute the given value in the formula.
Speed =  72 km/1/2 hours
Speed = 72 × 2 km/hours
The Speed = 144 km/hours
Speed = 144 km/hr
Reduced Speed = 144 km/hr – 18 km/hr = 126 km/hr
Time = distance/speed
The time = 72 km/126 km/hr
Time = 72/126 × 60 minutes
Time = 34.28 minutes (approximately).

The bus takes 34.28 minutes to cover the same distance if its speed is decreased by 18 km/hr.

4. A man is walking at a speed of 4 km per hour. After every km, he takes a rest for 2 minutes. How much time will it take to cover a distance of 6 km?

Solution:

Given that a man is walking at a speed of 4 km per hour.
Firstly, find the rest time to find the total time taken.
Rest time = Number of rests × time of each rest
Rest time = 5 × 2 minutes
The Rest time = 10 minutes
Time = distance/speed
Time = 6 km/4 km per hour
The time = 3/2 × 60 minutes = 90 minutes
Total time to cover 6 km = 90 minutes + 10 minutes = 100 minutes.

The man takes 100 minutes to cover a distance of 6 km.

5. Raj drives a bike at a uniform speed of 120 km/hr; Find the time taken by him to cover 560 km?

Solution:

Given that a car travels Raj drives a bike at a uniform speed of 120 km/hr and the distance = 560 km.
We know that Speed = distance/time
Substitute the given value in the formula.
The time = distance/speed
Time = 560 km/120 km/hr
Time = 14/3 hr
Since 1 hour = 60 minutes
The time = (14/3) × 60 minutes = 280 minutes.

Raj takes 280 minutes to cover a distance of 560 km.

6. The speed of sound in air is 165 m/sec. How long will the sound take to travel 99 km?

Solution:

Given that the speed of sound in air is 165 m/sec.
Convert speed into km/hr
m/sec × 18/5 = km/hr
165 m/sec × 18/5 = 594 km/hr
Given distance = 99 km
Time = distance/speed
Time = 99 km/ 594 km/hr
Time = 99/594 km/hr
The time = (99/594) × 60 minutes = 10 minutes.

Therefore, Sound takes 10 minutes to travel 99 km.

7. Ram walks 4 km in 60 minutes. How much time will he take to cover 800 m?

Solution:

Given that Ram walks 4 km in 60 minutes.
Firstly, find the speed.
Speed = distance/time
60 minutes = 1 hr
the distance = 4 km, the time = 1 hr
Speed = 4 km/hr.
Convert speed into m/sec
km/hr × 5/18 = m/sec
Speed = 4 km/hr × 5/18 = 20/18 m/sec.
Now, find the time that covers 800 m.
Time = Distance/Speed
Time = 800 m/20/18 m/sec.
The Time = 800 × 18/20 sec
Time = 720 sec = 720 × 60 minutes = 43200 minutes.

Ram takes 43200 minutes to cover 800m.

8. Arun cycle and covers a certain distance at the rate of 3 km/hr in 1 1/2 hours. How much time will be taken by him if he covers the same distance by scooter at the rate of 20 km/hr?

Solution:

Given that Arun cycle and covers a certain distance at the rate of 3 km/hr in 1 1/2 hours.
Speed = 3 km/hr, Time = 1 1/2 hours = 3/2 hours
Distance = Speed × Time
The distance = 3 km/hr × 3/2 hours = 9/2 km
If he covers the same distance by scooter at the rate of 20 km/hr,
Time = distance/speed
The time = (9/2 km)/20 km/hr = 9/40 hr
Time = 9/40 hr × 60 = 13.5 minutes.

The man takes 13.5 minutes to cover the same distance using a scooter at the rate of 20 km/hr.

9. A man is walking at a speed of 24 km/hr after every km, he takes rest for 2 minutes. How much time will it take to cover a distance of 8 km?

Solution:

Given that a man is walking at a speed of 24 km per hour.
Firstly, find the rest time to find the total time taken.
Rest time = Number of rests × time of each rest
Rest time = 7 × 2 minutes
The Rest time = 14 minutes
The time to cover 8 km is
Time = distance/speed.
Time = 8 km/24 km per hour
The time = 1/3 × 60 minutes = 20 minutes
Total time to cover 8 km = 20 minutes + 14 minutes = 34 minutes.

Therefore, the man takes 34 minutes to cover a distance of 8 Km.

10. A bike covers a distance of 15 km in 6 minutes. Find the time taken by it to cover the same distance if its speed is increased by 5 km/hr.

