SOLUTIONS TO MATHEMATICS WORKSHEET 2

Here are the solutions for the Mathematics Worksheet 2 (Problems).

1. 80% A = 50% (20% C) is just the same as 0.8 A = 0.5 (0.2 C). Simplifying the right side of the equation, it will be 0.8 A = 0.1 C. Divide both sides by 0.8, A= 1/8 C or C/8 (d).


2. Let x be the first integer. The sum of the first 5 of 10 consecutive integers: x + (x+1) + (x +2) + (x + 3) + (x + 4) =  265. Simplifying the left side of the equation, 5x + 10 = 265. Transpose 10 to the right side, it will be 5x = 255. Then, we get the sum of the last five integers which is (x +5) + (x + 6) + (x + 7) + (x + 8) + (x+ 9). Simplifying it will be 5x + 35. We substitute the value of 5x we got from the sum of the first 5 integers which is 255 to get the sum of the last 5 integers that is 5x + 35 = 255 + 35 = 290 (a).


3. Anna can do the typing job in 10 hours which means in 1 hour, she can finish 1/10 of the job. After she has worked for 2 hours, she already finished 2/10 or 1/5 of the job when Chester came to help her. So, 4/5 of the job is remaining for the two of them to finish the job. Let's say Chester can finish the job alone in x hours which means in 1 hour, he can finish 1/x of the job. In 3 hours, Anna can finish 3/10 [3 * 1/10] of the remaining job while Chester can do it 3/x [3* 1/x]. We put it in equation: 3/10 + 3/x = 4/5. Transpose 3/10 to the right, it will be 3/x = 4/5 - 3/10. Simplifying the right side, 3/x = 5/10. Cross multiply: 30 = 5x. Therefore, x = 6 (a).


4. 50 liters of 3% salt solution (97% water) minus X liters of 100% pure water = (50-X) liters of 5% salt solution (95% water). We put that in equation form: 50 (0.97)  - X = 0.95 (50-X). Simplifying both sides, we have 48.5 - X = 47.5 - 0.95 X. Segregating the X's and the constants, we have 1 = 0.05 X. Therefore X = 20 (d).


5. a = 3c + 1 and b = 2c -1. If c = 4, then a = 13 and b = 7. 2a = 26 and 3c = 21. We subtract 3c from 2a, we get 26 - 21 = 5 (d).


6. The equation to this problem is: (600/x) - [ 600/ (x + 3)] = 10. TO get rid of the denominators, let's multiply both sides by x (x +3). We will get 600(x+3) - 600x = 10x(x+3). Simplifying, we get 10 (x^2) + 30x - 1800 = 0. This is a quadratic equation. To solve this we use the method of factoring. Using trial and error, the factors we get are (x+15) (x-12) = 0. The answers are 12 and -15. We choose the positive number so the final answer is 12 (b).


7. In hour, 1/6 of the tank will be filled in water and 1/9 of it will be emptied. So, we subtract 1/9 from 1/6. And the answer is 1/18 which means 1/18 of the tank will be filled in water. So, it takes 18 hours for the tank to be full (c).


8. The speed is x/50. The formula is speed = distance/time. Time = distance/speed. So, time = 10/ (x/50). Simplifying, the time will be 500/x (a).


9. Let's say Michael is x years old now and Peter is 3x. Five years ago, Michael's age is x-5 while Peter's age is 3x-5. The equation will be 4 (x-5) = 3x-5. Simplifying the left side of the equation, it will be 4x - 20 = 3x - 5. Segregating the x's and the constants, it will be x = 15. So, Michael's age is 15 (c).


10. [a- (b-c)] - [(a-b)-c] = (a-b+c) - (a-b-c) = a - b + c - a + b + c = 2c  (c).

MATHEMATICS WORKSHEET 2 (PROBLEMS)

You have 5 minutes to solve these 10 items. Choose the letter of the correct answer. Good luck! The answers will be shown at the end of this post.

