## Week 10 Review: Practice Exercises and Problems

After learning how to solve motion problems, let’s answer some exercises and problems.

Exercises

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

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

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?

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 uses the same road, after how many hours will the two cars meet?

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?

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?

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?

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?

7.) A gray car leaves Batangas City for Quezon City at 11:00 am at an average speed of 50 kph. After one hour, a black car leaves Quezon City for Batangas at an average speed of 60 kph. If the two cars use the same route,  at exactly what time will the two cars pass each other?

Enjoy!

## PCSR REVIEW SERIES WEEK 10: Motion Problems

We have learned number problems and age problems. Now, we learn how to solve motion problems or problems involving distance, rate, and time. Below are the articles and videos that you can watch to learn about motion problems.

ARTICLES

VIDEOS

Enjoy!

## Week 9 Review: Practice Exercises and Problems

After learning how to solve age problems, let’s have some exercises.

Exercises

1.) Leah is 3 years older than Lanie. The sum of their ages is 29. What are their ages?

2.) Nico is 3 times as old as Jimmy. The sum of their ages is 48. How old are they?

3.) Mia’s age is 1 more than twice Lito’s age. The sum of their ages is 25. How old are they?

4.) Alfred’s thrice as old as Fely. The difference between their ages is 16. How old are they?

Problems

1.) Gina is 5 years older than Liezel. In 5 years, the sum of their ages will be 39. What are their ages?

2.) Alex is 7 years older than Ben. Three years ago, the sum of their ages was 29. What are their ages?

3.) Yna is 18 years older than Karl. In 8 years, she will be as twice as old as Karl. What are their ages?

4.) Peter’s age is thrice Amaya’s age. In 5 years, his age will be twice Amaya’s age. How old is Peter?

5.) Martin is thrice as old as Kaye. If 7 is subtracted from Martin’s age and 5 is added to Kaye’s age, then the sum of their ages is 34. What are their ages?

6.) James is 9 years older than Kevin. Two years ago, his age was twice that of Kevin’s age. How old is James?

7.) Mark is twice as old as Lorie. Rey is 6 years younger than Mark. Three years ago, the average of the ages of the three of them is 20. What are their present ages?

8.) Sam is thrice as old as Vina. Rio is half as old as Vina. The sum of their ages is 54. What are their ages?

9.) Four years from now, Tina’s age will be equal to Kris’ present age. Two years from now, Kris will be twice as old as Tina. What are their present ages?

In the next post, I am going to post the solutions to these problems.

## PCSR REVIEW SERIES WEEK 9: Age Problems

After learning number problems, let’s learn about age problems. Below are the articles and videos that you can use for studying age problems.

ARTICLES

VIDEOS

Good luck!

## Week 8 Review: Answers and Solutions

These are the solutions and answers to the problems in Week 8 Review on Number Problems.

Problem 1

One number is 3 more than the other. Their sum is 27. What are the numbers?

Let x – smaller number
x + 3 – larger number

Their sum is 27, so
x + (x + 3) = 27
2x + 3 = 27
2x = 27 – 3
2x = 24
x = 24/2
x = 12 (smaller number)
x + 3 = 15 (larger number).

Problem 2
One number is 5 less than the other. Their sum is 51. What are the numbers?

Let x – larger number
x – 5 –  smaller number

And their sum is 51. So,

x + (x – 5) = 51
2x – 5 = 51
2x = 51 + 5
2x = 56
x = 56/2
x = 28 (larger number)
x – 5 = 28 – 5 = 23 (smaller number).

Problem 3

One number is 3 times the other number. Their sum is 48. What are the numbers?

Let x – smaller number
3x – larger number

And their sum is 48. So,

x + 3x = 48
4x = 48
x = 48/4
x = 12(1st number)

2nd number = 3x
3(12) = 36

Problem 4
One number is 5 times the other number. Their difference is 52. What are the numbers?

Let x – smaller number
5x – larger number

And their difference is 52. So,

5x – x = 52
4x = 52
x = 52/4
x = 13 (smaller number)
5x = 5(13) = 65.

Checking: -13 – (-65)
-13 + (65) = 52

Problem 5
The sum of three numbers is 36. The second number is 5 more than the first number and the third number is 8 less than the first number. What are the three numbers?