Solution:

Given that a bike covers a distance of 15 km in 6 minutes.
the distance = 15 km, the time = 6 minutes = 6/60 = 1/10 hr
Calculate the speed.
We know that Speed = distance/time.
Speed = 15 km/ (1/10 hr)
The speed = 15 km × 10 hr = 150 km/hr
If the speed is increased by 5 km/hr, 150 km/hr + 5 km/hr = 155 km/hr
Time = Distance/Speed
The time = 15 km/ 155 km/hr
Time = 15/155 × 60 minutes
Time = 5.80 minutes (approximately)

11. A bus is running at 28 km/hr. What time will it take to cover 280 meters?

Solution:

Given that a bus is running at 28 km/hr.
Speed = 28 km/hr
Distance = 280 meters
Convert Speed into m/sec
km/hr × (5/18) = m/sec
28 km/hr × (5/18) = 140/18 m/sec
Speed = 140/18 m/sec
Time = Distance/Speed
The time = 280 meters/ (140/18 m/sec)
Time = 280 × 18/140 sec
Time = 36 sec
The time taken to cover 280 meters = 36 × 60 minutes = 2160 minutes.

## Worksheet on Calculating Speed | Speed Practice Problems Worksheet with Answers

Students who want to learn Time and Distance Problems can refer to Time and Distance Worksheets. In this article, we are including all the problems related to the calculation of speed. You can know the speed calculation process with the help of a Worksheet on Calculating Speed. Everything you have to know about speed problems is included in this article along with the extra questions. Speed Problems Worksheet will also let you know about the shortcut ways to calculate speed. Therefore, without skipping, practice every problem available in this article.

Our Calculating Speed Worksheets will create an interest for students to learn math. If you want to make your students show interest in math, you must make them use our Time and Distance Worksheets. All the real-time speed distance time problems are included with a clear explanation.

1. Sam walks 25 km in 10 hours, find her speed?

Solution:

Given that Sam walks 25 km in 10 hours.
Therefore, the distance = 25km, the time = 10 hours,
We know that Speed = distance/time.
Substitute the given value in the formula.
Speed = 25km/10hours
Speed = 25/10 km/hours
The Speed = 5/2 km/hours
Speed = 2.5 km/hr

2. A bus covers a distance of 650 m in 1 minute whereas a car covers 96 km in 15 minutes. Find the ratio of their speeds?

Solution:

Firstly, calculate the speed of the bus.
Given that A bus covers a distance of 650 m in 1 minute.
1 minute = 60 seconds
Therefore, the distance = 650 m, the time = 60 seconds,
We know that Speed = distance/time.
Substitute the given value in the formula.
Speed = 650 m/60 seconds
The Speed = 650/60 m/sec
Speed = 130/12 m/sec
Secondly, calculate the speed of the car.
Given that a car covers 96 km in 15 minutes.
Therefore, the distance = 96 km, the time = 15 minutes,
We know that Speed = distance/time.
Substitute the given value in the formula.
Speed = 96 km/15 minutes
Convert minutes to km
15 minutes = 15/60hr = 1/4 hr
Speed = 96 km/(1/4 hr)
Speed = 96 × 4 km/hr = 384 km/hr
Convert Km/hr to m/sec
Multiply 18/5 to km/hr to convert it into m/sec
384 km/hr = 384 (18/5) = 6912/5 m/sec
Find the ratio of speeds.
Speed of the bus/Speed of car = (130/12)/6912/5 = 10.833/1382.4.

3. Olivia travels a distance of 8 km from her house to the school by auto-rickshaw at 16 km/hr and returns on rickshaw at 20 km/hr. Find the average speed for the whole journey?

Solution:

Given that Olivia travels a distance of 8 km from her house to the school by auto-rickshaw at 16 km/hr and returns by a rickshaw at 20 km/hr.
Firstly, calculate the time taken to move from house to school.
Time taken by Olivia to reach school = Distance/Speed = 8/16hr = 1/2 hr
Next, calculate the time taken to move from school to house.
Time taken by Olivia to reach house = Distance/Speed = 8/20hr = 2/5hr
Total time of journey = (1/2 + 2/5)hr = 9/10 hr
Total distance = (8 + 8) km = 16 km
Average speed = Total Distance/Total Time
Average Speed = 16 km/(9/10) hr
The Average Speed = (16 × 10/9) km/hr
Average Speed = 17.77 km/hr (approximately)

4. A bus covers 18 km in 2 hours. Find its speed?

Solution:

Given that a bus covers 18 km in 2 hours.
Therefore, the distance =  18 km, the time = 2 hours,
We know that Speed = distance/time.
Substitute the given value in the formula.
Speed = 18 km/2 hours
The Speed = 18/2 km/hours
Speed = 9 km/hr

5. Michael traveled 70 km in 3 hours by train and then traveled 40 km in 2 hours by car and 30 km in 2 hours by cycle. What is the average speed during the whole journey?

Solution:

Given that Michael traveled 70 km in 3 hours by train and then traveled 40 km in 2 hours by car and 30 km in 2 hours by cycle
Total Distance = 70 km + 40 km + 30 km = 140 km
Total time = 3 hours + 2 hours + 2 hours = 7 hours
Average Speed = Total distance/Total time = 140 km/7 hours
The Average Speed = 140/7 km/hour
Average Speed = 20 km/hour
The average speed of the whole journey is 20 km/hr.

6. A car moves from B to C at a speed of 40 km/hr and comes back from C to B at a speed of 20 km/hr. Find its average speed during the journey?