1. If 80% of a number A is 50% of B and B is 20% of C, then what is A n terms of C?

a. C/4 b. C/5 c. 3C d. C/8

2. There are 10 consecutive integers. If the sum of the first 5 integers is 265, then what is the sum of the 5 last integers?

a. 290 b. 275 c. 285 d. 280

3. Anna can do the typing job in 10 hours. After she has worked for 2 hours, Chester helps her and together they finish the job in 3 more hours. In how many hours could Chester have done the job by himself?

a. 6 b. 8 c. 5 d. 7

4. How many liters of water must be evaporated from 50 liters of a 3% salt solution so that the remaining solution will be 5% salt?

a. 19.6 b. 15 c. 12.5 d. 20

5. If a is equal to 1 more than the product of 3 and b, and b is equal to 1 less than the product of of 2 and c, then 2a is how much greater than 3b when c=4?

a. 2 b. 3 c. 4 d. 5

6.A group of students equally paid $600 for a broken laboratory instrument. If there have been 3 more students in the group, the cost of each share would have been $10 less. How many students were in the group?

a. 15 b. 12 c. 16 c. 18

7. A tank has two pipes. One pipe can fill the tank in 6 hours while the other pipe can empty the tank in 9 hours. If both pipes are open, in how many hours can the tank be filled?

a. 15 b. 12 c. 18 d. 16

8. If John can drive the distance of x km in 50 minutes, how many minutes (in terms of x) will it take him to drive 10 km at the same speed?

a. 500/x b. x/50 c. 50x d. 10x

9. Michael is one-third as old as Peter. Five years ago, Peter was four times as old as Michael. How old is Michael now?

a. 12 b. 14 c. 15 c. 16

10. Solve for this: [a- (b-c)] - [(a-b)-c]

a. -2b-2c b. 0 c. 2c c. -2b



==============

Answers:

1.d, 2.a, 3.a, 4.d, 5.d, 6.b, 7.c, 8.a, 9.c, 10.c

Solutions to these problems can be found here.

LANGUAGE WORKSHEET 2 (ANALOGY)

1. PATRON : PROTECTION ;

a. restaurant : management
b. host : hostility
c. counselor : advice
d. spouse : divorce

2. PREAMBLE : CONSTITUTION ;

a. episode : serial
b. prologue : play
c. by-line : article
d. amendment : bill

3. PRICK : STAB ;

a. lend : borrow
b. thread : sew
c. sip : gulp
d. point : thrust

4. CIRCUITOUS : DIRECTNESS ;

a. religious : faith
b. faulty : impropriety
c. cautious : duplicity
d. inexact : accuracy

5. INDUSTRIOUS : ASSIDUOUS ;

a. mendacious : beggarly
b. affluent : impecunious
c. fortuitous : fortunate
d. impoverished : poor

6. TEAM : ATHLETES ;

a. alliance : nations
b. delegates : alternates
c. games : series
d. congregation : preachers

7. DISBAND : ARMY ;

a. muster : platoon
b. abandon : navy
c. dissolve : corporation
d. convene : assembly

8. INTREPID : VALOR ;

a. clever : ingenuity
b. timorous : haste
c. frivolous : fervor
d. boisterous : grief

9. DOCTOR : DISEASE ;

a. pedestrian : senility
b. broker : stocks
c. psychiatrist : maladjustments
d. moron : imbecility

10. KANGAROO : MARSUPIAL ;

a. antelope : gazelle
b. rose : hybrid
c. bee : drone
d. mushroom : fungus

===========

Answers:
1.c, 2.b, 3.c, 4.d, 5.d, 6.d, 7.c, 8.a, 9.c, 10.d

LANGUAGE WORKSHEET 1 (SENTENCE COMPLETION)

You're given 1 minute to answer 10 items. Choose the letter of the correct answer. Check your answers at the end of this worksheet.


1. Even if you don't _______________ what I have to say, I'd appreciate your listening to me with an open mind.
a. concur with
b. clarify
c. anticipate
d. deviate

2. Most mammals give birth to live offspring, but few rare species _____________ eggs in the soil until they hatch.
a. they incubate
b. to have incubated
c. to incubate
d. incubate

3. With age, veins and arteries lose their elasticity and ability to quickly transport blood throughout the body, ___________________ them less efficient.
a. which it makes
b. makes
c. thereby making
c. and to make

4. The explorers of the moon have gone _____________ earlier explorers.
a. further of
b. farther than
c. further than
d. farther on

5. Natural gas, _______________ of methane is considered the cleanest of all fossil fuels.
a. it is composed primarily
b. which is composed primarily
c. is primarily composed
d. that is a primarily composition

6. We had not realized how much people ___________________ the library's old borrowing policy until we received complaints once it had been _______________.
a. appreciated... superseded
b. disliked... administered
c. respected... irritated
d. enjoyed... continued

7. Many hours of practice are required of a successful musician , so it is often not so much ____________ as ____________, which distinguishes the professional from the amateur.
a. talent... discipline
b. technique... pomposity
c. genius... understanding
d. money... education

8. Psychologists agree that a good marriage is based on the _____________ of the partners as well as on their _________________ one another.
a. infallibility... manipulation of
b. conformity... denial of
c. financial security... animosity toward
d. compatibility... independence from