Let x – 1st number
x + 5 – 2nd number
x – 8 – 3rd number

Their sum is 36. So,
x + (x + 5) + (x – 8) = 36
3x – 3 = 36
3x = 36 + 3
3x = 39
x = 13 (1st number)
2nd number = x + 5 => (13) + 5 => 18
3rd number = x – 8 => (13) – 8 => 5

Checking: 13 + 18 + 5 = 36

Problem 6

The sum of three numbers is 98. The second number is twice the first number and the third number twice the second number. What are the three numbers?

14, 28 & 56

Let x = 1st number
2x = 2nd number (twice the first)
2(2x)=3rd number (twice the second)

And their sum is 98. So,

x + (2x) + 2(2x) = 98
x + 2x + 4x =98
7x = 98
x = 98/7
x = 14 (1st number)

2nd number = 2x => 2(14) => 28

3rd number = 2(2x) => 2(2(14)) => 2(28) => 56

Problem 7

One number is two more than thrice the other. Their sum is 26. What are the two numbers?

Let x – 1st number
3x + 2 = 2nd number (two more than thrice the other)

And their sum is 26.

x + (3x + 2) = 26
4x + 2 = 26
4x = 26 – 2
4x = 24
x = 24/4
x = 6 (1st number)

2nd number = (3x + 2) => 3(6) + 2 => 18 + 2 => 20

Problem 8

One number is thrice the other. When 3 is added to the larger and 7 is subtracted from the smaller, their sum becomes 32. What are the two numbers?

Let x – smaller number
3x – larger number (thrice the other)

When 3 is added to larger number… = 3x + 3

…and 7 is subtracted to smaller = x – 7

Their sum becomes 32. So,

(3x + 3) + (x – 7) = 32
4x – 4 = 32
4x = 32 + 4
4x = 36
x = 36/4
x = 9(smaller number)

Larger number = 3x = 3(9) = 27

Checking:
When 3 is added to larger number = 27 + 3 = 30
And 7 is subtracted to smaller number = 9-7 = 2

Their sum is 32 = 30 + 2 = 32

Problem 9

The sum of two consecutive numbers is 91. What are the two numbers?

Let x – first number
x + 1 – 2nd number

x + (x + 1) = 91
2x + 1 = 91
2x = 91 – 1
2x = 90
x = 90/2
x = 45 (1st number)

2nd number => x + 1 => 45 + 1 => 46

Problem 10
The sum of two positive consecutive EVEN integers is 66. What are the two numbers?

Let x – 1st number
x + 2 = 2nd number

x + (x + 2) = 66
2x + 2 = 66
2x = 66 – 2
2x = 64
x = 64/2
x = 32 (1st number)

2nd number => x + 2 => 32 + 2 => 34

PCSR Problem 11

The sum of two positive consecutive ODD integers is 36. What are the two numbers?

Let x – 1st odd number
x + 2 – 2nd odd number

And their sum is 36.

x + (x + 2) = 36
2x + 2 = 36
2x = 36-2
2x = 34
x = 34/2
x = 17(1st number)
x + 2 = 17 + 2 = 19 (2nd number)

Checking:

17 + 19 = 36
And 17 and 19 are both odd numbers

Problem 12

The sum of three positive consecutive ODD integers is 81. What are the three integers?

Let x – 1st odd integer
x + 2 – 2nd odd integer
x + 4 – 3rd odd integer

Their sum is 81.

x + (x + 2) + (x + 4) = 81
3x + 6 = 81
3x = 81 – 6
3x = 75
x = 75/3
x = 25 (1st int)

2nd int = (x + 2) => 25 + 2 => 27
3rd int = (x + 4) => 25 + 4 => 29

Checking:

25 + 27 + 29 = 81
They are consecutive ODD integers.

Problem 13

The sum of the smallest and the largest of five positive consecutive integers is 108. What is the third integer?

Let x – 1st integer
x + 1 = 2nd integer
x + 2 = 3rd integer
x + 3 = 4th integer
x + 4 = 5th integer

Since the sum of the first and the fifth is 108,

x + (x + 4) = 108
2x + 4 = 108
2x = 108 – 4
2x = 104
x = 104/2
x = 52 (smallest number).

2nd int. => (x + 1) => 52 + 1 => 53
3rd int => ( x + 2) => 52 + 2 => 54
4th int. => (x + 3) => 52 + 3 => 55
5th int. => (x + 4) => 52 + 4 => 56

Since we are looking for the third integer, the answer is 54.

Problem 14
The average of four positive consecutive EVEN integers is 19. What is the largest integer?

Let x – 1st even integer
x + 2 = 2nd even integer
x + 4 = 3rd even integer
x + 6 = 4th even integer

Their average is 19.