Solution:

Given that a car moves from B to C at a speed of 40 km/hr.
Here the distance = 40 km, time = 1 hr
The car comes back from C to B at a speed of 20 km/hr.
Here the distance = 20 km, time = 1 hr
The average speed = Total distance/Total time
Total Distance = 40 km + 20 km = 60 km
Total time = 1 hr  + 1 hr = 2 hours
Average Speed = Total distance/Total time = 60 km/2 hours
The Average Speed = 60/2 km/hour
Average Speed = 30 km/hour
The average speed during the journey is 30 km/hr.

7. A car covers a distance of 30 m in 2 minutes whereas a train covers a distance of 25 km in 5 minutes. Find the ratio of their speed?

Solution:

Given that a car covers a distance of 30 m in 2 minutes.
Speed = distance/time
The Speed = 30 m / 2 minutes
Speed = 30/2 m/minutes
2 minutes = 60 × 2 = 120 sec
Speed = 30/120 m/sec
Conver m/sec to km/hr
m/sec × 18/5 = km/hr
30/120 m/sec × 18/5 = 1/4  × 18/5  km/hr = 18/20 km/hr
Next, find the speed of the train
A train covers a distance of 25 km in 5 minutes
Conver minutes into hours
5 minutes = 5/60 hours
Speed = distance/time
The Speed = 25 km/ (5/60) hours
Speed = 25 × 60/5 km/hours = 5 × 60 km/hr
Speed = 300 km/hr
Ratio of speeds = (18/20) : 300 = 18 : 6000 = 9 : 3000 = 3 : 1000

8. A car covers a distance of 80 km in 4 hours. However, for the first 60 km it travels 30 km/hr. At what speed must it travel for the rest of the distance in order to complete the journey on time?

Solution:

A car covers a distance of 80 km in 4 hours.
If for the first 60 km it travels 10 km/hr, then time = distance/speed as speed = distance/time
Time = 60 km/30 km/hr
Time = 2 hr
The remaining time to travel the remaining distance of 20 km is 4 hours – 2 hours = 2 hours
Therefore, Speed = remaining distance/ remaining time = 20 km/2 hours
Speed = 10 km/hr
Therefore, the car must travel with a speed of 10 km/hr to travel the rest of the distance in order to complete the journey on time.

9. A bus covers a certain distance in 60 minutes if it runs at a speed of 60 km/hr. What must be the speed of the bus in order to reduce the time of the journey by 20 minutes?

Solution:

A bus covers a certain distance in 60 minutes if it runs at a speed of 60 km/hr.
Speed = distance/time km/hr
Distance = speed × time
60 minutes = 60/60 = 1 hr
Distance = 60 km/hr = 60 km
If the time of the journey reduced by 20 minutes, the remaining time = 40 minutes.
Speed = Distance/time
The Speed = 60 km/(40/60)hr
Speed = 60 × 60/40 km/ hr
Speed = 90 km/hr

## Worksheet on Conversion of Units of Speed | Speed Conversions Worksheets

Worksheet on Conversion of Units of Speed consists of converting the speeds from m/sec to km/hr and converting the speeds from km/hr to m/sec problems. If you are looking for the Practice Questions for Conversion of Units of Speed, you are at the correct place. Every question is given with the answer and explanation. So, without searching anywhere, you can clearly understand the complete concept of Conversion of Units of Speed.

Students can easily learn to convert different speed units by referring to Time and Distance Worksheets. Find the simple ways to convert km/hr to m/sec and m/sec to km/hr from this article. Simply, multiply 5/18 to km/hr to convert it into m/sec. Similarly, multiply 18/5 to m/sec to convert it into km/hr.

1. Convert each of the following speed into m/sec

(i) 162 km/hr (ii) 14.4 km/hr (iii) 216 km/hr (iv) 252 km/hr (v) 54 km/hr

Solution:

(i) 162 km/hr

Given that 162 km/hr
To convert km/hr to m/sec, you need to multiply the given number with 5/18
Therefore, multiply 162 km/hr to 5/18
162 km/hr = 162 × (5/18)
162 km/hr = 45 m/sec

(ii) 14.4 km/hr

Given that 14.4 km/hr
To convert km/hr to m/sec, you need to multiply the given number with 5/18
Therefore, multiply 14.4 km/hr to 5/18
14.4 km/hr = 14.4 × (5/18)
14.4 km/hr = 4 m/sec

(iii) 216 km/hr

Given that 216 km/hr
To convert km/hr to m/sec, you need to multiply the given number with 5/18
Therefore, multiply 216 km/hr to 5/18
216 km/hr = 216 × (5/18)
216 km/hr = 60 m/sec

(iv) 252 km/hr

Given that 252 km/hr
To convert km/hr to m/sec, you need to multiply the given number with 5/18
Therefore, multiply 252 km/hr to 5/18
252 km/hr = 252 × (5/18)
252 km/hr = 70 m/sec

(v) 54 km/hr

Given that 54 km/hr
To convert km/hr to m/sec, you need to multiply the given number with 5/18
Therefore, multiply 54 km/hr to 5/18
54 km/hr = 54 × (5/18)
54 km/hr = 15 m/sec