9. The _____________ treatment of the zoo animals resulted in a community-wide _______________.
a. adequate... revolution
b. popular... neglect
c. critical... distension
d. inhumane... criticism

10. Andrea had ________________ taste in clothing and always dressed fashionably but she was totally ____________________ her surroundings, and her apartment and office were drab and disorganized.
a. impeccable... indifferent to
b. bizarre... suspicious of
c. dreadful... dependent on
d. pedestrian... fascinated by


============

Answers:
1.a, 2.d, 3.b, 4.b, 5.b, 6.b, 7.a, 8.d, 9.d, 10.a

SOLUTIONS TO MATHEMATICS WORKSHEET 1

These are the solutions to MATHEMATICS WORKSHEET 1 Problems.

1. Andrew traveled 3/4 of his journey by plane, 1/6 by bus and the rest by walking. What part of his journey did he walk?

3/4 + 1/6 + ? = 1

To get the unknown fraction, let's add first 3/4 and 1/6. The sum is 11/12. Then we subtract 11/12 from 1. So, the answer is  1/12 (d).


2. In high school, the ratio of students who join basketball or volleyball to students who don't join in either game is 3:8. If there are 220 students in this school, how many of them do not join in either game?

For the ratio 3:8, the sum is 11. Let's find the proportion factor by dividing 220 by 11 and we get 20. Then, let's multiply 20 by 8 and we get the final answer 160 (c).


3. Three dogs bark regularly at night. Brownie barks every 12 minutes, Blackie barks every 15 minutes and Whitie barks every 18 minutes. All of them bark together at midnight. At what time will they bark together again?

To answer this problem, we must get the LCM (Least Common Multiple) of 12, 15, and 18. We come up with 180 minutes which is equal to 3 hours. The dogs started at midnight or 12am so they will bark together again after 3 hours or at 3:00 am (c).

4. In Limestone High School, 60% of the freshman class is male. If 90% of the females and 70% of the males are going on a freshman concert, then what percent of the freshman class is going to concert?

Let's first get the percentage of males who are going to the concert i.e. 70% of 60%. The answer is 42% (multiply 0.7 by 60). Next, let's compute the percentage of females i.e. 90% of 40% (since 60% are males). The answer is 36% (0.9 * 40). We take the sum of the percentages we get. 42% + 36% = 78%  (c).


5. A group of young men rented a bus for $600. Two men were unable to go so the rent had to be paid by the remaining men, each paying $25. How many men were originally in the group?

        We have to compute for the men who are present that time. 600/25 = 24. Then add 2 to 24. So, the original group was composed of 26 men (c).


6. A wealthy businessman divides his estate equally among his 4 grandchildren. One grandchild died and 1/2 of his share was equally divided to the remaining grandchildren and was added to each of their shares. What part of the entire estate did each of the 3 grandchildren receive?

The estate was divided by 4. So, each grandchild got 1/4 of the estate. Let's compute for 1/2 of share of dead grandchild. 1/2 * 1/4 = 1/8. Then divide this by 3. 1/8 * 1/3 = 1/24. Lastly, add this to original share of 1/4. 1/24 + 1/4 = 7/24 (c).


7. A boutique was selling a designer's bag and marked it down 20% for a summer sale. During Holidays, the bag was marked down another 20% from its summer price. If the bag was sold at the Holidays price, what percent of the original price did it sell for?

When the bag was marked down 20% the first time, the price became 80% of the original price. The second time around, it became 80% of 80% of the original price. That would be 0.8 * 80 = 64% (c).


8. The rich Prince sent 50 men to cut 100 trees in 4 weeks. At this rate, how many men should he send to cut 50 trees in 5 weeks?

At first, 100/4 = 25 trees per week. So, the ratio is 50 men is to 25 trees per week (50:25 = 2:1). For 50 trees in 5 weeks, that's 10 trees per week. Given the ratio 2:1, therefore we need 20 men to cut 10 trees per week (a).  [2:1 = 20:10]


9. In Algebra class in a College school, 20% are Civil Engineering majors, 30% are IT majors, and the rest are Physics majors. If there are 30 students are Physics majors, how many students are Civil Engineering majors?

20% + 30% + ? = 100%

So, 50% are Physics major and is equal to 30 students. There are 60 students all in all. Therefore, Civil Engineering students are 20% of 60 which is equal to 12 (b).


10. A company has 8 departments, each with 10-16 sub-departments. In each sub-department, there are at least 40 but no more than 60 employees. If 10% of the employees in each sub-department are encoders, what is the minimum number of encoders in this company?