(x + (x + 2) + (x + 4) + (x + 6))/4 = 19
(4x + 12)/4 = 19

Multiplying both sides of the equation by 4,

4x + 12 = 19(4)
4x + 12 = 76
4x = 76 – 12
4x = 64
x = 64/4
x = 16(1st even int).

2nd even int. = x + 2 => 16 + 2 => 18
3rd even int. = x + 4 => 16 + 4 => 20
4th even int. = x + 6 => 16 + 22 => 22

Checking:

(16 + 18 + 20 + 22)/4 = 19
(76)/4 = 19
19 = 19

PCSR Problem 15
The average of seven positive consecutive integers is 31. What is the smallest integer?

Let x – 1st integer
x + 1 = 2nd integer
x + 2 = 3rd integer
x + 3 = 4th integer
x + 4 = 5th integer
x + 5 = 6th integer
x + 6 = 7th integer

Their average is 31.

(x + (x + 1) + (x + 2) + (x +3) + (x + 4) + (x + 5) + (x + 6))/7 = 31
(7x + 21)/7 = 31
7x + 21 = 31(7)
7x + 21 = 217
7x = 217 – 21
7x = 196
x = 196/7
x = 28(1st integer)

2nd int. = x + 1 => 28 + 1 => 29
3rd int. = x + 2 => 28 + 2 => 30
4th int. = x + 3 => 28 + +3 => 31
5th int. = x + 4 => 28 + 4 => 32
6th int. = x + 5 => 28 + 5 => 33
7th int. = x + 6 => 29 + 6 => 34

Checking:
(28 + 29 + 30 + 31 + 32 + 33 + 34)/7 = 31
217/7 = 31
31 = 31

Since we are looking for the smallest integer, the answer is 28.

## Week 8 Review: Practice Exercises and Problems

After learning how to solve number problems, let’s have some practice exercises.

Week 8 Review: Practice Exercises and Problems

1.) One number is 3 more than the other. Their sum is 27. What are the numbers?

2.) One number is 5 less than the other. Their sum is 51. What are the numbers?

3.) One number is 3 times the other number. Their sum is 48. What are the numbers?

4.) One number is 5 times the other number. Their difference is 52. What are the numbers?

5.) The sum of three numbers is 36. The second number is 5 more than the first number and the third number is 8 less than the first number. What are the three numbers?

6.) The sum of three numbers is 98. The second number is twice the first number and the third number twice the second number. What are the three numbers?
7.) One number is two more than thrice the other. Their sum is 26. What are the two numbers?

8.) One number is thrice the other. When 3 is added to the larger and 7 is subtracted from the smaller, their sum becomes 32. What are the two numbers?

9.) The sum of two positive consecutive numbers is 91. What are the two numbers?

10.) The sum of two positive consecutive EVEN integers is 66. What are the two numbers?

11. ) The sum of two positive consecutive ODD integers is 36. What are the two numbers?

12.) The sum of three positive consecutive ODD integers is 81. What are the three integers?

13.) The sum of the smallest and the largest of five positive consecutive integers is 108. What is the third integer?

14.) The average of four positive consecutive EVEN integers is 19. What is the largest integer?

15.) The average of seven positive consecutive integers is 31. What is the smallest integer?

Enjoy solving!

Below are the solutions and answers to the Week 4 Practice Problems and Solutions.

Practice Exercises 1

Note: In multiplying fractions, we multiply the numerator by the numerator of the other fraction, and then multiply the denominator by the denominator of the other fraction. For whole numbers, we can put 1 as the denominator. All fractions must be in lowest terms.

A. 1/2 × 1/3 = 1/6
B. 2/3 × 4/5 = 8/15
C. 8/1 × 5/6 = 40/6 = 6 4/6 or 6 2/3
D. 2 5/8 × 3 = 21/8 × 3/1 = 63/8 = 7 7/8
E. 3 1/8 × 4/5 = 25/8 × 4/5 = 100/40 = 2 20/40 or 2 1/2
F. 1 2/3 × 2 3/4 = 5/3 × 11/4 = 55/12 = 4 7/12

Practice Exercises 2

A. When dividing fractions you get the reciprocal of the divisor, and then multiply. In 1/5÷ 3/10, the divisor 3/10 and the reciprocal of 3/10 is 10/3. So, 1/5 × 10/3 = 10/15 or 2/3

B. 1/2 ÷ 3/8 = 1/2 × 8/3 = 8/6 = 1 2/6 or 1 1/3

C. 9 ÷ 3/7 = 9 × 7/3 = 63/3 or 21

D. 2 5/8 ÷ 2

First, we convert 2 5/8 to improper fraction as follows. That is $2\frac{5}{8} = \frac{8 \times 2 + 5}{8} = \frac{21}{8}$. Don’t forget that the denominator of the mixed fraction is the same as the denominator of the improper fractions.