2. Convert each of the following speed into km/hr

(i) 75 m/sec (ii) 90 m/sec (iii) 3.5 m/sec (iv) 20.5 m/sec (v) 140 m/sec

Solution:

(i) 75 m/sec

Given that 75 m/sec
To convert m/sec to km/hr, you need to multiply the given number with 18/5
Therefore, multiply 75 m/sec to 18/5
75 m/sec = 75 × (18/5)
75 m/sec = 270 km/hr

(ii) 90 m/sec

Given that 90 m/sec
To convert m/sec to km/hr, you need to multiply the given number with 18/5
Therefore, multiply 90 m/sec to 18/5
90 m/sec = 90 × (18/5)
90 m/sec = 324 km/hr

(iii) 3.5 m/sec

Given that 3.5 m/sec
To convert m/sec to km/hr, you need to multiply the given number with 18/5
Therefore, multiply 3.5 m/sec to 18/5
3.5 m/sec = 3.5 × (18/5)
3.5 m/sec = 12.6 km/hr

(iv) 20.5 m/sec

Given that 20.5 m/sec
To convert m/sec to km/hr, you need to multiply the given number with 18/5
Therefore, multiply 20.5 m/sec to 18/5
20.5 m/sec = 20.5 × (18/5)
20.5 m/sec = 73.8 km/hr

(v) 140 m/sec

Given that 140 m/sec
To convert m/sec to km/hr, you need to multiply the given number with 18/5
Therefore, multiply 140 m/sec to 18/5
140 m/sec = 140 × (18/5)
140 m/sec = 504 km/hr

3. The speed of the bicycle is 72 km/hr, what is its speed in m/sec?

Solution:

The speed of the bicycle is 72 km/hr.
To convert km/hr to m/sec, multiply the given number with 5/18.
72 km/hr = 72 × (5/18)
72 km/hr = 20 m/sec

4. Sam covers a distance of 1600 m in 2 minutes. Find his speed in km/hr.

Solution:

Given that Sam covers a distance of 1600 m in 2 minutes.
Convert minutes into seconds.
1 minute = 60 seconds.
2 minutes = 2 × 60 = 120 seconds.
1600 m in 2 minutes = 1600 m in 120 seconds
1600/120 m/sec
To convert m/sec to km/hr, you need to multiply the given number with 18/5
Therefore, multiply 1600/120 m/sec to 18/5
1600/120 m/sec = (1600/120) × (18/5)
1600/120 m/sec = 48 km/hr

5. Which is greater – a speed of 288 km/hr or a speed of 85 m/sec?

Solution:

Given that 288 km/hr and 85 m/sec.
To find which is greater in speed convert either one from km/hr to m/sec or m/sec to km/hr.
Convert 288 km/hr to m/sec
288 km/hr
To convert km/hr to m/sec, you need to multiply the given number with 5/18
Therefore, multiply 288 km/hr to 5/18
288 km/hr = 288 × (5/18)
288 km/hr = 80 m/sec
85 m/sec > 80 m/sec.
Therefore, 85 m/sec is greater than the 288 km/hr.

6. A car covers a distance of 200 km in the first two hours, 110 km in the next 2 hours, and 28 km in the next 1/2 hour. Convert the speed into m/sec.

Solution:

The Total distance covered by the car = (200 + 110 + 28) km = 338 km
Total time taken = (2 + 2 + 1/2) hr = (4 + 4 + 1)/2 = 9/2 hr
Average speed = Distance covered/Time taken
= 338 km/(9/2 hr)
= 338 × (2/9) km/hr
= 75.11 km/hr
To convert km/hr to m/sec, multiply the given number with 5/18.
75.11 km/hr = 75.11× (5/18)
75.11 km/hr = 20.86 m/sec

7. The speed of a cyclist is 8 m/sec. find the speed in km/hr.

Solution:

The speed of a cyclist is 8 m/sec.
To convert m/sec to km/hr, you need to multiply the given number with 18/5
Therefore, multiply 8 m/sec to 18/5
8 m/sec = 8 × (18/5)
8 m/sec = 28.8 km/hr

## Worksheet on Cube Root | Cube Root Worksheets with Answers

Check out the Worksheet on Cube Root to get conceptual knowledge on the concept. You will find the cube and cube root problems quite easy after practicing from our worksheet. Learn the shortcut ways and easy process to solve cube and cube root problems.  All the Cube Root Questions and Answers are provided here along with the explanations that make your preparation easy.

Find the concept-wise Cube and Cube Root Worksheets and begin your practice. Get various questions from Cube Root like finding the cube root of a number, cube root of a number using prime factorization method, cube root of negative numbers, cube root of rational numbers, etc. Get Step by Step Solutions for the Cube Root Problems and learn different topics involved easily.

1. Find the Cube Root of the given numbers?
(i) 512 (ii) 1728 (iii) 216 (iv) 1331

Solution:

(i) 512

Write the product of primes of a given number 512 those form groups in triplets.
Cube Root of 512 = ∛512 = ∛(8 × 8 × 8)
Take one number from a group of triplets to find the cube root of 512.
Therefore, 8 is the cube root of a given number 512.