We multiply 8 by 10 by 40 in order to get the minimum employees. 8 * 10 * 40 = 3200. And get 10% of that. 0.10 * 3200 = 320 (c).

DIVISIBILITY RULES

In factoring numbers, you should know if a certain number is divisible to another number. That's why there are divisibility rules that you have to know by heart (yes, you don't just memorize it). Don't worry, they are so simple that you can easily master them by yourself. Here they are:

1. A number is divisible by 2 if it ends with 0, 2, 4, 6, and 8. In short, the number should be an even number to be divisible by 2.

Example: 42, 74, 66, 98, and 230 are all divisible by 2.

2. A number is divisible by 3 if the sum of the digits of the number is divisible by 3.

Example: 753 is divisible by 3 (the sum of 7,5 and 3 is 15 which is divisible by 3)

3. A number is divisible by 4 if the two last digits of the number are both zeros or divisible by 4.

Example: 46,500 is divisible by 4 (last two digits are both zeros)

         875,144 is divisible by 4 (last two digits in this case is 44 which is divisible by 4)

4. A number is divisible by 5 if it ends with 5 or 0.

Example: 65 and 7,500 are both divisible by 5.

5. A number is divisible by 6 if it is divisible by both 2 and 3.

Example:  132 is divisible by 6 (it is divisible by 2 as well as divisible by 3)

6. A number is divisible by 7 if the number without the unit's digit minus twice the unit's digit is divisible by 7.

Example: 924 [ 92 - (4 * 2) = 84 ] 84 is divisible by 7 therefore, 924 is divisible by 7.

7. A number is divisible by 8 if the last three digits are all zeros or divisible by 8.

Example: 872,000 [last three digits are all zeros] - divisible by 8

274,168 [168 is divisible by 8] - divisible by 8

8. A number is divisible by 9 if the sum of its digits is divisible by 9.

Example: 2,718 [2+7+1+8 = 18] 18 is divisible by 9 so, 2,718 is divisible by 9

9. A number is divisible by 10 if it ends with 0.

Example: 784,280 - divisible by 10.

10. A number is divisible by 12 if it is divisible by both 3 and 4.

Example: 7,500 [divisible by 3 and 4] - divisible by 12

11. Real number (10^n) - 1 [n is a natural number] is divisible by 9. If n is even, (10^n) - 1  is divisible by 11.

Example:  (10^5) - 1 = 100,000 - 1 = 99,999 is divisible by 9 [n=5, odd]

 (10^4) - 1 = 10,000 - 1 = 9,999 is divisible by 9 and 11 [n=4, even]


I'll be having worksheets for this. So, watch out for that. In the meantime, absorb everything you can from these basic rules.

MATHEMATICS WORKSHEET 1 (PROBLEMS)

In this worksheet, you will need the basic arithmetic lessons you learned from gradeschool and high school. Time limit is 5 minutes. The answers to this worksheet will be shown at the end of this post. Good luck!

1. Andrew traveled 3/4 of his journey by plane, 1/6 by bus and the rest by walking. What part of his journey did he walk?

a. 1/8 b. 1/9 c. 1/5 d. 1/12

2. In high school, the ratio of students who join basketball or volleyball to students who don't join in either game is 3:8. If there are 220 students in this school, how many of them do not join in either game?

a. 180 b. 100 c. 160 d. 60

3. Three dogs bark regularly at night. Brownie barks every 12 minutes, Blackie barks every 15 minutes and Whitie barks every 18 minutes. All of them bark together at midnight. At what time will they bark together again?

a. 2:45am b. 3:15 am c. 3:00 am d. 2:45 am

4. In Limestone High School, 60% of the freshman class is male. If 90% of the females and 70% of the males are going on a freshman concert, then what percent of the freshman class is going to concert?

a. 80% b. 76% c. 78% d. 82%

5. A group of young men rented a bus for $600. Two men were unable to go so the rent had to be paid by the remaining men, each paying $25. How many men were originally in the group?

a. 24 b. 28 c. 26 d. 22

6. A wealthy businessman divides his estate equally among his 4 grandchildren. One grandchild died and 1/2 of his share was equally divided to the remaining grandchildren and was added to each of their shares. What part of the entire estate did each of the 3 grandchildren receive?

a. 5/12 b. 7/12 c. 7/24 d. 1/24

7. A boutique was selling a designer's bag and marked it down 20% for a summer sale. During Holidays, the bag was marked down another 20% from its summer price. If the bag was sold at the Holidays price, what percent of the original price did it sell for?