Here, the reciprocal of 2 is 1/2. So, 21/8 × 1/2 = 21/16 = 1 5/16

E. 3 1/8 ÷ 3/5 = 25/8 × 5/3 = 125/24 = 5 5/24

F. 2 3/4 ÷ 1 1/8 = 11/4 ÷ 9/8 = 11/4 × 8/9 = 88/36 = 2 16/36 or 2 4/9

Practice Problems

Note: In multiplication and division of fractions, all mixed fractions must be converted to improper fractions (see Practice Problem 2D above).

1.) 2/3 × 1/4 = 2/12 or 1/6

2.) 3/5 × 35/1=105/5 = 21 (women)
2/5 × 35/1 = 70/5 = 14 (men)

3.) 2 3/4 × 7 = (11/4) ×  7 = 77/4 = 19 1/4

4.) A = L × W
A = (35 1/4) × (20 1/2) = (141/4) × (41/2)
= 5781/8= 722 5/8

5.) 2 4/5 × 5/1= 14/5 × 5/1 = 14

6.) 1 1/2 L juice is to be shared equally by 6 friends, so 1 1/2 ÷ 6.

The mixed fraction  1 1/2 is 3/2 in improper form.

Dividing by 6 is the same as multiplying by 1/6, so 3/2 × 1/6 = 3/12

3/12 is not yet in its lowest term. So get its lowest term, divide the numerator and denominator by their GCF which is 3.  So, 3/12 will become 1/4.

The final answer is 1/4 L.

7.) 8 ÷ 2/5 = 8 × 5/2 = 8/1 × 5/2 = 40/2 = 20.

8.) Bookshelf length divided by book’s thickness = number of books that will fit in the bookshelf

5 1/4 feet ÷ 1 1/2 inches = ?

Notice that the units are in feet and inches. We cannot proceed until the units are the same, so we need  to convert feet into inches. (1 ft = 12 in) So 5 ft × 12 in per ft = 60 in. We still have 1/4 ft, so 1/4 of 12 which is 3 in. All in all, we have 63 in. Now our equation is

63  ÷ 3/2 = 63/1 × 2/3

= 126/3 = 42.

9.) 5 pumpkin pies are to be shared equally among 12 persons, equals 5/12.

10.) We have 8 3/4 hectares ÷ 4 children

35/4 ÷ 4 =35/4 × 1/4 = 35/16

Converting 35/16 to mixed fraction, we have 2 3/16.

## Week 4 Review: Practice Exercises and Problems

In the previous post, you have learned about multiplication and division of fractions. Now, let’s solve some exercises and problems.

Practice Exercises 1

a.) 1/2 × 1/3
b.) 2/3 × 4/5
c.) 8 × 5/6
d.) 2 5/8 × 3
e.) 3 1/8 × 4/5
f.) 1 2/3 × 2 3/4

Practice Exercises 2
a.) 1/5 ÷ 3/10
b.) 1/2 ÷ 3/8
c.) 9 ÷ 3/7
d.) 2 5/8 ÷ 2
e.) 3 1/8 ÷ 3/5
f.) 2 3/4 ÷ 1 1/8

Practice Problems

1.) What is 2/3 of 1/4?

2.) In a dance studio, 3/5 are women and 2/5 are men. If there are 35 persons in the dance studio, how many are men? How many are women?

3.) 2 3/4 liters of water is needed to water a flower bed. How many liters is needed to water 7 flower beds?

4.) A rectangular fish pond is 35 1/4 feet long and 20 1/2 wide. What is its area?

5.) 2 4/5 deciliters or soda is needed to make a punch. How many deciliters of soda is needed to make 5 punches?

6.) A 1 1/2 L juice is to be shared equally by 6 friends. How many L of soda is the share of each person?

7.) Two-fifth cup of oil is needed to make a birthday cake. How many birthday cakes can be made using 8 cups?

8.) The length of a bookshelf is 5 1/4 feet long. Each book on the shelf is 1 1/2 inches thick. How many books will fit on the shelf?

9.) Five pumpkin pies are to be shared equally among 12 persons. How much pumpkin pie does each person get?

10.) Jessie has 8 3/4 hectares of land. He decided to divide it equally among his four children. How many hectares of land will each receive?

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