(ii) 1728

Write the product of primes of a given number 1728 those form groups in triplets.
Cube Root of 1728 = ∛1728 = ∛(12 × 12 × 12)
Take one number from a group of triplets to find the cube root of 1728.
Therefore, 12 is the cube root of a given number 1728.

(iii) 216

Write the product of primes of a given number 216 those form groups in triplets.
Cube Root of 216= ∛216= ∛(6 × 6 × 6)
Take one number from a group of triplets to find the cube root of 216.
Therefore, 6 is the cube root of a given number 216.

(iv) 1331

Write the product of primes of a given number 1331 those form groups in triplets.
Cube Root of 1331 = ∛1331 = ∛(11 × 11 × 11)
Take one number from a group of triplets to find the cube root of 1331.
Therefore, 11 is the cube root of a given number 1331.

2. Find the Cube Root of a Number by Prime Factorisation Method?

(i) 15625 (ii) 3375 (iii) 216 (iv) 13824

Solution:

(i) 15625

Firstly, find the prime factors of the given number.
15625 = 5 × 5 × 5 × 5 × 5 × 5
Group the prime factors into each triplet.
15625 = (5 × 5 × 5) × (5 × 5 × 5).
Collect each one factor from each group.
5 and 5
Finally, find the product of each one factor from each group.
∛15625= 5 × 5 = 25
25 is the cube root of 15625.

(ii) 3375

Firstly, find the prime factors of the given number.
3375 = 5 × 5 × 5 × 3 × 3 × 3
Group the prime factors into each triplet.
3375 = (5 × 5 × 5) × (3 × 3 × 3).
Collect each one factor from each group.
5 and 3
Finally, find the product of each one factor from each group.
∛3375= 5 × 3 = 15
15 is the cube root of 3375.

(iii) 216

Firstly, find the prime factors of the given number.
216 = 2 × 2 × 2 × 3 × 3 × 3
Group the prime factors into each triplet.
216 = (2 × 2 × 2) × (3 × 3 × 3).
Collect each one factor from each group.
2 and 3
Finally, find the product of each one factor from each group.
∛216 = 2 × 3 = 6
6 is the cube root of 216.

(iv) 13824

Firstly, find the prime factors of the given number.
13824 = 6 × 6 × 6 × 4 × 4 × 4
Group the prime factors into each triplet.
13824 = (6 × 6 × 6) × (4 × 4 × 4).
Collect each one factor from each group.
6 and 4
Finally, find the product of each one factor from each group.
∛13824 = 6 × 4 = 24
24 is the cube root of 13824.

3. Find the Cube Roots of Negative Numbers?

(i) (-27) (ii) (-1728) (iii) (-2744) (iv) (-512)

Solution:

Cube Root of a negative number is always negative. If -m be a negative number. Then, (-m)³ = -m³.
Therefore, ∛-m³ = -m.
cube root of (-m³) = -(cube root of m³).
∛-m = – ∛m

(i) (-27)

Find the prime factors of the number 27.
27 = 3 × 3 × 3
Group the prime factors into each triplet.
27 = (3 × 3 × 3)
Collect each one factor from each group.
3
Finally, find the product of each one factor from each group.
∛27= 3
∛-m = – ∛m
∛-27= – ∛27= -3
-3 is the cube root of (-27).

(ii) (-1728)

Find the prime factors of the number 1728.
1728 = 2 × 2 × 2 × 6 × 6 × 6
Group the prime factors into each triplet.
1728 = (2 × 2 × 2) × (6 × 6 × 6)
Collect each one factor from each group.
2 × 6
Finally, find the product of each one factor from each group.
∛1728= 12
∛-m = – ∛m
∛-1728= – ∛1728= -12
-12 is the cube root of (-1728).

(iii) (-2744)

Find the prime factors of the number 2744.
2744 = 14 × 14 × 14
Group the prime factors into each triplet.
2744 = (14 × 14 × 14)
Collect each one factor from each group.
14
Finally, find the product of each one factor from each group.
∛2744 = 14
∛-m = – ∛m
∛-2744 = – ∛2744= -14
-14 is the cube root of (-2744).

(iv) (-512)

Find the prime factors of the number 512.
512 = 8 × 8 × 8
Group the prime factors into each triplet.
512 = (8 × 8 × 8)
Collect each one factor from each group.
8
Finally, find the product of each one factor from each group.
∛512= 8
∛-m = – ∛m
∛-512= – ∛512= -8
-8 is the cube root of (-512).

4. Evaluate Cube Root of Product of Integers?

(i) ∛[27 × (-343)] (ii) ∛(64 × 216) (iii) ∛(125 × 216)

Solution:

(i) ∛[27 × (-343)]

Firstly, apply the cube root to both integers.
∛ab = (∛a × ∛b)
∛[27 × (-343)] = ∛27 × ∛-343
Then, find the prime factors for each integer separately.
[∛{3 × 3 × 3}] × [∛{(-7) × (-7) × (-7)}]
Take each integer from the group in triplets and multiply them to get the cube root of a given number.
(3 × (-7)) = -21
-21 is the cube root of ∛[27 × (-343)].