a. 60% b. 67% c. 64% d. 40%

8. The rich Prince sent 50 men to cut 100 trees in 4 weeks. At this rate, how many men should he send to cut 50 trees in 5 weeks?

a. 20 b. 30 c. 40 d. 60

9. In Algebra class in a College school, 20% are Civil Engineering majors, 30% are IT majors, and the rest are Physics majors. If there are 30 students are Physics majors, how many students are Civil Engineering majors?

a. 18 b. 12 c. 10 d. 20

10. A company has 8 departments, each with 10-16 sub-departments. In each sub-department, there are at least 40 but no more than 60 employees. If 10% of the employees in each sub-department are encoders, what is the minimum number of encoders in this company?

a. 96 b. 768 c. 320 d. 65

==========




Answers:

1.d, 2.c, 3.c, 4.c, 5.c, 6.c, 7.c, 8.a, 9.b, 10.c


The solutions will be shown in the next post. Check it here.

MATHEMATICAL OPERATIONS ON INTEGERS

Mathematical operations are addition, subtraction, multiplication, and division. These are all basic and it is expected that the high school students already mastered these operations involving whole numbers even integers both positive and negative. But some students get confused when they see different signs (positive and negative). It's ok because I'm here to help you out. So, let's get started.

Let's start with addition first. There are only 3 rules in adding integers.

1) To add both (+) integers, add them and the answer is also (+).

Example:  5 + 7 = 12

2) To add both (-) integers: add them and the answer is also (-).

Example: -5 + (-7) = -12

3) To add (+) and (-) integers: subtract them and the answer should follow the sign of the larger number.

Example 1: -5 + 7 = 2
Example 2: 5 + (-7) = -2


Now, we go to multiplication and/or division of integers.

1) To multiply/divide two same signs i.e. both (+) or both (-) integers, multiply/divide the numbers and the answer is (+).

Example 1: 5 * 7 = 35

Example 2: (-49)/(-7) = 7

2) To multiply/divide two different signs, i.e. a (+) and a (-) integers, multiply/divide the numbers and the answer is (-).

Example 1: 5 * (-7) = -35

Example 2: (-49)/ 7 = -7


3) To multiply/divide two or more different signs, multiply/divide the numbers and count the (-) integers. If the number of (-) integers is even, the answer should be (+). If the number (-) integers is odd, then the answer should be (-).

Example 1: (-5) * (-7) * 1 * (-2) * (-1) = 70   [there are 4 (-) integers, 4 is even so the answer is (+)]

Example 2: (-5) * (-7) * 1 * (-2) = 70 [there are 3 (-) integers, 3 is odd so the answer is (-)]


For subtraction of integers, we have to follow the rules in multiplication and addition of integers.

Example 1: -7 - (-5) [since there are two (-) signs before 5, that will be (+) as per rule 1 of multiplication/division of integers] To make it simpler, it will be:
           -7 + 5 = -2 [from the rule 3 of addition of integers]

Example 2: -7 - 5  [will turn this into addition] To make it simpler, it will be
  -7 + (-5) = -12  [since there are (+) and (-) signs before 5, that will be (-) as per rule 2 of multiplication/division of integers]


To add/subtract a string of (+) and (-) integers, separate the (+) integers from the (-) integers first. Then, add all (+) integers and do the same for the (-) integers. You will come up with a (+) and a (-) integers so just do the rule 3 of addition of integers to come up with the final answer.

There you have it. Hope this will help high school students especially the seniors.

ROUNDING OFF NUMBERS

Rounding off numbers is so basic but then most students hardly remember the simple rule. So to refresh your memory, here it is. 

Let's say the number is 157 and it must be rounded off to the nearest tens. I assume you remember the place values. What's the digit in the tens' place? Right, it is 5. Then we look at the number on its right. If the digit is less than 5, drop that number and replace it with zero/zeros. If the digit is 5 or more, then add 1 to the digit in the ten's place before you drop the number and replace it with zero/zeros. In this case, since the digit on the right of 5 (which is in the ten's place) is 7, let's add 1 to 5 which makes it 6 and then drop 7 and replace it with zero/zeros.

What if the number is 4,999 rounded off to the nearest hundreds. What would be the answer? The correct answer is 5000. Why? Because 9 is in the hundreds' place and on its right is the digit 9 so we add 1 to 9 (hundreds' place) which makes it 10 (meaning, we add 1 to digit 4). That's why we came up with 5000.

More of basic arithmetic pointers to come. Also, I will be giving free worksheets in the coming days so better stay tuned.

If you have questions, please feel free to leave comments. I would love to hear from you. :)