Motion Problem Exercise 1

Solve the following motion problems.

1.) A plane is traveling at a rate of 400 kilometers per hour. How many kilometers can it travel in 4.5 hours?

2.) Lance’s house is 3 kilometers away from his school. He goes to school every day using a bicycle. If it takes him 15 minutes to go to school using his bicycle, what is his average speed in km/hr?

3.) The distance between Alma’s house and her office is 8 km. If she leaves her house at 7:00 am and drives her car to the office at an average rate of 40 km per hour, what time will she reach the office?

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Work Problem Exercise

Solve the following.

1) Manny can do a job in 8 days. What part of the job is finished after he worked for 4 days?

2.) Josie can do a job in 7 days. Danna can do the same job in 12 days. If they both worked for 1 day, what part of the job is finished?

3.) Daniel can paint a house in 8 days. Vic can do the same job in 10 days. If they both worked for 2 days, what part of the job is done?

4.) One hose can fill a pool in 4 hours and a smaller hose can fill the same pool in 6 hours. How long will it take the two hoses to fill the entire pool?

5.) Lucian can dig a ditch in 6 hours. Lucian and Michael can do it in 3 hours. How long would it take Jun to dig the same ditch alone?

Answer key:

1.) \frac{1}{2}
2.) 1\frac{9}{84}
3.) \frac{9}{20}
4.) 2 hrs and 24 minutes
5.) 6 hrs

Age Word Problem

Solve the following problems.

1.) Jeff is 8 years older than Mira. The sum of their ages is 60. What are their ages?

2.) Angie is 12 years younger than Jaime. The sum of their ages is 44. How old is Jaime?

3.) Leo is 7 years older than Karen. In 5 years, the sum of their ages will be 45. How old is Karen?

4.) Arvin is twice as old as Cory. The sum of their ages is 42. What are their ages?

5.) Rashid is four times as old as Soraya. Three years ago, the sum of their ages was 39. How old is Soraya now?

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Number Word Problem

Solve the following problems.


1.) One number is 3 greater than the other number. Their sum is 47. What are the two numbers?

2.) One number is 12 less than the other number. Their sum is 64. What are the two numbers?

3.) One number is twice the other number. Their sum is 84. What are the two numbers?

4.) One number is five times the other number. Their sum is 72. What are the two numbers?

5.) One number is four times the other number. Their difference is 45. What are the two numbers?

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LCM and GCD Exercises Set 2

Here are some Civil Service exam exercises on GCD and LCM.

1.) What is the GCD of 8, 20, and 28?

2.) What is the GCD of 21, 35, and 56?

3.) What is 18/54 in lowest terms?

4.) What is 38/95 in lowest terms?

5.) What is the LCM of 6 and 8?

6.) What is the LCM of 5, 6, and 12?

7.) What is the product of the LCM and the GCD of 4, 8, and 20?

8.) There are 18 red marbles and 27 blue marbles to be distributed among children. What is the maximum number of children that can receive the marbles if each kid receives the same number of marbles for each color and no marble is to be left over?

9.) In a school sportsfest, there are 60 Grade 4 pupils, 48 Grade 5 pupils and 36 Grade 6 pupils. What is the largest number of teams that can be formed if the pupils in each Grade level are equally distributed and no pupil is left without a team?

10.) In a disco, the red lights blink every 3 seconds and the blue lights blink every 5 seconds. If the two colored lights blink at the same time if you turn them on, they will blink at the same time every ___ seconds.

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Week 11 Review: Answers and Solutions

Below are the solutions to the exercises and about work problems.

Problem 1
Aria can do a job in 7 days. What part of the job is finished after she worked for 3 days?

Aria can do a job in 7 days. So meaning, she can do 1/7 job each day.

1st day – 1/7
2nd day – 1/7
3rd day – 1/7

So, 3(1/7) = 3/7

Answer: 3/7

Problem 2
Katya can do a job in 5 days. Marie can do the same job in 6 days. If they both worked for 1 day, what part of the job is finished?

Katya can do a job in 5 days which means each day 1/5 of the job.
Marie can do a job in 6 days which means each day 1/6 of the job.

Let x = part of the job finished for 1 day

1/5 + 1/6 = 1/x
LCD: 30x

Multiplying both sides of the equation by 30, we have
(30x)(1/5 + 1/6) = (30x)(1/x)
(30x/5) + 30x(1/6) = (30x/x)
6x + 5x = 30
11x = 30
x = 30/11 or 2 8/11
Answer: 2 and 8/11 days.

Problem 3
Ramon can paint a house in 6 days. Ralph can do the same job in 10 days. If they both worked for 2 days, what part of the job is done if they were to work together the whole time?

Ramon = 6 days (or 1/6 part each day)
Ralph = 10 days (or 1/10 part each day)

Let w – part of the job done for 2 days

w = 2(1/6) + 2(1/10)
w = 2/6 + 2/10

LCD: 30

30(2/6 + 2/10)
((30/6)*2)/30 + ((30/10)*2)/30
(5*2)/30 + (3*2)/30
10/30 + 6/30
16/30 or 8/15

8/15 – work done for 2 days.

If they were to worked together, they will finish the work in x days using the equation

1/6 + 1/10 = 1/x

We multiply both sides of the equation which is 30x giving us

30x(1/6) + 30x(1/10) = 30x(1/x)
(30x)/6 + (30x)/10 = (30x/x)
5x + 3x = 30
8x = 30
x = 30/8
x = 15/4

This means that both of them will finish the work in 15/4 days. This means that the amount of worked finished is (8/15)/(15/4) = (8/15)(4/15) = 32/225.

Problem 4
One hose can fill a pool in 3 hours and a smaller hose can fill the same pool in 4 hours. How long will it take the two hoses to fill the entire pool?