(ii) ∛(64 × 216)

Firstly, apply the cube root to both integers.
∛ab = (∛a × ∛b)
∛(64 × 216) = ∛64 × ∛216
Then, find the prime factors for each integer separately.
[∛{4 × 4 × 4}] × [∛{6 × 6 × 6}]
Take each integer from the group in triplets and multiply them to get the cube root of a given number.
(4 × 6) = 24
24 is the cube root of ∛(64 × 216).

(iii) ∛(125 × 216)

Firstly, apply the cube root to both integers.
∛ab = (∛a × ∛b)
∛(125 × 216) = ∛125 × ∛216
Then, find the prime factors for each integer separately.
[∛{5 × 5 × 5}] × [∛{6 × 6 × 6}]
Take each integer from the group in triplets and multiply them to get the cube root of a given number.
(5 × 6) = 30
30 is the cube root of ∛(125 × 216).

5. Find the Cube Root of Decimal Number 4.096?

Solution:

Convert the given decimal 4.096 into a fraction.
4.096 = 4096/1000
Now, apply the cube root to the fraction.
∛4096/1000
Apply the cube root to both integers.
∛(a/b) = (∛a)/(∛b)
∛4096/1000 = ∛4096/∛1000.
Then, find the prime factors for each integer separately.
∛(16 × 16 × 16)/∛(2 × 2 × 2 × 5 × 5 × 5)
Take each integer from the group in triplets to get the cube root of a given number.
(16)/(2 × 5) = 16/10
Convert the fraction into a decimal
16/10 = 1.6
1.6 is the cube root of 4.096.

6. Find the Cube Root of a Rational Number ∛(216/27)?

Solution:

Firstly, apply the cube root to both integers.
∛(a/b) = (∛a)/(∛b)
∛(216/27) = ∛216/∛27
Then, find the prime factors for each integer separately.
[∛(6 × 6 × 6)]/[ ∛(3 × 3 × 3)]
Take each integer from the group in triplets to get the cube root of a given number.
6/3
6/3 is the cube root of ∛(216/2197).

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1. Find which of the following numbers is a perfect cube?

(a) 256
(b) 81
(c) 2744
(d) 16

Solution:

(a) 256
Find the Prime Factors of the given number 256
256 prime factors are 4, 4, 4, 4
256 = 4 × 4 × 4 × 4
Form the triplets with the group of the same prime factor numbers.
256 = (4 × 4 × 4) × 4
There is only one triplet and one single factor available.
Therefore, the given number is not a perfect cube.

(b) 81
Find the Prime Factors of the given number 81
81 prime factors are 3, 3, 3, 3
81 = 3 × 3 × 3 × 3
Form the triplets with the group of the same prime factor numbers.
81 = (3 × 3 × 3) × 3
There is only one triplet and one single factor available.
Therefore, the given number is not a perfect cube.

(c) 2744
Find the Prime Factors of the given number 2744
2744 prime factors are 14, 14, 14
2744 = 14 × 14 × 14
Form the triplets with the group of the same prime factor numbers.
2744 = (14 × 14 × 14)
There is only one triplet available

(d) 16
Find the Prime Factors of the given number 16
16 prime factors are 2, 2, 2, 2
16 = 2 × 2 × 2 × 2
Form the triplets with the group of the same prime factor numbers.
16 = (2 × 2 × 2) × 2
There is only one triplet and one single factor available.
Therefore, the given number is not a perfect cube.

Therefore, (c) 2744 is a perfect cube.

2. Which of the following numbers is not a perfect cube?

(a) 1331
(b) 729
(c) 4096
(d) 2744

Solution:

(a) 1331
Find the Prime Factors of the given number 1331
1331 prime factors are 11, 11, 11
1331 = 11 × 11 × 11
Form the triplets with the group of the same prime factor numbers.
1331 = (11 × 11 × 11)
There is only one triplet available.
Therefore, the given number is a perfect cube.

(b) 729
Find the Prime Factors of the given number 729
729 prime factors are 9, 9, 9
729 = 9 × 9 × 9
Form the triplets with the group of the same prime factor numbers.
729 = (9 × 9 × 9)
There is only one triplet available.
Therefore, the given number is a perfect cube.

(c) 4096
Find the Prime Factors of the given number 4096
4096 prime factors are 8, 8, 8, 8
4096 = 8 × 8 × 8 × 8
Form the triplets with the group of the same prime factor numbers.
4096 = (8 × 8 × 8) × 8
There is only one triplet and one single factor available.
Therefore, the given number is not a perfect cube.

(d) 2744
Find the Prime Factors of the given number 2744
2744 prime factors are 14, 14, 14
2744 = 14 × 14 × 14
Form the triplets with the group of the same prime factor numbers.
2744 = (14 × 14 × 14)
There is only one triplet available.
Therefore, the given number is a perfect cube.

(c) 4096 is not a perfect cube.

3. What least number must be multiplied to 8575 so that the product becomes a perfect cube?
(a) 4
(b) 2
(c) 7
(d) 5

Solution:

Find the Prime Factors of the given number 8575
8575 prime factors are 7, 7, 7, 5, 5
8575 = 7 × 7 × 7 × 5 × 5
Form the triplets with the group of the same prime factor numbers.
8575 = (7 × 7 × 7) × (5 × 5)
Multiply 5 to the given number to make it a perfect cube.

4. What is the least number by which 12096 must be divided so that the quotient is a perfect cube?
(a) 5
(b) 7
(c) 3
(d) 4

Solution:

Find the Prime Factors of the given number 12096
12096 prime factors are 12, 12, 12, 7
12096 = 12 × 12 × 12 × 7
Form the triplets with the group of the same prime factor numbers.
12096 = (12 × 12 × 12) × 7
Divide 7 to the given number to make it a perfect cube.

5. ∛4913

(a) 18
(b) 21
(c) 14
(d) 17

Solution:

Write the product of primes of a given number 4913 those form groups in triplets.
Cube Root of 4913 = ∛4913 = ∛(17 ×17 × 17)
Take one number from a group of triplets to find the cube root of 4913.
Therefore, 17 is the cube root of a given number 4913.
(d) 17 is the cube root of 4913

6. Evaluate: ∛[(2197)/216]
(a) 6/13
(b) 12/11
(c) 11/12
(d) 13/6

Solution:

Firstly, apply the cube root to both integers.
∛(a/b) = (∛a)/(∛b)
∛(2197/216) = ∛2197/∛216
Then, find the prime factors for each integer separately.
[∛(13 × 13 × 13)]/[ ∛(6 × 6 × 6)]
Take each integer from the group in triplets to get the cube root of a given number.
13/6
13/6 is the cube root of ∛(2197/216).

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1. Evaluate the cube of a number

(i) (9)³ (ii) (14)³ (iii) (26)³ (iv) (50)³

Solution:

(i) (9)³

The given number is 9.
Multiply 9 three times to get a Cube of 9.
Cube of 9 = (9)³ = 9 × 9 × 9
Cube of 9 = 729

(ii) (14)³

The given number is 14.
Multiply 14 three times to get a Cube of 14.
Cube of 14 = (14)³ = 14 × 14 × 14
Cube of 14 = 2744

(iii) (26)³

The given number is 26.
Multiply 26 three times to get a Cube of 26.
Cube of 26 = (26)³ = 26 × 26 × 26
Cube of 26 = 17576

(iv) (50)³

The given number is 50.
Multiply 50 three times to get a Cube of 50.
Cube of 50 = (50)³ = 50 × 50 × 50
Cube of 50 = 125000

2. Find the cube of a decimal number

(i) (1.3)³ (ii) (3.6)³ (iii) (0.7)³ (iv) (0.03)³

Solution:

(i) (1.3)³

The given decimal number is 1.3.
Multiply 1.3 three times to get a Cube of 1.3.
Cube of 1.3 = (1.3)³ = (1.3) × (1.3) × (1.3)
Cube of 1.3 = 2.197

(ii) (3.6)³

The given decimal number is 3.6.
Multiply 3.6 three times to get a Cube of 3.6.
Cube of 3.6 = (3.6)³ = (3.6) × (3.6) × (3.6)
Cube of 3.6 = 46.656

(iii) (0.7)³

The given decimal number is 0.7.
Multiply 0.7 three times to get a Cube of 0.7.
Cube of 0.7 = (0.7)³ = (0.7) × (0.7) × (0.7)
Cube of 0.7 = 0.343

(iv) (0.03)³

The given decimal number is 0.03.
Multiply 0.03 three times to get a Cube of 0.03.
Cube of 0.03 = (0.03)³ = (0.03) × (0.03) × (0.03)
Cube of 0.03 = 0.000027

3. Evaluate the cube of a fraction number

(i) (3/7)³ (ii) (11/10)³ (iii) (1/12)³ (iv) (1(4/10))³

Solution:

(i) (3/7)³

The given fraction number is 3/7.
Multiply 3/7 three times to get a Cube of 3/7
Cube of 3/7 = (3/7)³ = (3/7) × (3/7) × (3/7)
Cube of 3/7 = (3 × 3 × 3)/(7 × 7 × 7)
Cube of 3/7 = 27/343

(ii) (11/10)³

The given fraction number is 11/10.
Multiply 11/10 three times to get a Cube of 11/10
Cube of 11/10 = (11/10)³ = (11/10) × (11/10) × (11/10)
Cube of 11/10 = (11 × 11 × 11)/(10 × 10 × 10)
Cube of 11/10 = 1331/1000

(iii) (1/12)³

The given fraction number is 1/12.
Multiply 1/12 three times to get a Cube of 1/12
Cube of 1/12 = (1/12)³ = (1/12) × (1/12) × (1/12)
Cube of 1/12 = (1 × 1 × 1)/(12 × 12 × 12)
Cube of 1/12 = 1/1728