Let x – total time to fill the pool

1/3 + 1/4 = 1/x
LCD: 12x

Multiplying both sides by 12x, we have

12x(1/3 + 1/4 = 1/x)12
(12x)/3 + (12x)/4 = 12
4x + 3x = 12
7x = 12
x = 12/7 or 1 5/7 hours

Answer: 1 5/7 hours

Problem 5
Marco can dig a ditch in 5 hours and he and Jimmy can do it in 2 hours. How long would it take Jimmy to dig the same ditch alone?

Let 1/x – Jimmy’s time alone
1/5 – Marco’s time

1/x + 1/5 = 1/2
LCD: 10x
10x(1/x + 1/5) = (10x)(1/2)
10 + 2x = 5x
2x – 5x = -10
-3x = -10
x = -10/-3 or 3 1/3 hrs

Problem 6
Maria can paint a fence in 6 days and Leonora can do the same job in 7 days. They start to paint it together, but after two days, Leonora left, and Maria finishes the job alone. How many days will it take Leonora to finish the job?

Job done by Maria in 2 days = 2/6
Job done by Leonora in 2 days = 2/7
Job done by Maria in the remaining days = x/6

2/6 + 2/7 + x/6 = 1

LCD:42x

Multiplying both sides of the equation by 42, we have

42(2/6 + 2/7 + x/6) = (42)(1)
84/6 + 84/7 + (42x)/6 = 42
14 + 12 + 7x = 42
26 + 7x = 42
7x = 42 – 26
7x = 16
x = 16/7 or 2 2/7 days

Problem 7
An inlet pipe can fill a pool in 4 hours. An outlet pipe can fill the same pool in 6 hours. One day, the pool was empty. The owner opened the inlet pipe but forgot to close the outlet pipe. How long will it take to fill the pool?

Let x – hours to fill the pool
1/4 – 1/6 = 1/x
LCD: 12x

(12x)(1/4) – (12x)(1/6) = (12x)(1/x)
(12x)/4 – (12x)/6 = (12x)/x
3x – 2x = 12
x = 12 hours

Week 10 Review: Answers and Solutions

After learning how to solve motion problems, let’s answer some exercises and problems. In the solutions, we let d = distance, r = rate, and t = time.

Exercises

1.)   A car travels and average speed of 75 kph. If it traveled for 3.5 hours, what is the total distance traveled?

d = rt
d = (75 kph)(3.5 hrs) =
d = 262.5 km

2.) A bus traveled 4 hours from City A to City B which is 450 kilometers apart. What is its average speed?

d = rt
450 km = (4 hrs)(r)
r = (450 kph)/(4 hrs)
r = 112.5 km

Problem

1.) Two cars left City A at 8:00 am going to City B using the same route. Car 1 traveled at the average speed of 60 kph while Car 2 traveled at an average speed of 50kph. At what time were the two cars 25 kilometers apart?

Let x = time traveled by the two cars
60x – 50x = 25
10x = 25
x = 25/10
x = 2.5 hours

2.5 hours = 2hours and 30 mins
2 hours and 30 minutes after 8:00 am is 10:30 am.

Answer: 10:30 AM

2.) The road distance from Sapiro City to Lireo City is 195 km. Car 1 left Sapiro City going to Lireo City at an average speed of 70kph. Car 2 left City Lireo City going to Sapiro City at an average speed of 60 kph. If both cars left the two cities at the same time and use the same road, after how many hours will the two cars meet?

Car 1
Rate = 70kph
Time = x
Distance = 70x

Car 2
Rate = 60kph
Time = x
Distance = 60x

Total distance = 195kph

70x + 60x = 195
130x = 195
x = 195/130

x = 1.5hrs

3.) A red car left Vigan at 9:00 AM and traveled to Manila at an average speed of 45 kph. After one hour, a white car left the same place for Manila using the same route at an average speed of 60 kph. At what time will the white car overtake the red car?

RED CAR
Rate = 45kph
Time = x+1
Distance = 45(x+1)

WHITE CAR
Rage = 60kph
Time = x
Distance = 60x

60x = 45(x+1)
60x = 45x+45
60x – 45x = 45
15x = 45
x = 45/15
x = 3hours

3 hours after 10:00am is 1:00 p.m.

Note: We add 3 hours to 10:00 am because the second car left at 10:00 am.

Answer: 1:00pm noon

4.) Two cars started from the same point, at 12nn, traveling to opposite directions at 50 and 60 kph, respectively. What is the distance between them at exactly 3:30 PM?

CAR 1
Rate = 50kph
Time = 3.5 hrs
Distance = (50 kph)(3.5 hrs) = 175

CAR 2
Rate = 60kph
Time = 3.5 hrs
Distance = (60 kph)( 3.5 hrs) = 210

What is the distance between them at exactly 3:30 PM?
175 km + 210 km = 385 km

5.) Two cars from the same point traveling to opposite directions at 75 and 85 kph, respectively. After how many hours will they be 240 kilometers apart?

75x + 85x = 240
160x = 240
240/160 = x
x = 1.5 hours

6.) A blue car leaves City A for City B at exactly 8:00 AM traveling at average speed of 55 kph. A gray car leaves City B for City A at the same time traveling at an average speed of 45 kph. The distance between the two cities is 75 kilometers.

If the two cars use the same route, what time will they pass each other?

let x be the time
d = r x t

sum of the distance traveled by 2 cars is equal to 75km
so

55x + 45x = 75
100x = 75
x = 75/100 or 0.75 hours
0.75 hours (45 mins)

They will pass each other 45 mins after 8:00am so the answer is 8:45 am.

Answer: 8:45 am

Enjoy!



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