(iv) (1(4/10))³

The given fraction number is 1(4/10).
1(4/10) = 14/10
Multiply 1(4/10) three times to get a Cube of 1(4/10)
Cube of 1(4/10) = (14/10)³ = (14/10) × (14/10) × (14/10)
Cube of 1(4/10) = (14 × 14 × 14)/(10 × 10 × 10)
Cube of 1(4/10) = 2744/1000

4. Find whether the given numbers are perfect cubes or not?
(i) 512 (ii) 5488 (iii) 686 (iv) 216

Solution:

(i) 512

The given number is 512.
Separate 512 into different prime factors
The prime factors for 512 are 8, 8, 8.
512 = 8 × 8 × 8
Group prime factors of a 512 in triples of equal factors.
512 = (8 × 8 × 8)
There is one triple factor available in the prime factors of the given number.
Therefore, the given number is a perfect cube.

ii) 5488

The given number is 5488.
Separate 5488 into different prime factors
The prime factors for 5488 are 7, 7, 7, 4, 4.
5488 = 7 × 7 × 7 × 4 × 4
Group prime factors of a 5488 in triples of equal factors.
5488 = (7 × 7 × 7) × (4 × 4)
There are one triple factor and double factors available in the prime factors of the given number.
Therefore, the given number is not a perfect cube.

(iii) 686

The given number is 686.
Separate 686 into different prime factors
The prime factors for 686 are 7, 7, 7, 2.
686 = 7 × 7 × 7 × 2
Group prime factors of a 686 in triples of equal factors.
686 = (7 × 7 × 7) × (2)
There are one triple factor and single factor available in the prime factors of the given number.
Therefore, the given number is not a perfect cube.

(iv) 216

The given number is 216.
Separate 216 into different prime factors
The prime factors for 216 are 6, 6, 6.
216 = 6 × 6 × 6
Group prime factors of a 216 in triples of equal factors.
216 = (6 × 6 × 6)
There is one triple available in the prime factors of the given number.
Therefore, the given number is a perfect cube.

5. Which of the following are Cubes of Even Numbers and Odd Numbers?

(i) 64 (ii) 125 (iii) 1728 (iv) 1331

Solution:

(i) 64

The given number is 64.
Find the prime factors of the given number 64.
The prime factors of 64 are 4, 4, 4
64 = 4 × 4 × 4
64 = 4³
4 is an even number
64 is a cube of an even number.

(ii) 125

The given number is 125.
Find the prime factors of the given number 125.
The prime factors of 125 are 4, 4, 4
125 = 5 × 5 × 5
125 = 5³
5 is an odd number
125 is a cube of an odd number.

(iii) 1728

The given number is 1728.
Find the prime factors of the given number 1728.
The prime factors of 1728 are 12, 12, 12
64 = 12 × 12 × 12
64 = (12)³
12 is an even number
1728 is a cube of an even number.

(iv) 1331

The given number is 1331.
Find the prime factors of the given number 1331.
The prime factors of 125 are 4, 4, 4
1331 = 5 × 5 × 5
1331 = 5³
5 is an odd number
1331 is a cube of an odd number.

6. What is the smallest number by which 84672 must be multiplied so that the product is a perfect cube?

Solution:

To find the smallest number by which 84672 must be multiplied so that the product is a perfect cube, you need to find the prime factors of the given number.
The prime factors of the given number 84672 are 12, 12, 12, 7, 7
84672 = 12 × 12 × 12 × 7 × 7
84672 = (12)³ × 7 × 7
From above, by multiplying 7 the number 84672 becomes the perfect cube.
Therefore, the smallest number is 7.

7. What is the smallest number by which 36000 must be multiplied so that the product is a perfect cube?

Solution:

To find the smallest number by which 36000 must be multiplied so that the product is a perfect cube, you need to find the prime factors of the given number.
The prime factors of the given number 36000 are 10, 10, 10, 6, 6
36000 = 10 × 10 × 10 × 6 × 6
36000 = (10)³ × 6 × 6
From above, by multiplying 6 the number 36000 becomes the perfect cube.
Therefore, the smallest number is 6.

8. Find the smallest number by which 2916 must be divided so that the quotient is a perfect cube?

Solution:

To find the smallest number by which 2916 must be divided so that the product is a perfect cube, you need to find the prime factors of the given number.
The prime factors of the given number 2916 are 9, 9, 9, 2, 2
2916 = 9 × 9 × 9 × 2 × 2
2916 = (9)³ × 2 × 2
From above, by dividing 4 the number 2916 becomes the perfect cube.
Therefore, the smallest number is 4.

9. Find the smallest number by which 3456 must be divided so that the quotient is a perfect cube?

Solution:

To find the smallest number by which 3456 must be divided so that the product is a perfect cube, you need to find the prime factors of the given number.
The prime factors of the given number 3456 are 6, 6, 6, 4, 4
3456 = 6 × 6 × 6 × 4 × 4
3456 = (6)³ × 4 × 4
From above, by dividing 16 the number 3456 becomes the perfect cube.
Therefore, the smallest number is 16.